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Hypothesis testing, sometimes called significance testing, is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used and the reason for the analysis.
Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data. Such data may come from a larger population or a data-generating process. The word "population" will be used for both of these cases in the following descriptions.
In hypothesis testing, an analyst tests a statistical sample, intending to provide evidence on the plausibility of the null hypothesis. Statistical analysts measure and examine a random sample of the population being analyzed. All analysts use a random population sample to test two different hypotheses: the null hypothesis and the alternative hypothesis.
The null hypothesis is usually a hypothesis of equality between population parameters; e.g., a null hypothesis may state that the population mean return is equal to zero. The alternative hypothesis is effectively the opposite of a null hypothesis. Thus, they are mutually exclusive , and only one can be true. However, one of the two hypotheses will always be true.
The null hypothesis is a statement about a population parameter, such as the population mean, that is assumed to be true.
If an individual wants to test that a penny has exactly a 50% chance of landing on heads, the null hypothesis would be that 50% is correct, and the alternative hypothesis would be that 50% is not correct. Mathematically, the null hypothesis is represented as Ho: P = 0.5. The alternative hypothesis is shown as "Ha" and is identical to the null hypothesis, except with the equal sign struck-through, meaning that it does not equal 50%.
A random sample of 100 coin flips is taken, and the null hypothesis is tested. If it is found that the 100 coin flips were distributed as 40 heads and 60 tails, the analyst would assume that a penny does not have a 50% chance of landing on heads and would reject the null hypothesis and accept the alternative hypothesis.
If there were 48 heads and 52 tails, then it is plausible that the coin could be fair and still produce such a result. In cases such as this where the null hypothesis is "accepted," the analyst states that the difference between the expected results (50 heads and 50 tails) and the observed results (48 heads and 52 tails) is "explainable by chance alone."
Some statisticians attribute the first hypothesis tests to satirical writer John Arbuthnot in 1710, who studied male and female births in England after observing that in nearly every year, male births exceeded female births by a slight proportion. Arbuthnot calculated that the probability of this happening by chance was small, and therefore it was due to “divine providence.”
Hypothesis testing helps assess the accuracy of new ideas or theories by testing them against data. This allows researchers to determine whether the evidence supports their hypothesis, helping to avoid false claims and conclusions. Hypothesis testing also provides a framework for decision-making based on data rather than personal opinions or biases. By relying on statistical analysis, hypothesis testing helps to reduce the effects of chance and confounding variables, providing a robust framework for making informed conclusions.
Hypothesis testing relies exclusively on data and doesn’t provide a comprehensive understanding of the subject being studied. Additionally, the accuracy of the results depends on the quality of the available data and the statistical methods used. Inaccurate data or inappropriate hypothesis formulation may lead to incorrect conclusions or failed tests. Hypothesis testing can also lead to errors, such as analysts either accepting or rejecting a null hypothesis when they shouldn’t have. These errors may result in false conclusions or missed opportunities to identify significant patterns or relationships in the data.
Hypothesis testing refers to a statistical process that helps researchers determine the reliability of a study. By using a well-formulated hypothesis and set of statistical tests, individuals or businesses can make inferences about the population that they are studying and draw conclusions based on the data presented. All hypothesis testing methods have the same four-step process, which includes stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result.
Sage. " Introduction to Hypothesis Testing ," Page 4.
Elder Research. " Who Invented the Null Hypothesis? "
Formplus. " Hypothesis Testing: Definition, Uses, Limitations and Examples ."
Hypothesis testing is as old as the scientific method and is at the heart of the research process.
Research exists to validate or disprove assumptions about various phenomena. The process of validation involves testing and it is in this context that we will explore hypothesis testing.
A hypothesis is a calculated prediction or assumption about a population parameter based on limited evidence. The whole idea behind hypothesis formulation is testing—this means the researcher subjects his or her calculated assumption to a series of evaluations to know whether they are true or false.
Typically, every research starts with a hypothesis—the investigator makes a claim and experiments to prove that this claim is true or false . For instance, if you predict that students who drink milk before class perform better than those who don’t, then this becomes a hypothesis that can be confirmed or refuted using an experiment.
Read: What is Empirical Research Study? [Examples & Method]
1. simple hypothesis.
Also known as a basic hypothesis, a simple hypothesis suggests that an independent variable is responsible for a corresponding dependent variable. In other words, an occurrence of the independent variable inevitably leads to an occurrence of the dependent variable.
Typically, simple hypotheses are considered as generally true, and they establish a causal relationship between two variables.
Examples of Simple Hypothesis
A complex hypothesis is also known as a modal. It accounts for the causal relationship between two independent variables and the resulting dependent variables. This means that the combination of the independent variables leads to the occurrence of the dependent variables .
Examples of Complex Hypotheses
As the name suggests, a null hypothesis is formed when a researcher suspects that there’s no relationship between the variables in an observation. In this case, the purpose of the research is to approve or disapprove this assumption.
Examples of Null Hypothesis
Read: Research Report: Definition, Types + [Writing Guide]
To disapprove a null hypothesis, the researcher has to come up with an opposite assumption—this assumption is known as the alternative hypothesis. This means if the null hypothesis says that A is false, the alternative hypothesis assumes that A is true.
An alternative hypothesis can be directional or non-directional depending on the direction of the difference. A directional alternative hypothesis specifies the direction of the tested relationship, stating that one variable is predicted to be larger or smaller than the null value while a non-directional hypothesis only validates the existence of a difference without stating its direction.
Examples of Alternative Hypotheses
Logical hypotheses are some of the most common types of calculated assumptions in systematic investigations. It is an attempt to use your reasoning to connect different pieces in research and build a theory using little evidence. In this case, the researcher uses any data available to him, to form a plausible assumption that can be tested.
Examples of Logical Hypothesis
After forming a logical hypothesis, the next step is to create an empirical or working hypothesis. At this stage, your logical hypothesis undergoes systematic testing to prove or disprove the assumption. An empirical hypothesis is subject to several variables that can trigger changes and lead to specific outcomes.
Examples of Empirical Testing
When forming a statistical hypothesis, the researcher examines the portion of a population of interest and makes a calculated assumption based on the data from this sample. A statistical hypothesis is most common with systematic investigations involving a large target audience. Here, it’s impossible to collect responses from every member of the population so you have to depend on data from your sample and extrapolate the results to the wider population.
Examples of Statistical Hypothesis
Hypothesis testing is an assessment method that allows researchers to determine the plausibility of a hypothesis. It involves testing an assumption about a specific population parameter to know whether it’s true or false. These population parameters include variance, standard deviation, and median.
Typically, hypothesis testing starts with developing a null hypothesis and then performing several tests that support or reject the null hypothesis. The researcher uses test statistics to compare the association or relationship between two or more variables.
Explore: Research Bias: Definition, Types + Examples
Researchers also use hypothesis testing to calculate the coefficient of variation and determine if the regression relationship and the correlation coefficient are statistically significant.
The basis of hypothesis testing is to examine and analyze the null hypothesis and alternative hypothesis to know which one is the most plausible assumption. Since both assumptions are mutually exclusive, only one can be true. In other words, the occurrence of a null hypothesis destroys the chances of the alternative coming to life, and vice-versa.
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To successfully confirm or refute an assumption, the researcher goes through five (5) stages of hypothesis testing;
Like we mentioned earlier, hypothesis testing starts with creating a null hypothesis which stands as an assumption that a certain statement is false or implausible. For example, the null hypothesis (H0) could suggest that different subgroups in the research population react to a variable in the same way.
Once you know the variables for the null hypothesis, the next step is to determine the alternative hypothesis. The alternative hypothesis counters the null assumption by suggesting the statement or assertion is true. Depending on the purpose of your research, the alternative hypothesis can be one-sided or two-sided.
Using the example we established earlier, the alternative hypothesis may argue that the different sub-groups react differently to the same variable based on several internal and external factors.
Many researchers create a 5% allowance for accepting the value of an alternative hypothesis, even if the value is untrue. This means that there is a 0.05 chance that one would go with the value of the alternative hypothesis, despite the truth of the null hypothesis.
Something to note here is that the smaller the significance level, the greater the burden of proof needed to reject the null hypothesis and support the alternative hypothesis.
Explore: What is Data Interpretation? + [Types, Method & Tools]
Test statistics in hypothesis testing allow you to compare different groups between variables while the p-value accounts for the probability of obtaining sample statistics if your null hypothesis is true. In this case, your test statistics can be the mean, median and similar parameters.
If your p-value is 0.65, for example, then it means that the variable in your hypothesis will happen 65 in100 times by pure chance. Use this formula to determine the p-value for your data:
After conducting a series of tests, you should be able to agree or refute the hypothesis based on feedback and insights from your sample data.
Hypothesis testing isn’t only confined to numbers and calculations; it also has several real-life applications in business, manufacturing, advertising, and medicine.
In a factory or other manufacturing plants, hypothesis testing is an important part of quality and production control before the final products are approved and sent out to the consumer.
During ideation and strategy development, C-level executives use hypothesis testing to evaluate their theories and assumptions before any form of implementation. For example, they could leverage hypothesis testing to determine whether or not some new advertising campaign, marketing technique, etc. causes increased sales.
In addition, hypothesis testing is used during clinical trials to prove the efficacy of a drug or new medical method before its approval for widespread human usage.
An employer claims that her workers are of above-average intelligence. She takes a random sample of 20 of them and gets the following results:
Mean IQ Scores: 110
Standard Deviation: 15
Mean Population IQ: 100
Step 1: Using the value of the mean population IQ, we establish the null hypothesis as 100.
Step 2: State that the alternative hypothesis is greater than 100.
Step 3: State the alpha level as 0.05 or 5%
Step 4: Find the rejection region area (given by your alpha level above) from the z-table. An area of .05 is equal to a z-score of 1.645.
Step 5: Calculate the test statistics using this formula
Z = (110–100) ÷ (15÷√20)
10 ÷ 3.35 = 2.99
If the value of the test statistics is higher than the value of the rejection region, then you should reject the null hypothesis. If it is less, then you cannot reject the null.
In this case, 2.99 > 1.645 so we reject the null.
The most significant benefit of hypothesis testing is it allows you to evaluate the strength of your claim or assumption before implementing it in your data set. Also, hypothesis testing is the only valid method to prove that something “is or is not”. Other benefits include:
Several limitations of hypothesis testing can affect the quality of data you get from this process. Some of these limitations include:
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Simple guide on pure or basic research, its methods, characteristics, advantages, and examples in science, medicine, education and psychology
This article will discuss the two different types of errors in hypothesis testing and how you can prevent them from occurring in your research
In this article, we will discuss the concept of internal validity, some clear examples, its importance, and how to test it.
We are going to discuss alternative hypotheses and null hypotheses in this post and how they work in research.
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Hypothesis testing, p values, confidence intervals, and significance.
Jacob Shreffler ; Martin R. Huecker .
Last Update: March 13, 2023 .
Medical providers often rely on evidence-based medicine to guide decision-making in practice. Often a research hypothesis is tested with results provided, typically with p values, confidence intervals, or both. Additionally, statistical or research significance is estimated or determined by the investigators. Unfortunately, healthcare providers may have different comfort levels in interpreting these findings, which may affect the adequate application of the data.
Without a foundational understanding of hypothesis testing, p values, confidence intervals, and the difference between statistical and clinical significance, it may affect healthcare providers' ability to make clinical decisions without relying purely on the research investigators deemed level of significance. Therefore, an overview of these concepts is provided to allow medical professionals to use their expertise to determine if results are reported sufficiently and if the study outcomes are clinically appropriate to be applied in healthcare practice.
Hypothesis Testing
Investigators conducting studies need research questions and hypotheses to guide analyses. Starting with broad research questions (RQs), investigators then identify a gap in current clinical practice or research. Any research problem or statement is grounded in a better understanding of relationships between two or more variables. For this article, we will use the following research question example:
Research Question: Is Drug 23 an effective treatment for Disease A?
Research questions do not directly imply specific guesses or predictions; we must formulate research hypotheses. A hypothesis is a predetermined declaration regarding the research question in which the investigator(s) makes a precise, educated guess about a study outcome. This is sometimes called the alternative hypothesis and ultimately allows the researcher to take a stance based on experience or insight from medical literature. An example of a hypothesis is below.
Research Hypothesis: Drug 23 will significantly reduce symptoms associated with Disease A compared to Drug 22.
The null hypothesis states that there is no statistical difference between groups based on the stated research hypothesis.
Researchers should be aware of journal recommendations when considering how to report p values, and manuscripts should remain internally consistent.
Regarding p values, as the number of individuals enrolled in a study (the sample size) increases, the likelihood of finding a statistically significant effect increases. With very large sample sizes, the p-value can be very low significant differences in the reduction of symptoms for Disease A between Drug 23 and Drug 22. The null hypothesis is deemed true until a study presents significant data to support rejecting the null hypothesis. Based on the results, the investigators will either reject the null hypothesis (if they found significant differences or associations) or fail to reject the null hypothesis (they could not provide proof that there were significant differences or associations).
To test a hypothesis, researchers obtain data on a representative sample to determine whether to reject or fail to reject a null hypothesis. In most research studies, it is not feasible to obtain data for an entire population. Using a sampling procedure allows for statistical inference, though this involves a certain possibility of error. [1] When determining whether to reject or fail to reject the null hypothesis, mistakes can be made: Type I and Type II errors. Though it is impossible to ensure that these errors have not occurred, researchers should limit the possibilities of these faults. [2]
Significance
Significance is a term to describe the substantive importance of medical research. Statistical significance is the likelihood of results due to chance. [3] Healthcare providers should always delineate statistical significance from clinical significance, a common error when reviewing biomedical research. [4] When conceptualizing findings reported as either significant or not significant, healthcare providers should not simply accept researchers' results or conclusions without considering the clinical significance. Healthcare professionals should consider the clinical importance of findings and understand both p values and confidence intervals so they do not have to rely on the researchers to determine the level of significance. [5] One criterion often used to determine statistical significance is the utilization of p values.
P values are used in research to determine whether the sample estimate is significantly different from a hypothesized value. The p-value is the probability that the observed effect within the study would have occurred by chance if, in reality, there was no true effect. Conventionally, data yielding a p<0.05 or p<0.01 is considered statistically significant. While some have debated that the 0.05 level should be lowered, it is still universally practiced. [6] Hypothesis testing allows us to determine the size of the effect.
An example of findings reported with p values are below:
Statement: Drug 23 reduced patients' symptoms compared to Drug 22. Patients who received Drug 23 (n=100) were 2.1 times less likely than patients who received Drug 22 (n = 100) to experience symptoms of Disease A, p<0.05.
Statement:Individuals who were prescribed Drug 23 experienced fewer symptoms (M = 1.3, SD = 0.7) compared to individuals who were prescribed Drug 22 (M = 5.3, SD = 1.9). This finding was statistically significant, p= 0.02.
For either statement, if the threshold had been set at 0.05, the null hypothesis (that there was no relationship) should be rejected, and we should conclude significant differences. Noticeably, as can be seen in the two statements above, some researchers will report findings with < or > and others will provide an exact p-value (0.000001) but never zero [6] . When examining research, readers should understand how p values are reported. The best practice is to report all p values for all variables within a study design, rather than only providing p values for variables with significant findings. [7] The inclusion of all p values provides evidence for study validity and limits suspicion for selective reporting/data mining.
While researchers have historically used p values, experts who find p values problematic encourage the use of confidence intervals. [8] . P-values alone do not allow us to understand the size or the extent of the differences or associations. [3] In March 2016, the American Statistical Association (ASA) released a statement on p values, noting that scientific decision-making and conclusions should not be based on a fixed p-value threshold (e.g., 0.05). They recommend focusing on the significance of results in the context of study design, quality of measurements, and validity of data. Ultimately, the ASA statement noted that in isolation, a p-value does not provide strong evidence. [9]
When conceptualizing clinical work, healthcare professionals should consider p values with a concurrent appraisal study design validity. For example, a p-value from a double-blinded randomized clinical trial (designed to minimize bias) should be weighted higher than one from a retrospective observational study [7] . The p-value debate has smoldered since the 1950s [10] , and replacement with confidence intervals has been suggested since the 1980s. [11]
Confidence Intervals
A confidence interval provides a range of values within given confidence (e.g., 95%), including the accurate value of the statistical constraint within a targeted population. [12] Most research uses a 95% CI, but investigators can set any level (e.g., 90% CI, 99% CI). [13] A CI provides a range with the lower bound and upper bound limits of a difference or association that would be plausible for a population. [14] Therefore, a CI of 95% indicates that if a study were to be carried out 100 times, the range would contain the true value in 95, [15] confidence intervals provide more evidence regarding the precision of an estimate compared to p-values. [6]
In consideration of the similar research example provided above, one could make the following statement with 95% CI:
Statement: Individuals who were prescribed Drug 23 had no symptoms after three days, which was significantly faster than those prescribed Drug 22; there was a mean difference between the two groups of days to the recovery of 4.2 days (95% CI: 1.9 – 7.8).
It is important to note that the width of the CI is affected by the standard error and the sample size; reducing a study sample number will result in less precision of the CI (increase the width). [14] A larger width indicates a smaller sample size or a larger variability. [16] A researcher would want to increase the precision of the CI. For example, a 95% CI of 1.43 – 1.47 is much more precise than the one provided in the example above. In research and clinical practice, CIs provide valuable information on whether the interval includes or excludes any clinically significant values. [14]
Null values are sometimes used for differences with CI (zero for differential comparisons and 1 for ratios). However, CIs provide more information than that. [15] Consider this example: A hospital implements a new protocol that reduced wait time for patients in the emergency department by an average of 25 minutes (95% CI: -2.5 – 41 minutes). Because the range crosses zero, implementing this protocol in different populations could result in longer wait times; however, the range is much higher on the positive side. Thus, while the p-value used to detect statistical significance for this may result in "not significant" findings, individuals should examine this range, consider the study design, and weigh whether or not it is still worth piloting in their workplace.
Similarly to p-values, 95% CIs cannot control for researchers' errors (e.g., study bias or improper data analysis). [14] In consideration of whether to report p-values or CIs, researchers should examine journal preferences. When in doubt, reporting both may be beneficial. [13] An example is below:
Reporting both: Individuals who were prescribed Drug 23 had no symptoms after three days, which was significantly faster than those prescribed Drug 22, p = 0.009. There was a mean difference between the two groups of days to the recovery of 4.2 days (95% CI: 1.9 – 7.8).
Recall that clinical significance and statistical significance are two different concepts. Healthcare providers should remember that a study with statistically significant differences and large sample size may be of no interest to clinicians, whereas a study with smaller sample size and statistically non-significant results could impact clinical practice. [14] Additionally, as previously mentioned, a non-significant finding may reflect the study design itself rather than relationships between variables.
Healthcare providers using evidence-based medicine to inform practice should use clinical judgment to determine the practical importance of studies through careful evaluation of the design, sample size, power, likelihood of type I and type II errors, data analysis, and reporting of statistical findings (p values, 95% CI or both). [4] Interestingly, some experts have called for "statistically significant" or "not significant" to be excluded from work as statistical significance never has and will never be equivalent to clinical significance. [17]
The decision on what is clinically significant can be challenging, depending on the providers' experience and especially the severity of the disease. Providers should use their knowledge and experiences to determine the meaningfulness of study results and make inferences based not only on significant or insignificant results by researchers but through their understanding of study limitations and practical implications.
All physicians, nurses, pharmacists, and other healthcare professionals should strive to understand the concepts in this chapter. These individuals should maintain the ability to review and incorporate new literature for evidence-based and safe care.
Disclosure: Jacob Shreffler declares no relevant financial relationships with ineligible companies.
Disclosure: Martin Huecker declares no relevant financial relationships with ineligible companies.
This book is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ), which permits others to distribute the work, provided that the article is not altered or used commercially. You are not required to obtain permission to distribute this article, provided that you credit the author and journal.
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About hypothesis testing.
Contents (Click to skip to the section):
What is hypothesis testing.
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A hypothesis is an educated guess about something in the world around you. It should be testable, either by experiment or observation. For example:
It can really be anything at all as long as you can put it to the test.
If you are going to propose a hypothesis, it’s customary to write a statement. Your statement will look like this: “If I…(do this to an independent variable )….then (this will happen to the dependent variable ).” For example:
A good hypothesis statement should:
Hypothesis testing can be one of the most confusing aspects for students, mostly because before you can even perform a test, you have to know what your null hypothesis is. Often, those tricky word problems that you are faced with can be difficult to decipher. But it’s easier than you think; all you need to do is:
If you trace back the history of science, the null hypothesis is always the accepted fact. Simple examples of null hypotheses that are generally accepted as being true are:
You won’t be required to actually perform a real experiment or survey in elementary statistics (or even disprove a fact like “Pluto is a planet”!), so you’ll be given word problems from real-life situations. You’ll need to figure out what your hypothesis is from the problem. This can be a little trickier than just figuring out what the accepted fact is. With word problems, you are looking to find a fact that is nullifiable (i.e. something you can reject).
A researcher thinks that if knee surgery patients go to physical therapy twice a week (instead of 3 times), their recovery period will be longer. Average recovery times for knee surgery patients is 8.2 weeks.
The hypothesis statement in this question is that the researcher believes the average recovery time is more than 8.2 weeks. It can be written in mathematical terms as: H 1 : μ > 8.2
Next, you’ll need to state the null hypothesis . That’s what will happen if the researcher is wrong . In the above example, if the researcher is wrong then the recovery time is less than or equal to 8.2 weeks. In math, that’s: H 0 μ ≤ 8.2
Ten or so years ago, we believed that there were 9 planets in the solar system. Pluto was demoted as a planet in 2006. The null hypothesis of “Pluto is a planet” was replaced by “Pluto is not a planet.” Of course, rejecting the null hypothesis isn’t always that easy— the hard part is usually figuring out what your null hypothesis is in the first place.
The one sample z test isn’t used very often (because we rarely know the actual population standard deviation ). However, it’s a good idea to understand how it works as it’s one of the simplest tests you can perform in hypothesis testing. In English class you got to learn the basics (like grammar and spelling) before you could write a story; think of one sample z tests as the foundation for understanding more complex hypothesis testing. This page contains two hypothesis testing examples for one sample z-tests .
A principal at a certain school claims that the students in his school are above average intelligence. A random sample of thirty students IQ scores have a mean score of 112.5. Is there sufficient evidence to support the principal’s claim? The mean population IQ is 100 with a standard deviation of 15.
Step 1: State the Null hypothesis . The accepted fact is that the population mean is 100, so: H 0 : μ = 100.
Step 2: State the Alternate Hypothesis . The claim is that the students have above average IQ scores, so: H 1 : μ > 100. The fact that we are looking for scores “greater than” a certain point means that this is a one-tailed test.
Step 4: State the alpha level . If you aren’t given an alpha level , use 5% (0.05).
Step 5: Find the rejection region area (given by your alpha level above) from the z-table . An area of .05 is equal to a z-score of 1.645.
Step 6: If Step 6 is greater than Step 5, reject the null hypothesis. If it’s less than Step 5, you cannot reject the null hypothesis. In this case, it is more (4.56 > 1.645), so you can reject the null.
Blood glucose levels for obese patients have a mean of 100 with a standard deviation of 15. A researcher thinks that a diet high in raw cornstarch will have a positive or negative effect on blood glucose levels. A sample of 30 patients who have tried the raw cornstarch diet have a mean glucose level of 140. Test the hypothesis that the raw cornstarch had an effect.
*This process is made much easier if you use a TI-83 or Excel to calculate the z-score (the “critical value”). See:
You can use the TI 83 calculator for hypothesis testing, but the calculator won’t figure out the null and alternate hypotheses; that’s up to you to read the question and input it into the calculator.
Example problem : A sample of 200 people has a mean age of 21 with a population standard deviation (σ) of 5. Test the hypothesis that the population mean is 18.9 at α = 0.05.
Step 1: State the null hypothesis. In this case, the null hypothesis is that the population mean is 18.9, so we write: H 0 : μ = 18.9
Step 2: State the alternative hypothesis. We want to know if our sample, which has a mean of 21 instead of 18.9, really is different from the population, therefore our alternate hypothesis: H 1 : μ ≠ 18.9
Step 3: Press Stat then press the right arrow twice to select TESTS.
Step 4: Press 1 to select 1:Z-Test… . Press ENTER.
Step 5: Use the right arrow to select Stats .
Step 6: Enter the data from the problem: μ 0 : 18.9 σ: 5 x : 21 n: 200 μ: ≠μ 0
Step 7: Arrow down to Calculate and press ENTER. The calculator shows the p-value: p = 2.87 × 10 -9
This is smaller than our alpha value of .05. That means we should reject the null hypothesis .
Bayesian hypothesis testing helps to answer the question: Can the results from a test or survey be repeated? Why do we care if a test can be repeated? Let’s say twenty people in the same village came down with leukemia. A group of researchers find that cell-phone towers are to blame. However, a second study found that cell-phone towers had nothing to do with the cancer cluster in the village. In fact, they found that the cancers were completely random. If that sounds impossible, it actually can happen! Clusters of cancer can happen simply by chance . There could be many reasons why the first study was faulty. One of the main reasons could be that they just didn’t take into account that sometimes things happen randomly and we just don’t know why.
It’s good science to let people know if your study results are solid, or if they could have happened by chance. The usual way of doing this is to test your results with a p-value . A p value is a number that you get by running a hypothesis test on your data. A P value of 0.05 (5%) or less is usually enough to claim that your results are repeatable. However, there’s another way to test the validity of your results: Bayesian Hypothesis testing. This type of testing gives you another way to test the strength of your results.
Traditional testing (the type you probably came across in elementary stats or AP stats) is called Non-Bayesian. It is how often an outcome happens over repeated runs of the experiment. It’s an objective view of whether an experiment is repeatable. Bayesian hypothesis testing is a subjective view of the same thing. It takes into account how much faith you have in your results. In other words, would you wager money on the outcome of your experiment?
Traditional testing (Non Bayesian) requires you to repeat sampling over and over, while Bayesian testing does not. The main different between the two is in the first step of testing: stating a probability model. In Bayesian testing you add prior knowledge to this step. It also requires use of a posterior probability , which is the conditional probability given to a random event after all the evidence is considered.
Many researchers think that it is a better alternative to traditional testing, because it:
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Hypothesis testing is a form of statistical inference that uses data from a sample to draw conclusions about a population parameter or a population probability distribution . First, a tentative assumption is made about the parameter or distribution. This assumption is called the null hypothesis and is denoted by H 0 . An alternative hypothesis (denoted H a ), which is the opposite of what is stated in the null hypothesis, is then defined. The hypothesis-testing procedure involves using sample data to determine whether or not H 0 can be rejected. If H 0 is rejected, the statistical conclusion is that the alternative hypothesis H a is true.
For example, assume that a radio station selects the music it plays based on the assumption that the average age of its listening audience is 30 years. To determine whether this assumption is valid, a hypothesis test could be conducted with the null hypothesis given as H 0 : μ = 30 and the alternative hypothesis given as H a : μ ≠ 30. Based on a sample of individuals from the listening audience, the sample mean age, x̄ , can be computed and used to determine whether there is sufficient statistical evidence to reject H 0 . Conceptually, a value of the sample mean that is “close” to 30 is consistent with the null hypothesis, while a value of the sample mean that is “not close” to 30 provides support for the alternative hypothesis. What is considered “close” and “not close” is determined by using the sampling distribution of x̄ .
Ideally, the hypothesis-testing procedure leads to the acceptance of H 0 when H 0 is true and the rejection of H 0 when H 0 is false. Unfortunately, since hypothesis tests are based on sample information, the possibility of errors must be considered. A type I error corresponds to rejecting H 0 when H 0 is actually true, and a type II error corresponds to accepting H 0 when H 0 is false. The probability of making a type I error is denoted by α, and the probability of making a type II error is denoted by β.
In using the hypothesis-testing procedure to determine if the null hypothesis should be rejected, the person conducting the hypothesis test specifies the maximum allowable probability of making a type I error, called the level of significance for the test. Common choices for the level of significance are α = 0.05 and α = 0.01. Although most applications of hypothesis testing control the probability of making a type I error, they do not always control the probability of making a type II error. A graph known as an operating-characteristic curve can be constructed to show how changes in the sample size affect the probability of making a type II error.
A concept known as the p -value provides a convenient basis for drawing conclusions in hypothesis-testing applications. The p -value is a measure of how likely the sample results are, assuming the null hypothesis is true; the smaller the p -value, the less likely the sample results. If the p -value is less than α, the null hypothesis can be rejected; otherwise, the null hypothesis cannot be rejected. The p -value is often called the observed level of significance for the test.
A hypothesis test can be performed on parameters of one or more populations as well as in a variety of other situations. In each instance, the process begins with the formulation of null and alternative hypotheses about the population. In addition to the population mean, hypothesis-testing procedures are available for population parameters such as proportions, variances , standard deviations , and medians .
Hypothesis tests are also conducted in regression and correlation analysis to determine if the regression relationship and the correlation coefficient are statistically significant (see below Regression and correlation analysis ). A goodness-of-fit test refers to a hypothesis test in which the null hypothesis is that the population has a specific probability distribution, such as a normal probability distribution. Nonparametric statistical methods also involve a variety of hypothesis-testing procedures.
The methods of statistical inference previously described are often referred to as classical methods. Bayesian methods (so called after the English mathematician Thomas Bayes ) provide alternatives that allow one to combine prior information about a population parameter with information contained in a sample to guide the statistical inference process. A prior probability distribution for a parameter of interest is specified first. Sample information is then obtained and combined through an application of Bayes’s theorem to provide a posterior probability distribution for the parameter. The posterior distribution provides the basis for statistical inferences concerning the parameter.
A key, and somewhat controversial, feature of Bayesian methods is the notion of a probability distribution for a population parameter. According to classical statistics, parameters are constants and cannot be represented as random variables. Bayesian proponents argue that, if a parameter value is unknown, then it makes sense to specify a probability distribution that describes the possible values for the parameter as well as their likelihood . The Bayesian approach permits the use of objective data or subjective opinion in specifying a prior distribution. With the Bayesian approach, different individuals might specify different prior distributions. Classical statisticians argue that for this reason Bayesian methods suffer from a lack of objectivity. Bayesian proponents argue that the classical methods of statistical inference have built-in subjectivity (through the choice of a sampling plan) and that the advantage of the Bayesian approach is that the subjectivity is made explicit.
Bayesian methods have been used extensively in statistical decision theory (see below Decision analysis ). In this context , Bayes’s theorem provides a mechanism for combining a prior probability distribution for the states of nature with sample information to provide a revised (posterior) probability distribution about the states of nature. These posterior probabilities are then used to make better decisions.
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About this unit.
Significance tests give us a formal process for using sample data to evaluate the likelihood of some claim about a population value. Learn how to conduct significance tests and calculate p-values to see how likely a sample result is to occur by random chance. You'll also see how we use p-values to make conclusions about hypotheses.
Data preprocessing.
Hypothesis testing involves formulating assumptions about population parameters based on sample statistics and rigorously evaluating these assumptions against empirical evidence. This article sheds light on the significance of hypothesis testing and the critical steps involved in the process.
Hypothesis testing is a statistical method that is used to make a statistical decision using experimental data. Hypothesis testing is basically an assumption that we make about a population parameter. It evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data.
Example: You say an average height in the class is 30 or a boy is taller than a girl. All of these is an assumption that we are assuming, and we need some statistical way to prove these. We need some mathematical conclusion whatever we are assuming is true.
Hypothesis testing is an important procedure in statistics. Hypothesis testing evaluates two mutually exclusive population statements to determine which statement is most supported by sample data. When we say that the findings are statistically significant, thanks to hypothesis testing.
One tailed test focuses on one direction, either greater than or less than a specified value. We use a one-tailed test when there is a clear directional expectation based on prior knowledge or theory. The critical region is located on only one side of the distribution curve. If the sample falls into this critical region, the null hypothesis is rejected in favor of the alternative hypothesis.
There are two types of one-tailed test:
A two-tailed test considers both directions, greater than and less than a specified value.We use a two-tailed test when there is no specific directional expectation, and want to detect any significant difference.
In hypothesis testing, Type I and Type II errors are two possible errors that researchers can make when drawing conclusions about a population based on a sample of data. These errors are associated with the decisions made regarding the null hypothesis and the alternative hypothesis.
Null Hypothesis is True | Null Hypothesis is False | |
---|---|---|
Null Hypothesis is True (Accept) | Correct Decision | Type II Error (False Negative) |
Alternative Hypothesis is True (Reject) | Type I Error (False Positive) | Correct Decision |
Step 1: define null and alternative hypothesis.
We first identify the problem about which we want to make an assumption keeping in mind that our assumption should be contradictory to one another, assuming Normally distributed data.
Gather relevant data through observation or experimentation. Analyze the data using appropriate statistical methods to obtain a test statistic.
The data for the tests are evaluated in this step we look for various scores based on the characteristics of data. The choice of the test statistic depends on the type of hypothesis test being conducted.
There are various hypothesis tests, each appropriate for various goal to calculate our test. This could be a Z-test , Chi-square , T-test , and so on.
We have a smaller dataset, So, T-test is more appropriate to test our hypothesis.
T-statistic is a measure of the difference between the means of two groups relative to the variability within each group. It is calculated as the difference between the sample means divided by the standard error of the difference. It is also known as the t-value or t-score.
In this stage, we decide where we should accept the null hypothesis or reject the null hypothesis. There are two ways to decide where we should accept or reject the null hypothesis.
Comparing the test statistic and tabulated critical value we have,
Note: Critical values are predetermined threshold values that are used to make a decision in hypothesis testing. To determine critical values for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.
We can also come to an conclusion using the p-value,
Note : The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed in the sample, assuming the null hypothesis is true. To determine p-value for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.
At last, we can conclude our experiment using method A or B.
To validate our hypothesis about a population parameter we use statistical functions . We use the z-score, p-value, and level of significance(alpha) to make evidence for our hypothesis for normally distributed data .
When population means and standard deviations are known.
T test is used when n<30,
t-statistic calculation is given by:
Chi-Square Test for Independence categorical Data (Non-normally distributed) using:
Let’s examine hypothesis testing using two real life situations,
Imagine a pharmaceutical company has developed a new drug that they believe can effectively lower blood pressure in patients with hypertension. Before bringing the drug to market, they need to conduct a study to assess its impact on blood pressure.
Let’s consider the Significance level at 0.05, indicating rejection of the null hypothesis.
If the evidence suggests less than a 5% chance of observing the results due to random variation.
Using paired T-test analyze the data to obtain a test statistic and a p-value.
The test statistic (e.g., T-statistic) is calculated based on the differences between blood pressure measurements before and after treatment.
t = m/(s/√n)
then, m= -3.9, s= 1.8 and n= 10
we, calculate the , T-statistic = -9 based on the formula for paired t test
The calculated t-statistic is -9 and degrees of freedom df = 9, you can find the p-value using statistical software or a t-distribution table.
thus, p-value = 8.538051223166285e-06
Step 5: Result
Conclusion: Since the p-value (8.538051223166285e-06) is less than the significance level (0.05), the researchers reject the null hypothesis. There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.
Let’s create hypothesis testing with python, where we are testing whether a new drug affects blood pressure. For this example, we will use a paired T-test. We’ll use the scipy.stats library for the T-test.
Scipy is a mathematical library in Python that is mostly used for mathematical equations and computations.
We will implement our first real life problem via python,
In the above example, given the T-statistic of approximately -9 and an extremely small p-value, the results indicate a strong case to reject the null hypothesis at a significance level of 0.05.
Data: A sample of 25 individuals is taken, and their cholesterol levels are measured.
Cholesterol Levels (mg/dL): 205, 198, 210, 190, 215, 205, 200, 192, 198, 205, 198, 202, 208, 200, 205, 198, 205, 210, 192, 205, 198, 205, 210, 192, 205.
Populations Mean = 200
Population Standard Deviation (σ): 5 mg/dL(given for this problem)
As the direction of deviation is not given , we assume a two-tailed test, and based on a normal distribution table, the critical values for a significance level of 0.05 (two-tailed) can be calculated through the z-table and are approximately -1.96 and 1.96.
Step 4: Result
Since the absolute value of the test statistic (2.04) is greater than the critical value (1.96), we reject the null hypothesis. And conclude that, there is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL
Hypothesis testing stands as a cornerstone in statistical analysis, enabling data scientists to navigate uncertainties and draw credible inferences from sample data. By systematically defining null and alternative hypotheses, choosing significance levels, and leveraging statistical tests, researchers can assess the validity of their assumptions. The article also elucidates the critical distinction between Type I and Type II errors, providing a comprehensive understanding of the nuanced decision-making process inherent in hypothesis testing. The real-life example of testing a new drug’s effect on blood pressure using a paired T-test showcases the practical application of these principles, underscoring the importance of statistical rigor in data-driven decision-making.
1. what are the 3 types of hypothesis test.
There are three types of hypothesis tests: right-tailed, left-tailed, and two-tailed. Right-tailed tests assess if a parameter is greater, left-tailed if lesser. Two-tailed tests check for non-directional differences, greater or lesser.
Null Hypothesis ( ): No effect or difference exists. Alternative Hypothesis ( ): An effect or difference exists. Significance Level ( ): Risk of rejecting null hypothesis when it’s true (Type I error). Test Statistic: Numerical value representing observed evidence against null hypothesis.
Statistical method to evaluate the performance and validity of machine learning models. Tests specific hypotheses about model behavior, like whether features influence predictions or if a model generalizes well to unseen data.
Pytest purposes general testing framework for Python code while Hypothesis is a Property-based testing framework for Python, focusing on generating test cases based on specified properties of the code.
Similar reads.
Descriptive statistics, inferential statistics, stat reference, statistics - hypothesis testing a mean.
A population mean is an average of value a population.
Hypothesis tests are used to check a claim about the size of that population mean.
The following steps are used for a hypothesis test:
For example:
And we want to check the claim:
"The average age of Nobel Prize winners when they received the prize is more than 55"
By taking a sample of 30 randomly selected Nobel Prize winners we could find that:
The mean age in the sample (\(\bar{x}\)) is 62.1
The standard deviation of age in the sample (\(s\)) is 13.46
From this sample data we check the claim with the steps below.
The conditions for calculating a confidence interval for a proportion are:
A moderately large sample size, like 30, is typically large enough.
In the example, the sample size was 30 and it was randomly selected, so the conditions are fulfilled.
Note: Checking if the data is normally distributed can be done with specialized statistical tests.
We need to define a null hypothesis (\(H_{0}\)) and an alternative hypothesis (\(H_{1}\)) based on the claim we are checking.
The claim was:
In this case, the parameter is the mean age of Nobel Prize winners when they received the prize (\(\mu\)).
The null and alternative hypothesis are then:
Null hypothesis : The average age was 55.
Alternative hypothesis : The average age was more than 55.
Which can be expressed with symbols as:
\(H_{0}\): \(\mu = 55 \)
\(H_{1}\): \(\mu > 55 \)
This is a ' right tailed' test, because the alternative hypothesis claims that the proportion is more than in the null hypothesis.
If the data supports the alternative hypothesis, we reject the null hypothesis and accept the alternative hypothesis.
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The significance level (\(\alpha\)) is the uncertainty we accept when rejecting the null hypothesis in a hypothesis test.
The significance level is a percentage probability of accidentally making the wrong conclusion.
Typical significance levels are:
A lower significance level means that the evidence in the data needs to be stronger to reject the null hypothesis.
There is no "correct" significance level - it only states the uncertainty of the conclusion.
Note: A 5% significance level means that when we reject a null hypothesis:
We expect to reject a true null hypothesis 5 out of 100 times.
The test statistic is used to decide the outcome of the hypothesis test.
The test statistic is a standardized value calculated from the sample.
The formula for the test statistic (TS) of a population mean is:
\(\displaystyle \frac{\bar{x} - \mu}{s} \cdot \sqrt{n} \)
\(\bar{x}-\mu\) is the difference between the sample mean (\(\bar{x}\)) and the claimed population mean (\(\mu\)).
\(s\) is the sample standard deviation .
\(n\) is the sample size.
In our example:
The claimed (\(H_{0}\)) population mean (\(\mu\)) was \( 55 \)
The sample mean (\(\bar{x}\)) was \(62.1\)
The sample standard deviation (\(s\)) was \(13.46\)
The sample size (\(n\)) was \(30\)
So the test statistic (TS) is then:
\(\displaystyle \frac{62.1-55}{13.46} \cdot \sqrt{30} = \frac{7.1}{13.46} \cdot \sqrt{30} \approx 0.528 \cdot 5.477 = \underline{2.889}\)
You can also calculate the test statistic using programming language functions:
With Python use the scipy and math libraries to calculate the test statistic.
With R use built-in math and statistics functions to calculate the test statistic.
There are two main approaches for making the conclusion of a hypothesis test:
Note: The two approaches are only different in how they present the conclusion.
For the critical value approach we need to find the critical value (CV) of the significance level (\(\alpha\)).
For a population mean test, the critical value (CV) is a T-value from a student's t-distribution .
This critical T-value (CV) defines the rejection region for the test.
The rejection region is an area of probability in the tails of the standard normal distribution.
Because the claim is that the population mean is more than 55, the rejection region is in the right tail:
The student's t-distribution is adjusted for the uncertainty from smaller samples.
This adjustment is called degrees of freedom (df), which is the sample size \((n) - 1\)
In this case the degrees of freedom (df) is: \(30 - 1 = \underline{29} \)
Choosing a significance level (\(\alpha\)) of 0.01, or 1%, we can find the critical T-value from a T-table , or with a programming language function:
With Python use the Scipy Stats library t.ppf() function find the T-Value for an \(\alpha\) = 0.01 at 29 degrees of freedom (df).
With R use the built-in qt() function to find the t-value for an \(\alpha\) = 0.01 at 29 degrees of freedom (df).
Using either method we can find that the critical T-Value is \(\approx \underline{2.462}\)
For a right tailed test we need to check if the test statistic (TS) is bigger than the critical value (CV).
If the test statistic is bigger than the critical value, the test statistic is in the rejection region .
When the test statistic is in the rejection region, we reject the null hypothesis (\(H_{0}\)).
Here, the test statistic (TS) was \(\approx \underline{2.889}\) and the critical value was \(\approx \underline{2.462}\)
Here is an illustration of this test in a graph:
Since the test statistic was bigger than the critical value we reject the null hypothesis.
This means that the sample data supports the alternative hypothesis.
And we can summarize the conclusion stating:
The sample data supports the claim that "The average age of Nobel Prize winners when they received the prize is more than 55" at a 1% significance level .
For the P-value approach we need to find the P-value of the test statistic (TS).
If the P-value is smaller than the significance level (\(\alpha\)), we reject the null hypothesis (\(H_{0}\)).
The test statistic was found to be \( \approx \underline{2.889} \)
For a population proportion test, the test statistic is a T-Value from a student's t-distribution .
Because this is a right tailed test, we need to find the P-value of a t-value bigger than 2.889.
The student's t-distribution is adjusted according to degrees of freedom (df), which is the sample size \((30) - 1 = \underline{29}\)
We can find the P-value using a T-table , or with a programming language function:
With Python use the Scipy Stats library t.cdf() function find the P-value of a T-value bigger than 2.889 at 29 degrees of freedom (df):
With R use the built-in pt() function find the P-value of a T-Value bigger than 2.889 at 29 degrees of freedom (df):
Using either method we can find that the P-value is \(\approx \underline{0.0036}\)
This tells us that the significance level (\(\alpha\)) would need to be bigger than 0.0036, or 0.36%, to reject the null hypothesis.
This P-value is smaller than any of the common significance levels (10%, 5%, 1%).
So the null hypothesis is rejected at all of these significance levels.
The sample data supports the claim that "The average age of Nobel Prize winners when they received the prize is more than 55" at a 10%, 5%, or 1% significance level .
Note: An outcome of an hypothesis test that rejects the null hypothesis with a p-value of 0.36% means:
For this p-value, we only expect to reject a true null hypothesis 36 out of 10000 times.
Many programming languages can calculate the P-value to decide outcome of a hypothesis test.
Using software and programming to calculate statistics is more common for bigger sets of data, as calculating manually becomes difficult.
The P-value calculated here will tell us the lowest possible significance level where the null-hypothesis can be rejected.
With Python use the scipy and math libraries to calculate the P-value for a right tailed hypothesis test for a mean.
Here, the sample size is 30, the sample mean is 62.1, the sample standard deviation is 13.46, and the test is for a mean bigger than 55.
With R use built-in math and statistics functions find the P-value for a right tailed hypothesis test for a mean.
This was an example of a right tailed test, where the alternative hypothesis claimed that parameter is bigger than the null hypothesis claim.
You can check out an equivalent step-by-step guide for other types here:
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Teach yourself statistics
This lesson explains how to conduct a hypothesis test of a mean, when the following conditions are met:
Generally, the sampling distribution will be approximately normally distributed if any of the following conditions apply.
This approach consists of four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results.
Every hypothesis test requires the analyst to state a null hypothesis and an alternative hypothesis . The hypotheses are stated in such a way that they are mutually exclusive. That is, if one is true, the other must be false; and vice versa.
The table below shows three sets of hypotheses. Each makes a statement about how the population mean μ is related to a specified value M . (In the table, the symbol ≠ means " not equal to ".)
Set | Null hypothesis | Alternative hypothesis | Number of tails |
---|---|---|---|
1 | μ = M | μ ≠ M | 2 |
2 | μ M | μ < M | 1 |
3 | μ M | μ > M | 1 |
The first set of hypotheses (Set 1) is an example of a two-tailed test , since an extreme value on either side of the sampling distribution would cause a researcher to reject the null hypothesis. The other two sets of hypotheses (Sets 2 and 3) are one-tailed tests , since an extreme value on only one side of the sampling distribution would cause a researcher to reject the null hypothesis.
The analysis plan describes how to use sample data to accept or reject the null hypothesis. It should specify the following elements.
Using sample data, conduct a one-sample t-test. This involves finding the standard error, degrees of freedom, test statistic, and the P-value associated with the test statistic.
SE = s * sqrt{ ( 1/n ) * [ ( N - n ) / ( N - 1 ) ] }
SE = s / sqrt( n )
t = ( x - μ) / SE
As you probably noticed, the process of hypothesis testing can be complex. When you need to test a hypothesis about a mean score, consider using the Sample Size Calculator. The calculator is fairly easy to use, and it is free. You can find the Sample Size Calculator in Stat Trek's main menu under the Stat Tools tab. Or you can tap the button below.
If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. Typically, this involves comparing the P-value to the significance level , and rejecting the null hypothesis when the P-value is less than the significance level.
In this section, two sample problems illustrate how to conduct a hypothesis test of a mean score. The first problem involves a two-tailed test; the second problem, a one-tailed test.
Problem 1: Two-Tailed Test
An inventor has developed a new, energy-efficient lawn mower engine. He claims that the engine will run continuously for 5 hours (300 minutes) on a single gallon of regular gasoline. From his stock of 2000 engines, the inventor selects a simple random sample of 50 engines for testing. The engines run for an average of 295 minutes, with a standard deviation of 20 minutes. Test the null hypothesis that the mean run time is 300 minutes against the alternative hypothesis that the mean run time is not 300 minutes. Use a 0.05 level of significance. (Assume that run times for the population of engines are normally distributed.)
Solution: The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:
Null hypothesis: μ = 300
Alternative hypothesis: μ ≠ 300
SE = s / sqrt(n) = 20 / sqrt(50) = 20/7.07 = 2.83
DF = n - 1 = 50 - 1 = 49
t = ( x - μ) / SE = (295 - 300)/2.83 = -1.77
where s is the standard deviation of the sample, x is the sample mean, μ is the hypothesized population mean, and n is the sample size.
Since we have a two-tailed test , the P-value is the probability that the t statistic having 49 degrees of freedom is less than -1.77 or greater than 1.77. We use the t Distribution Calculator to find P(t < -1.77) is about 0.04.
Note: If you use this approach on an exam, you may also want to mention why this approach is appropriate. Specifically, the approach is appropriate because the sampling method was simple random sampling, the population was normally distributed, and the sample size was small relative to the population size (less than 5%).
Problem 2: One-Tailed Test
Bon Air Elementary School has 1000 students. The principal of the school thinks that the average IQ of students at Bon Air is at least 110. To prove her point, she administers an IQ test to 20 randomly selected students. Among the sampled students, the average IQ is 108 with a standard deviation of 10. Based on these results, should the principal accept or reject her original hypothesis? Assume a significance level of 0.01. (Assume that test scores in the population of engines are normally distributed.)
Null hypothesis: μ >= 110
Alternative hypothesis: μ < 110
SE = s / sqrt(n) = 10 / sqrt(20) = 10/4.472 = 2.236
DF = n - 1 = 20 - 1 = 19
t = ( x - μ) / SE = (108 - 110)/2.236 = -0.894
Here is the logic of the analysis: Given the alternative hypothesis (μ < 110), we want to know whether the observed sample mean is small enough to cause us to reject the null hypothesis.
The observed sample mean produced a t statistic test statistic of -0.894. We use the t Distribution Calculator to find P(t < -0.894) is about 0.19.
Here's a look at the foundation of doing science — the scientific method.
Hypothesis, theory and law, a brief history of science, additional resources, bibliography.
Science is a systematic and logical approach to discovering how things in the universe work. It is also the body of knowledge accumulated through the discoveries about all the things in the universe.
The word "science" is derived from the Latin word "scientia," which means knowledge based on demonstrable and reproducible data, according to the Merriam-Webster dictionary . True to this definition, science aims for measurable results through testing and analysis, a process known as the scientific method. Science is based on fact, not opinion or preferences. The process of science is designed to challenge ideas through research. One important aspect of the scientific process is that it focuses only on the natural world, according to the University of California, Berkeley . Anything that is considered supernatural, or beyond physical reality, does not fit into the definition of science.
When conducting research, scientists use the scientific method to collect measurable, empirical evidence in an experiment related to a hypothesis (often in the form of an if/then statement) that is designed to support or contradict a scientific theory .
"As a field biologist, my favorite part of the scientific method is being in the field collecting the data," Jaime Tanner, a professor of biology at Marlboro College, told Live Science. "But what really makes that fun is knowing that you are trying to answer an interesting question. So the first step in identifying questions and generating possible answers (hypotheses) is also very important and is a creative process. Then once you collect the data you analyze it to see if your hypothesis is supported or not."
The steps of the scientific method go something like this, according to Highline College :
Some key underpinnings to the scientific method:
The process of generating and testing a hypothesis forms the backbone of the scientific method. When an idea has been confirmed over many experiments, it can be called a scientific theory. While a theory provides an explanation for a phenomenon, a scientific law provides a description of a phenomenon, according to The University of Waikato . One example would be the law of conservation of energy, which is the first law of thermodynamics that says that energy can neither be created nor destroyed.
A law describes an observed phenomenon, but it doesn't explain why the phenomenon exists or what causes it. "In science, laws are a starting place," said Peter Coppinger, an associate professor of biology and biomedical engineering at the Rose-Hulman Institute of Technology. "From there, scientists can then ask the questions, 'Why and how?'"
Laws are generally considered to be without exception, though some laws have been modified over time after further testing found discrepancies. For instance, Newton's laws of motion describe everything we've observed in the macroscopic world, but they break down at the subatomic level.
This does not mean theories are not meaningful. For a hypothesis to become a theory, scientists must conduct rigorous testing, typically across multiple disciplines by separate groups of scientists. Saying something is "just a theory" confuses the scientific definition of "theory" with the layperson's definition. To most people a theory is a hunch. In science, a theory is the framework for observations and facts, Tanner told Live Science.
The earliest evidence of science can be found as far back as records exist. Early tablets contain numerals and information about the solar system , which were derived by using careful observation, prediction and testing of those predictions. Science became decidedly more "scientific" over time, however.
1200s: Robert Grosseteste developed the framework for the proper methods of modern scientific experimentation, according to the Stanford Encyclopedia of Philosophy. His works included the principle that an inquiry must be based on measurable evidence that is confirmed through testing.
1400s: Leonardo da Vinci began his notebooks in pursuit of evidence that the human body is microcosmic. The artist, scientist and mathematician also gathered information about optics and hydrodynamics.
1500s: Nicolaus Copernicus advanced the understanding of the solar system with his discovery of heliocentrism. This is a model in which Earth and the other planets revolve around the sun, which is the center of the solar system.
1600s: Johannes Kepler built upon those observations with his laws of planetary motion. Galileo Galilei improved on a new invention, the telescope, and used it to study the sun and planets. The 1600s also saw advancements in the study of physics as Isaac Newton developed his laws of motion.
1700s: Benjamin Franklin discovered that lightning is electrical. He also contributed to the study of oceanography and meteorology. The understanding of chemistry also evolved during this century as Antoine Lavoisier, dubbed the father of modern chemistry , developed the law of conservation of mass.
1800s: Milestones included Alessandro Volta's discoveries regarding electrochemical series, which led to the invention of the battery. John Dalton also introduced atomic theory, which stated that all matter is composed of atoms that combine to form molecules. The basis of modern study of genetics advanced as Gregor Mendel unveiled his laws of inheritance. Later in the century, Wilhelm Conrad Röntgen discovered X-rays , while George Ohm's law provided the basis for understanding how to harness electrical charges.
1900s: The discoveries of Albert Einstein , who is best known for his theory of relativity, dominated the beginning of the 20th century. Einstein's theory of relativity is actually two separate theories. His special theory of relativity, which he outlined in a 1905 paper, " The Electrodynamics of Moving Bodies ," concluded that time must change according to the speed of a moving object relative to the frame of reference of an observer. His second theory of general relativity, which he published as " The Foundation of the General Theory of Relativity ," advanced the idea that matter causes space to curve.
In 1952, Jonas Salk developed the polio vaccine , which reduced the incidence of polio in the United States by nearly 90%, according to Britannica . The following year, James D. Watson and Francis Crick discovered the structure of DNA , which is a double helix formed by base pairs attached to a sugar-phosphate backbone, according to the National Human Genome Research Institute .
2000s: The 21st century saw the first draft of the human genome completed, leading to a greater understanding of DNA. This advanced the study of genetics, its role in human biology and its use as a predictor of diseases and other disorders, according to the National Human Genome Research Institute .
Merriam-Webster Dictionary, Scientia. 2022. https://www.merriam-webster.com/dictionary/scientia
University of California, Berkeley, "Understanding Science: An Overview." 2022. https://undsci.berkeley.edu/article/0_0_0/intro_01
Highline College, "Scientific method." July 12, 2015. https://people.highline.edu/iglozman/classes/astronotes/scimeth.htm
North Carolina State University, "Science Scripts." https://projects.ncsu.edu/project/bio183de/Black/science/science_scripts.html
University of California, Santa Barbara. "What is an Independent variable?" October 31,2017. http://scienceline.ucsb.edu/getkey.php?key=6045
Encyclopedia Britannica, "Control group." May 14, 2020. https://www.britannica.com/science/control-group
The University of Waikato, "Scientific Hypothesis, Theories and Laws." https://sci.waikato.ac.nz/evolution/Theories.shtml
Stanford Encyclopedia of Philosophy, Robert Grosseteste. May 3, 2019. https://plato.stanford.edu/entries/grosseteste/
Encyclopedia Britannica, "Jonas Salk." October 21, 2021. https://www.britannica.com/ biography /Jonas-Salk
National Human Genome Research Institute, "Phosphate Backbone." https://www.genome.gov/genetics-glossary/Phosphate-Backbone
National Human Genome Research Institute, "What is the Human Genome Project?" https://www.genome.gov/human-genome-project/What
Live Science contributor Ashley Hamer updated this article on Jan. 16, 2022.
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Statistics By Jim
Making statistics intuitive
By Jim Frost 5 Comments
Ordinal data have at least three categories that have a natural rank order. The categories are ranked, but the differences between ranks may not be equal. These data indicate the order of values but not the degree of difference between them. For example, first, second, and third places in a race are ordinal data. You can clearly understand the order of finishes. However, the time difference between first and second place might not be the same as between second and third place.
Ordinal data are prevalent in social science and survey research. These variables are relatively convenient for respondents to choose even when the underlying variable is complex, allowing you to compare the participants. For example, subject-area expertise can be tricky to measure using a continuous scale. However, ordinal data can make this evaluation much easier by using Beginner, Intermediate, and Expert ranking choices in a survey.
Likert scale items in a survey are ordinal data. These items typically have 5 or 7 possible responses.
While this data type is expedient, it has downsides that limit the valid summary values and analyses you can use. More on this later!
Learn more about Likert Scale: Survey Use & Examples .
The key concept behind ordinal data is that it ranks observations. However, these ranks don’t indicate the relative degree of difference between two observations. For instance, you know that a high-income person earns more than a middle-income individual, but you don’t know how much more they make. Keep that in mind as you consider the following examples.
Expertise | |
Education level | |
Income | |
Agreement level | |
Frequency of activity |
Ordinal data share properties with both nominal and continuous variables yet are distinct from either.
Ordinal and nominal data are discrete variables that define categories. Consequently, statisticians consider both types to be qualitative data.
However, you can rank ordinal data, which is impossible with nominal data.
For example, college major is nominal data; you can’t rank those categories using that variable alone. They’re simply names of distinct groups, such as statistics, political science, and psychology.
Conversely, ordinal data form groups that you can inherently rank. For example, the relative size of college majors at an institution can be small, medium, or large.
Related post : Discrete vs Continuous Variables
Ordinal and continuous data (both interval and ratio scale) can rank observations on a scale. In other words, you can record that one observation has more of a characteristic than another observation. However, as discussed earlier, ordinal data can’t describe the degree of difference between values, while a continuous variable can.
For example, the size of a college major at an institution can be small, medium, or large. If one major is large and another medium, you see that the former is larger than the latter. However, you don’t know the degree of difference.
Conversely, if you measure size using a continuous variable such as the number of students or budget, you can determine the degree of difference between two observations.
In some cases, you can choose to measure a variable either as continuous or ordinal data. Whenever practical, choose the continuous form because it retains more information and gives you more options during the analysis.
Amongst the various measurement scales, ordinal data fall between the nominal and interval scales. For more information, read Nominal, Ordinal, Interval, and Ratio Scales .
The inability to know the precise differences between observations limits the mathematical functions and summary statistics you can calculate for ordinal data.
While analysts often record values for these variables using numbers, such as 1-5 for a Likert scale of agreement, that doesn’t indicate all numeric calculations are valid.
You cannot meaningfully add and subtract values. For example, if you take ordinal data values of 1 and 2, you can’t trust that summing them to 3 is a valid result. Why?
When adding 1 and 2 to get 3, you’re assuming the difference between 1 and 2 equals the difference between 2 and 3 because they’re both one unit apart. However, that is not a safe assumption with this data type.
Because addition isn’t valid, you can’t subtract because it’s the inverse function. Also, calculating the mean is invalid because it involves addition and division (also invalid). Division is valid only for continuous variables using a ratio scale.
So, what can you do with these variables? Which summary statistics are valid? And what kind of analyses can you perform?
Bar graphs are great for displaying discrete variables. Consequently, they’re an excellent choice for visually understanding ordinal data.
The bar chart below displays a Likert scale item for service ratings from Very Poor to Very Good.
It’s easy to see that most patrons rated the service as Good.
Learn more about Bar Charts and Data Types and How to Graph Them .
Measures of central tendency and variability are two standard summary statistics to report with your results. However, the mean and standard deviation are questionable for ordinal data. Consequently, consider using the following alternatives:
Click the links to learn more about these concepts and statistics.
Similarly, the standard hypothesis tests for the mean (e.g., t-tests and ANOVA) are questionable for this type of variable. Means tests are parametric hypothesis tests.
Instead, consider using nonparametric hypothesis tests as an alternative. They assess medians and ranks, making them perfect for ordinal data. These tests include Mood’s Median, Mann-Whitney, Wilcoxon, Friedman’s Test, and Spearman’s rho.
For more information, read my post about Parametric vs Nonparametric Hypothesis Tests and Spearman’s rho .
Finally, statisticians have some disputes over using hypothesis tests for the mean with Likert scale items. I discussed the critical reason in this post—the mean is not valid. However, some make the case that for specific Likert scales the differences between values are equal by design. If that is true, then the mean might be valid.
However, for parametric hypothesis testing, there are additional concerns for ordinal data. Specifically, these variables are less likely to satisfy the analysis’ assumptions. To learn more about this issue, including some answers about what to do, read my post about Analyzing Likert Scale Data .
November 29, 2023 at 10:37 am
Hi Jim, I have a question regarding a variable I am working on. What type of variable would you consider “grade average in sience” to be? I can see it being both ordinal and continious (interval). In my study I am going to investigate whether students confidence in science, and the amount of time they spend reading can predict their grade average at the end of the school year.
October 18, 2022 at 3:23 pm
Hi, thank you for the great post! I also have a question. You say that mean and standard deviation are questionable for ordinal data. Although my groups are statistically different, the median for all groups is equal (=2), meaning that all my graph bars, which show the median, look the same. What should I do instead? I would have preferred to use the mean and SD since it’s easier for the reader to see a difference among the groups.
October 19, 2022 at 1:30 am
When you say that your groups are statistically significant, what do you mean? How are they different? That’ll help me answer your question.
If the frequency distributions for both groups look similar, they might not be significantly different. You can perform a nonparametric test to see test whether the means are different . If the distribution looks the same as you say, the means might not be different either.
If you can safely assume that the differences between each value in your ordinal scale are spaced equidistantly, some analysts think it’s ok to assess the mean. There is some debate among analysts over that, but some do that. So, if you can make that assumption, you might be ok with the mean and standard deviation.
September 6, 2022 at 2:22 am
Great post! Question. I have several ordinal variables which theoretically are all associated with well-being. They are measured on different scales – some 5 point and some 7 points. I want to create a composite variable. Your post made me wonder about the use of z scores for ordinal variables? I had thought about creating z scores for each variable and summing to create one score.
October 3, 2022 at 4:15 pm
Sorry about the delay in replying. Sometimes things slip through the cracks!
Z-scores are specifically for continuous data that follow a normal distribution. So, it’s not appropriate for ordinal data. I suppose if your ordinal data followed a normal distribution, you might be able to use it for that purpose, but they are frequently non-normal.
Frequently, analysts will add a set of ordinal/Likert scores together, or average them, to create a composite variable. I’d probably go with that approach.
Although there is debate about whether an average is appropriate with ordinal data unless you’re sure that the values are equidistant. If the values are not equidistant, the average can be meaningless. The median is universally accepted as appropriate for ordinal data.
The technical details of ordinal/Likert can be a bit complicated and there is some debate about which measures and analyses are acceptable. But often analysts use a composite sum, average, or median of a set of Likert/ordinal values.
Please read my post about analyzing Likert scale data for a comparison between parametric vs. nonparametric analyses. Although, that post applies to individual items rather than a sum.
One of the reasons I point out these complications is because what is considered acceptable for ordinal data can vary by subject area and analysts. So, I’d also look into finding the acceptable norms for your situation.
Amazon (Amazon.com) is the world's largest online retailer and a prominent cloud service provider .
Originally started as an online bookselling company, Amazon has morphed into an internet-based business enterprise that is largely focused on providing e-commerce, cloud computing, digital streaming and artificial intelligence ( AI ) services.
Following an Amazon-to-buyer sales approach, the company offers a monumental product range and inventory, enabling consumers to buy just about anything, including clothing, beauty supplies, gourmet food, jewelry, books, movies, electronics, pet supplies, furniture, toys, garden supplies and household goods.
Headquartered in Seattle, Amazon has individual websites, software development centers, customer service centers, data centers and fulfillment centers around the world.
Amazon has come a long way since it was founded by Jeff Bezos in his garage in Bellevue, Wash., on July 5, 1994.
The following is a brief history and timeline of events that have evolved Amazon from its humble beginnings to a multinational business empire.
Amazon officially opened for business as an online bookseller on July 16, 1995. Originally, Bezos had incorporated the company as Cadabra but later changed the name to Amazon. Bezos is said to have browsed a dictionary for a word beginning with A for the value of alphabetic placement. He selected the name Amazon because it was exotic and different and as a reference to his plan for the company's size to reflect that of the Amazon River, one of the largest rivers in the world. Since its inception, the company's motto has always been "get big fast."
In 2005, Amazon Prime This membership-based service for Amazon customers offers free two-day shipping within the contiguous U.S., as well as streaming, shopping and reading benefits. According to Amazon's website, current Amazon Prime membership rates are $14.99 a month or $139 per year.
This comprehensive and evolving cloud computing platform was also born in the 2000s. The first Amazon Web Services ( AWS ) offerings were launched in 2006 to provide online services for websites and client-side applications. Amazon Elastic Compute Cloud ( EC2 ) and Simple Storage Service ( S3 ) are the backbones of the company's growing collection of web services. The same year, Amazon also launched a cloud computing and video-on-demand service known at the time as Unbox. By changing the way people bought books, Amazon also shaped how they read them with the launch of its first Kindle e-reader in 2007. This device helps users browse, buy and read e-books, magazines and newspapers from the Kindle Store.
Amazon debuted its first tablet computer, the Kindle Fire, in 2011 and the Amazon Fire TV Stick, which is part of Amazon's extensive line of streaming media devices, in 2014. Amazon also started an online Amazon Art marketplace for fine arts in 2013, which has featured original works by famous artists such as Claude Monet and Norman Rockwell. The popular in-home virtual assistant Amazon Alexa was rolled out to consumers in 2015 and was followed by the Alexa-equipped Echo Dot in 2016. Amazon acquired the organic grocery store Whole Foods in 2017 and launched Amazon Go, a chain of cashierless grocery stores in 2018. The rise of in-home shopping during the COVID-19 pandemic made consumers rely on Amazon even more, and the trend is likely to keep growing.
Amazon offers an ever-expanding portfolio of services and products. Following is a list of its noteworthy offerings.
From healthcare to entertainment, Amazon has acquired multiple companies by tapping into a variety of sectors over time.
Following is a list of Amazon's notable acquisitions and subsidiary companies:
Amazon has suffered a massive backlash over the years from multiple sources. The tech giant is also being held responsible for creating the Amazon effect -- the evolution and disruption of the retail market due to the company exhibiting monopolistic behaviors.
Following are a few concerns and allegations that Amazon has faced over time:
According to a news release posted on Amazon's investor relations website , Amazon experienced a significant increase in net sales but a decrease in operating income in the first quarter of 2022.
Following are some notable statistics from the release:
Besides being recognized as a company with business interests in e-commerce, cloud computing and AI services, Amazon also offers an extensive list of subscription services . Learn about these services and the perks they offer.
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C-reactive protein (CRP) is a protein made by the liver. The level of CRP increases when there's inflammation in the body. A simple blood test can check your C-reactive protein level.
A high-sensitivity C-reactive protein (hs-CRP) test is more sensitive than a standard C-reactive protein test. That means the high-sensitivity test can find smaller increases in C-reactive protein than a standard test can.
The hs-CRP test can help show the risk of getting coronary artery disease. In coronary artery disease, the arteries of the heart narrow. Narrowed arteries can lead to a heart attack.
Your health care provider might order a C-reactive protein test to:
A high level of hs-CRP in the blood has been linked to an increased risk of heart attacks. Also, people who have had a heart attack are more likely to have another heart attack if they have a high hs-CRP level. But their risk goes down when their hs-CRP level is in the typical range.
An hs-CRP test isn't for everyone. The test doesn't show the cause of inflammation. So it's possible to have a high hs-CRP level without it affecting the heart.
An hs-CRP test may be most useful for people who have a 10% to 20% chance of having a heart attack within the next 10 years. This is known as intermediate risk. A health care provider can determine your risk using tests that look at your lifestyle choices, family history and overall health.
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Hard exercise, such as intense weight training or a long run, can cause a sudden jump in the C-reactive protein level. Your health care provider might ask you to avoid such activities before the test.
Some medicines can affect CRP level. Tell your care provider about the medicines you take, including those you bought without a prescription.
If your blood sample will be used for other tests, you may need to avoid food or drink for a period before the test. For example, if you're having an hs-CRP test to check for heart disease, you might have a cholesterol test, which requires fasting, at the same time.
Your health care provider tells you how to prepare for your test.
To take a sample of your blood, a health care provider places a needle into a vein in your arm, usually at the bend of the elbow. The blood sample goes to a lab for analysis. You can return to your usual activities right away.
It can take a few days to get results. Your health care provider can explain what the test results mean.
C-reactive protein is measured in milligrams per liter (mg/L). Results equal to or greater than 8 mg/L or 10 mg/L are considered high. Range values vary depending on the lab doing the test.
A high test result is a sign of inflammation. It may be due to serious infection, injury or chronic disease. Your health care provider may recommend other tests to determine the cause.
Results for an hs-CRP test are usually given as follows:
A person's CRP levels vary over time. A coronary artery disease risk assessment should be based on the average of two hs-CRP tests. It's best if they're taken two weeks apart. Values above 2.0 mg/L may mean an increased risk of heart attacks or risk of a repeat heart attack.
Hs-CRP level is only one risk factor for coronary artery disease. Having a high hs-CRP level doesn't always mean a higher risk of developing heart disease. Other tests results can help determine the risk.
Talk to your health care provider about your risk factors for heart disease and ways to try to prevent it. Lifestyle changes or medicines might help lower the risk of a heart attack.
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Learning Objectives
We finish our discussion of the hypothesis test for a population mean with a review of the meaning of the P-value, along with a review of type I and type II errors.
At this point, we assume you know how to use a P-value to make a decision in a hypothesis test. The logic is always the same. If we pick a level of significance (α), then we compare the P-value to α.
In fact, we find that we treat these as “rules” and apply them without thinking about what the P-value means. So let’s pause here and review the meaning of the P-value, since it is the connection between probability and decision-making in inference.
Let’s return to the familiar context of birth weights for babies in a town. Suppose that babies in the town had a mean birth weight of 3,500 grams in 2010. This year, a random sample of 50 babies has a mean weight of about 3,400 grams with a standard deviation of about 500 grams. Here is the distribution of birth weights in the sample.
Obviously, this sample weighs less on average than the population of babies in the town in 2010. A decrease in the town’s mean birth weight could indicate a decline in overall health of the town. But does this sample give strong evidence that the town’s mean birth weight is less than 3,500 grams this year?
We now know how to answer this question with a hypothesis test. Let’s use a significance level of 5%.
Let μ = mean birth weight in the town this year. The null hypothesis says there is “no change from 2010.”
Since the sample is large, we can conduct the T-test (without worrying about the shape of the distribution of birth weights for individual babies.)
Statistical software tells us the P-value is 0.082 = 8.2%. Since the P-value is greater than 0.05, we fail to reject the null hypothesis.
Our conclusion: This sample does not suggest that the mean birth weight this year is less than 3,500 grams ( P -value = 0.082). The sample from this year has a mean of 3,400 grams, which is 100 grams lower than the mean in 2010. But this difference is not statistically significant. It can be explained by the chance fluctuation we expect to see in random sampling.
A simulation can help us understand the P-value. In a simulation, we assume that the population mean is 3,500 grams. This is the null hypothesis. We assume the null hypothesis is true and select 1,000 random samples from a population with a mean of 3,500 grams. The mean of the sampling distribution is at 3,500 (as predicted by the null hypothesis.) We see this in the simulated sampling distribution.
In the simulation, we can see that about 8.6% of the samples have a mean less than 3,400. Since probability is the relative frequency of an event in the long run, we say there is an 8.6% chance that a random sample of 500 babies has a mean less than 3,400 if the population mean is 3,500. We can see that the corresponding area to the left of T = −1.41 in the T-model (with df = 49) also gives us a good estimate of the probability. This area is the P-value, about 8.2%.
If we generalize this statement, we say the P-value is the probability that random samples have results more extreme than the data if the null hypothesis is true. (By more extreme, we mean further from value of the parameter, in the direction of the alternative hypothesis.) We can also describe the P-value in terms of T-scores. The P-value is the probability that the test statistic from a random sample has a value more extreme than that associated with the data if the null hypothesis is true.
Do women who smoke run the risk of shorter pregnancy and premature birth? The mean pregnancy length is 266 days. We test the following hypotheses.
Suppose a random sample of 40 women who smoke during their pregnancy have a mean pregnancy length of 260 days with a standard deviation of 21 days. The P-value is 0.04.
What probability does the P-value of 0.04 describe? Label each of the following interpretations as valid or invalid.
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We know that statistical inference is based on probability, so there is always some chance of making a wrong decision. Recall that there are two types of wrong decisions that can be made in hypothesis testing. When we reject a null hypothesis that is true, we commit a type I error. When we fail to reject a null hypothesis that is false, we commit a type II error.
The following table summarizes the logic behind type I and type II errors.
It is possible to have some influence over the likelihoods of committing these errors. But decreasing the chance of a type I error increases the chance of a type II error. We have to decide which error is more serious for a given situation. Sometimes a type I error is more serious. Other times a type II error is more serious. Sometimes neither is serious.
Recall that if the null hypothesis is true, the probability of committing a type I error is α. Why is this? Well, when we choose a level of significance (α), we are choosing a benchmark for rejecting the null hypothesis. If the null hypothesis is true, then the probability that we will reject a true null hypothesis is α. So the smaller α is, the smaller the probability of a type I error.
It is more complicated to calculate the probability of a type II error. The best way to reduce the probability of a type II error is to increase the sample size. But once the sample size is set, larger values of α will decrease the probability of a type II error (while increasing the probability of a type I error).
General Guidelines for Choosing a Level of Significance
Let’s return to the investigation of the impact of smoking on pregnancy length.
Recap of the hypothesis test: The mean human pregnancy length is 266 days. We test the following hypotheses.
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In this “Hypothesis Test for a Population Mean,” we looked at the four steps of a hypothesis test as they relate to a claim about a population mean.
Since the hypothesis test is based on probability, random selection or assignment is essential in data production. Additionally, we need to check whether the t-model is a good fit for the sampling distribution of sample means. To use the t-model, the variable must be normally distributed in the population or the sample size must be more than 30. In practice, it is often impossible to verify that the variable is normally distributed in the population. If this is the case and the sample size is not more than 30, researchers often use the t-model if the sample is not strongly skewed and does not have outliers.
The logic of the hypothesis test is always the same. To state a conclusion about H 0 , we compare the P-value to the significance level, α.
COMMENTS
Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is most often used by scientists to test specific predictions, called hypotheses, that arise from theories. ... Stating results in a statistics assignment In our comparison of mean height between men and women we found an average difference ...
Hypothesis testing is a technique that is used to verify whether the results of an experiment are statistically significant. It involves the setting up of a null hypothesis and an alternate hypothesis. There are three types of tests that can be conducted under hypothesis testing - z test, t test, and chi square test.
Test Statistic: z = x¯¯¯ −μo σ/ n−−√ z = x ¯ − μ o σ / n since it is calculated as part of the testing of the hypothesis. Definition 7.1.4 7.1. 4. p - value: probability that the test statistic will take on more extreme values than the observed test statistic, given that the null hypothesis is true.
Hypothesis testing is a crucial procedure to perform when you want to make inferences about a population using a random sample. These inferences include estimating population properties such as the mean, differences between means, proportions, and the relationships between variables. This post provides an overview of statistical hypothesis testing.
Components of a Formal Hypothesis Test. The null hypothesis is a statement about the value of a population parameter, such as the population mean (µ) or the population proportion (p).It contains the condition of equality and is denoted as H 0 (H-naught).. H 0: µ = 157 or H0 : p = 0.37. The alternative hypothesis is the claim to be tested, the opposite of the null hypothesis.
In hypothesis testing, the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis.The null hypothesis is usually denoted \(H_0\) while the alternative hypothesis is usually denoted \(H_1\). An hypothesis test is a statistical decision; the conclusion will either be to reject the null hypothesis in favor ...
Hypothesis Testing Significance levels. The level of statistical significance is often expressed as the so-called p-value. Depending on the statistical test you have chosen, you will calculate a probability (i.e., the p-value) of observing your sample results (or more extreme) given that the null hypothesis is true. Another way of phrasing this ...
The general idea of hypothesis testing involves: Making an initial assumption. Collecting evidence (data). Based on the available evidence (data), deciding whether to reject or not reject the initial assumption. Every hypothesis test — regardless of the population parameter involved — requires the above three steps.
Hypothesis testing is a method of statistical inference that considers the null hypothesis H ₀ vs. the alternative hypothesis H a, where we are typically looking to assess evidence against H ₀. Such a test is used to compare data sets against one another, or compare a data set against some external standard. The former being a two sample ...
A statistical hypothesis is an assumption about a population parameter.. For example, we may assume that the mean height of a male in the U.S. is 70 inches. The assumption about the height is the statistical hypothesis and the true mean height of a male in the U.S. is the population parameter.. A hypothesis test is a formal statistical test we use to reject or fail to reject a statistical ...
Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence.
Hypothesis testing is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used ...
What is Hypothesis Testing? Hypothesis testing is an assessment method that allows researchers to determine the plausibility of a hypothesis. It involves testing an assumption about a specific population parameter to know whether it's true or false. These population parameters include variance, standard deviation, and median.
Hypothesis Testing. Investigators conducting studies need research questions and hypotheses to guide analyses. Starting with broad research questions (RQs), investigators then identify a gap in current clinical practice or research. ... There was a mean difference between the two groups of days to the recovery of 4.2 days (95% CI: 1.9 - 7.8 ...
Step 2: State the Alternate Hypothesis. The claim is that the students have above average IQ scores, so: H 1: μ > 100. The fact that we are looking for scores "greater than" a certain point means that this is a one-tailed test. Step 3: Draw a picture to help you visualize the problem. Step 4: State the alpha level.
Hypothesis testing. Hypothesis testing is a form of statistical inference that uses data from a sample to draw conclusions about a population parameter or a population probability distribution.First, a tentative assumption is made about the parameter or distribution. This assumption is called the null hypothesis and is denoted by H 0.An alternative hypothesis (denoted H a), which is the ...
Unit 12: Significance tests (hypothesis testing) Significance tests give us a formal process for using sample data to evaluate the likelihood of some claim about a population value. Learn how to conduct significance tests and calculate p-values to see how likely a sample result is to occur by random chance. You'll also see how we use p-values ...
Hypothesis testing is a statistical method that is used to make a statistical decision using experimental data. Hypothesis testing is basically an assumption that we make about a population parameter. It evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data.
Hypothesis testing involves the formulate two hypothesis to test against the measured data: (1) ... The right tail describes the probability of observing such a large sample mean if the null hypothesis is true. The shaded tail in Figure 4.15 represents the chance of observing such a large mean, conditional on the null hypothesis being true. ...
The test statistic is used to decide the outcome of the hypothesis test. The test statistic is a standardized value calculated from the sample. The formula for the test statistic (TS) of a population mean is: x ¯ − μ s ⋅ n. x ¯ − μ is the difference between the sample mean ( x ¯) and the claimed population mean ( μ ).
The first set of hypotheses (Set 1) is an example of a two-tailed test, since an extreme value on either side of the sampling distribution would cause a researcher to reject the null hypothesis. The other two sets of hypotheses (Sets 2 and 3) are one-tailed tests, since an extreme value on only one side of the sampling distribution would cause a researcher to reject the null hypothesis.
A scientific hypothesis is a tentative, testable explanation for a phenomenon in the natural world. It's the initial building block in the scientific method. Many describe it as an "educated guess ...
A confidence interval (CI) is a range of values that is likely to contain the value of an unknown population parameter. These intervals represent a plausible domain for the parameter given the characteristics of your sample data. Confidence intervals are derived from sample statistics and are calculated using a specified confidence level.
Test the hypothesis and predictions in an experiment that can be reproduced. Analyze the data and draw conclusions; accept or reject the hypothesis or modify the hypothesis if necessary.
Hypothesis Testing. Similarly, the standard hypothesis tests for the mean (e.g., t-tests and ANOVA) are questionable for this type of variable. Means tests are parametric hypothesis tests. Instead, consider using nonparametric hypothesis tests as an alternative. They assess medians and ranks, making them perfect for ordinal data.
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A standardized test statistic for a hypothesis test is the statistic that is formed by subtracting from the statistic of interest its mean and dividing by its standard deviation. For example, reviewing Example 8.1.3, if instead of working with the sample mean ¯ X we instead work with the test statistic.
It can take a few days to get results. Your health care provider can explain what the test results mean. C-reactive protein is measured in milligrams per liter (mg/L). Results equal to or greater than 8 mg/L or 10 mg/L are considered high. Range values vary depending on the lab doing the test. A high test result is a sign of inflammation.
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Step 1: Determine the hypotheses. The hypotheses are claims about the population mean, µ. The null hypothesis is a hypothesis that the mean equals a specific value, µ 0. The alternative hypothesis is the competing claim that µ is less than, greater than, or not equal to the .