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Hypothesis testing is a statistical technique that is used in a variety of situations. Though the technical details differ from situation to situation, all hypothesis tests use the same core set of terms and concepts. The following descriptions of common terms and concepts refer to a hypothesis test in which the means of two populations are being compared.
The null hypothesis is a clear statement about the relationship between two (or more) statistical objects. These objects may be measurements, distributions, or categories. Typically, the null hypothesis, as the name implies, states that there is no relationship.
In the case of two population means, the null hypothesis might state that the means of the two populations are equal.
Once the null hypothesis has been stated, it is easy to construct the alternative hypothesis. It is essentially the statement that the null hypothesis is false. In our example, the alternative hypothesis would be that the means of the two populations are not equal.
The significance level is a measure of the statistical strength of the hypothesis test. It is often characterized as the probability of incorrectly concluding that the null hypothesis is false.
The significance level is something that you should specify up front. In applications, the significance level is typically one of three values: 10%, 5%, or 1%. A 1% significance level represents the strongest test of the three. For this reason, 1% is a higher significance level than 10%.
Related to significance, the power of a test measures the probability of correctly concluding that the null hypothesis is true. Power is not something that you can choose. It is determined by several factors, including the significance level you select and the size of the difference between the things you are trying to compare.
Unfortunately, significance and power are inversely related. Increasing significance decreases power. This makes it difficult to design experiments that have both very high significance and power.
The test statistic is a single measure that captures the statistical nature of the relationship between observations you are dealing with. The test statistic depends fundamentally on the number of observations that are being evaluated. It differs from situation to situation.
The whole notion of hypothesis rests on the ability to specify (exactly or approximately) the distribution that the test statistic follows. In the case of this example, the difference between the means will be approximately normally distributed (assuming there are a relatively large number of observations).
Depending on the situation, you may want (or need) to employ a one- or two-tailed test. These tails refer to the right and left tails of the distribution of the test statistic. A two-tailed test allows for the possibility that the test statistic is either very large or very small (negative is small). A one-tailed test allows for only one of these possibilities.
In an example where the null hypothesis states that the two population means are equal, you need to allow for the possibility that either one could be larger than the other. The test statistic could be either positive or negative. So, you employ a two-tailed test.
The null hypothesis might have been slightly different, namely that the mean of population 1 is larger than the mean of population 2. In that case, you don't need to account statistically for the situation where the first mean is smaller than the second. So, you would employ a one-tailed test.
The critical value in a hypothesis test is based on two things: the distribution of the test statistic and the significance level. The critical value(s) refer to the point in the test statistic distribution that give the tails of the distribution an area (meaning probability) exactly equal to the significance level that was chosen.
Your decision to reject or accept the null hypothesis is based on comparing the test statistic to the critical value. If the test statistic exceeds the critical value, you should reject the null hypothesis. In this case, you would say that the difference between the two population means is significant. Otherwise, you accept the null hypothesis.
The p-value of a hypothesis test gives you another way to evaluate the null hypothesis. The p-value represents the highest significance level at which your particular test statistic would justify rejecting the null hypothesis. For example, if you have chosen a significance level of 5%, and the p-value turns out to be .03 (or 3%), you would be justified in rejecting the null hypothesis.
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What is a hypothesis test.
A hypothesis test is carried out at the 5% level of significance to test if a normal coin is fair or not.Â
How do we decide whether to reject or accept the null hypothesis.
For the following situations, state at the 1% and 5% significance levels whether the null hypothesis should be rejected or not.
How is a hypothesis test carried out.
A teacher carried out a hypothesis test at the 10% significance level to test if her students perform better in exams after using a new revision technique. The p â value for her test statistic is 0.09142. Write a conclusion for her hypothesis test.
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Easy to understand info about the main types of hypothesis tests.
This FREE PDF cheat sheet will show you the differences between all of the main types of hypothesis testing. Including examples on when to use the, the equations used, and how to easily implement them in Excel!
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1.2 - the 7 step process of statistical hypothesis testing.
We will cover the seven steps one by one.
The null hypothesis can be thought of as the opposite of the "guess" the researchers made. In the example presented in the previous section, the biologist "guesses" plant height will be different for the various fertilizers. So the null hypothesis would be that there will be no difference among the groups of plants. Specifically, in more statistical language the null for an ANOVA is that the means are the same. We state the null hypothesis as:
\(H_0 \colon \mu_1 = \mu_2 = ⯠= \mu_T\)
for T levels of an experimental treatment.
\(H_A \colon \text{ treatment level means not all equal}\)
The alternative hypothesis is stated in this way so that if the null is rejected, there are many alternative possibilities.
For example, \(\mu_1\ne \mu_2 = ⯠= \mu_T\) is one possibility, as is \(\mu_1=\mu_2\ne\mu_3= ⯠=\mu_T\). Many people make the mistake of stating the alternative hypothesis as \(\mu_1\ne\mu_2\neâŻ\ne\mu_T\) which says that every mean differs from every other mean. This is a possibility, but only one of many possibilities. A simple way of thinking about this is that at least one mean is different from all others. To cover all alternative outcomes, we resort to a verbal statement of "not all equal" and then follow up with mean comparisons to find out where differences among means exist. In our example, a possible outcome would be that fertilizer 1 results in plants that are exceptionally tall, but fertilizers 2, 3, and the control group may not differ from one another.
If we look at what can happen in a hypothesis test, we can construct the following contingency table:
Decision | In Reality | |
---|---|---|
\(H_0\) is TRUE | \(H_0\) is FALSE | |
Accept \(H_0\) | correct | Type II Error \(\beta\) = probability of Type II Error |
Reject \(H_0\) | Type I Error | correct |
You should be familiar with Type I and Type II errors from your introductory courses. It is important to note that we want to set \(\alpha\) before the experiment ( a-priori ) because the Type I error is the more grievous error to make. The typical value of \(\alpha\) is 0.05, establishing a 95% confidence level. For this course, we will assume \(\alpha\) =0.05, unless stated otherwise.
Remember the importance of recognizing whether data is collected through an experimental design or observational study.
For categorical treatment level means, we use an F- statistic, named after R.A. Fisher. We will explore the mechanics of computing the F- statistic beginning in Lesson 2. The F- value we get from the data is labeled \(F_{\text{calculated}}\).
As with all other test statistics, a threshold (critical) value of F is established. This F- value can be obtained from statistical tables or software and is referred to as \(F_{\text{critical}}\) or \(F_\alpha\). As a reminder, this critical value is the minimum value of the test statistic (in this case \(F_{\text{calculated}}\)) for us to reject the null.
The F- distribution, \(F_\alpha\), and the location of acceptance/rejection regions are shown in the graph below:
If \(F_{\text{calculated}}\) is larger than \(F_\alpha\), then you are in the rejection region and you can reject the null hypothesis with \(\left(1-\alpha \right)\) level of confidence.
Note that modern statistical software condenses Steps 6 and 7 by providing a p -value. The p -value here is the probability of getting an \(F_{\text{calculated}}\) even greater than what you observe assuming the null hypothesis is true. If by chance, the \(F_{\text{calculated}} = F_\alpha\), then the p -value would be exactly equal to \(\alpha\). With larger \(F_{\text{calculated}}\) values, we move further into the rejection region and the p- value becomes less than \(\alpha\). So, the decision rule is as follows:
If the p- value obtained from the ANOVA is less than \(\alpha\), then reject \(H_0\) in favor of \(H_A\).
A statement that we assume to be correct Choose matching term 1 2 Null Hypothesis 3 Hypothesis 4 Alternative Hypothesis Don't know?
How science REALLY works...
Misconception: Â Science proves ideas.
Misconception: Â Science can only disprove ideas.
Correction:  Science neither proves nor disproves. It accepts or rejects ideas based on supporting and refuting evidence, but may revise those conclusions if warranted by new evidence or perspectives. Read more about it.
Testing ideas about childbed fever.
As a simple example of how scientific testing works, consider the case of Ignaz Semmelweis, who worked as a doctor on a maternity ward in the 1800s. In his ward, an unusually high percentage of new mothers died of what was then called childbed fever. Semmelweis considered many possible explanations for this high death rate. Two of the many ideas that he considered were (1) that the fever was caused by mothers giving birth lying on their backs (as opposed to on their sides) and (2) that the fever was caused by doctors’ unclean hands (the doctors often performed autopsies immediately before examining women in labor). He tested these ideas by considering what expectations each idea generated. If it were true that childbed fever were caused by giving birth on one’s back, then changing procedures so that women labored on their sides should lead to lower rates of childbed fever. Semmelweis tried changing the position of labor, but the incidence of fever did not decrease; the actual observations did not match the expected results. If, however, childbed fever were caused by doctors’ unclean hands, having doctors wash their hands thoroughly with a strong disinfecting agent before attending to women in labor should lead to lower rates of childbed fever. When Semmelweis tried this, rates of fever plummeted; the actual observations matched the expected results, supporting the second explanation.
Let’s take a look at another, very different, example of scientific testing: investigating the origins of coral atolls in the tropics. Consider the atoll Eniwetok (Anewetak) in the Marshall Islands â an oceanic ring of exposed coral surrounding a central lagoon. From the 1800s up until today, scientists have been trying to learn what supports atoll structures beneath the water’s surface and exactly how atolls form. Coral only grows near the surface of the ocean where light penetrates, so Eniwetok could have formed in several ways:
Hypothesis 2: The coral that makes up Eniwetok might have grown in a ring atop an underwater mountain already near the surface. The key to this hypothesis is the idea that underwater mountains donât sink; instead the remains of dead sea animals (shells, etc.) accumulate on underwater mountains, potentially assisted by tectonic uplifting. Eventually, the top of the mountain/debris pile would reach the depth at which coral grow, and the atoll would form.
Which is a better explanation for Eniwetok? Did the atoll grow atop a sinking volcano, forming an underwater coral tower, or was the mountain instead built up until it neared the surface where coral were eventually able to grow? Which of these explanations is best supported by the evidence? We canât perform an experiment to find out. Instead, we must figure out what expectations each hypothesis generates, and then collect data from the world to see whether our observations are a better match with one of the two ideas.
If Eniwetok grew atop an underwater mountain, then we would expect the atoll to be made up of a relatively thin layer of coral on top of limestone or basalt. But if it grew upwards around a subsiding island, then we would expect the atoll to be made up of many hundreds of feet of coral on top of volcanic rock. When geologists drilled into Eniwetok in 1951 as part of a survey preparing for nuclear weapons tests, the drill bored through more than 4000 feet (1219 meters) of coral before hitting volcanic basalt! The actual observation contradicted the underwater mountain explanation and matched the subsiding island explanation, supporting that idea. Of course, many other lines of evidence also shed light on the origins of coral atolls, but the surprising depth of coral on Eniwetok was particularly convincing to many geologists.
Visit the NOAA website to see an animation of coral atoll formation according to Hypothesis 1.
Scientists test hypotheses and theories. They are both scientific explanations for what we observe in the natural world, but theories deal with a much wider range of phenomena than do hypotheses. To learn more about the differences between hypotheses and theories, jump ahead to Science at multiple levels .
Observation beyond our eyes
The logic of scientific arguments
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Exclusive hypothesis testing is a new and special class of hypothesis testing. This kind of testing can be applied in survival analysis to understand the association between genomics information and clinical information about the survival time. Besides, it is well known that Coxâs proportional hazards model is the most commonly used model for regression analysis of failure time. In this paper, the authors consider doing the exclusive hypothesis testing for Coxâs proportional hazards model with right-censored data. The authors propose the comprehensive test statistics to make decision, and show that the corresponding decision rule can control the asymptotic Type I errors and have good powers in theory. The numerical studies indicate that the proposed approach works well for practical situations and it is applied to a set of real data arising from Rotterdam Breast Cancer Data study that motivated this study.
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School of Statistics, Beijing Normal University, Beijing, 100875, China
Qiang Wu, Xingwei Tong & Xiaogang Duan
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Correspondence to Xiaogang Duan .
The authors declare no conflict of interest.
This research was supported by the National Natural Science Foundation of China under Grant Nos. 11971064, 12371262, and 12171374.
This paper was recommended for publication by Editor SUN Liuquan.
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Wu, Q., Tong, X. & Duan, X. Exclusive Hypothesis Testing for Coxâs Proportional Hazards Model. J Syst Sci Complex 37 , 2157â2172 (2024). https://doi.org/10.1007/s11424-024-3283-0
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Received : 24 July 2023
Revised : 25 September 2023
Published : 30 August 2024
Issue Date : October 2024
DOI : https://doi.org/10.1007/s11424-024-3283-0
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Table of contents. Step 1: State your null and alternate hypothesis. Step 2: Collect data. Step 3: Perform a statistical test. Step 4: Decide whether to reject or fail to reject your null hypothesis. Step 5: Present your findings. Other interesting articles. Frequently asked questions about hypothesis testing.
hypothesis, H 0. p-value Probability of obtaining a sample "more extreme" than the ones observed in your data, assuming H 0 is true. Hypothesis A premise or claim that we want to test. Null Hypothesis: H 0 Currently accepted value for a parameter (middle of the distribution). Is assumed true for the purpose of carrying out the hypothesis test.
Likelihood ratio. In the likelihood ratio test, we reject the null hypothesis if the ratio is above a certain value i.e, reject the null hypothesis if L(X) > đ, else accept it. đ is called the critical ratio.. So this is how we can draw a decision boundary: we separate the observations for which the likelihood ratio is greater than the critical ratio from the observations for which it ...
Ranganathan P, Pramesh CS. An Introduction to Statistics: Understanding Hypothesis Testing and Statistical Errors. Indian J Crit Care Med 2019;23 (Suppl 3):S230-S231. Keywords: Biostatistics, Research design, Statistical bias. Two papers quoted in this issue of the Indian Journal of Critical Care Medicine report.
Photo from StepUp Analytics. Hypothesis testing is a method of statistical inference that considers the null hypothesis Hâ vs. the alternative hypothesis Ha, where we are typically looking to assess evidence against Hâ. 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 test (independent or ...
Hypothesis testing is a statistical method to determine whether a hypothesis that you have holds true or not. The hypothesis can be with respect to two variables within a dataset, an association between two groups or a situation. The method evaluates two mutually exclusive statements (two events that cannot occur simultaneously) to determine ...
A statistical hypothesis test has a null hypothesis, the status quo, what we assume to be true. Notation is H 0, read as "H naught". The alternative hypothesis is what you are trying to prove (mentioned in your research question), H 1 or H A. All hypothesis tests must include a null and an alternative hypothesis.
Step 7: Based on steps 5 and 6, draw a conclusion about H0. If the F\calculated from the data is larger than the Fα, then you are in the rejection region and you can reject the null hypothesis with (1 â α) level of confidence. Note that modern statistical software condenses steps 6 and 7 by providing a p -value.
A hypothesis test consists of five steps: 1. State the hypotheses. State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false. 2. Determine a significance level to use for the hypothesis. Decide on a significance level.
Hypothesis testing is an indispensable tool in data science, allowing us to make data-driven decisions with confidence. By understanding its principles, conducting tests properly, and considering real-world applications, you can harness the power of hypothesis testing to unlock valuable insights from your data.
Below these are summarized into six such steps to conducting a test of a hypothesis. Set up the hypotheses and check conditions: Each hypothesis test includes two hypotheses about the population. One is the null hypothesis, notated as H 0, which is a statement of a particular parameter value. This hypothesis is assumed to be true until there is ...
Frequentist Hypothesis Testing . The frequentist hypothesis or the traditional approach to hypothesis testing is a hypothesis testing method that aims on making assumptions by considering current data. The supposed truths and assumptions are based on the current data and a set of 2 hypotheses are formulated. A very popular subtype of the ...
In other words, a hypothesis test at the 0.05 level will virtually always fail to reject the null hypothesis if the 95% confidence interval contains the predicted value. A hypothesis test at the 0.05 level will nearly certainly reject the null hypothesis if the 95% confidence interval does not include the hypothesized parameter.
The significance level is a measure of the statistical strength of the hypothesis test. It is often characterized as the probability of incorrectly concluding that the null hypothesis is false. The significance level is something that you should specify up front. In applications, the significance level is typically one of three values: 10%, 5% ...
A hypothesis test is carried out at the 5% level of significance to test if a normal coin is fair or not. (i) Describe what the population parameter could be for the hypothesis test. (ii) State whether the hypothesis test should be a one-tailed test or a two-tailed test, give a reason for your answer. (iii)
Easy To Understand Info About The Main Types Of Hypothesis Tests. This FREE PDF cheat sheet will show you the differences between all of the main types of hypothesis testing. Including examples on when to use the, the equations used, and how to easily implement them in Excel!
Step 7: Based on Steps 5 and 6, draw a conclusion about H 0. If F calculated is larger than F α, then you are in the rejection region and you can reject the null hypothesis with ( 1 â α) level of confidence. Note that modern statistical software condenses Steps 6 and 7 by providing a p -value. The p -value here is the probability of getting ...
Terms in this set (8) Hypothesis Test. A test that uses a sample or an experiment to determine whether or not to reject the hypothesis. Null Hypothesis. A statement that we assume to be correct. Alternative Hypothesis. A hypothesis that describes the parameter if our assumption is proven wrong. Test Statistic.
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It tests the null hypothesis that the population variances are equal (called homogeneity of variance or homoscedasticity). Suppose the resulting p-value of Levene's test is less than the significance level (typically 0.05).In that case, the obtained differences in sample variances are unlikely to have occurred based on random sampling from a population with equal variances.
Steps to Follow. Define the null and alternative hypothesis. Conduct the test. Using data from the test: Calculate the test statistic (i.e. F) and the critical value (i.e. F crit). Calculate a p value and compare it to a significance level (α) or confidence level (1-α). For example, if the significance level = 5%, then the confidence level = 95%.
Testing ideas with evidence is at the heart of the process of science. ... Hypothesis 2: The coral that makes up Eniwetok might have grown in a ring atop an underwater mountain already near the surface. The key to this hypothesis is the idea that underwater mountains don't sink; instead the remains of dead sea animals (shells, etc ...
Hypothesis testing is a statistical method used to draw conclusions about populations from sample data, typically represented in tables. With the prevalence of graph representations in real-life applications, hypothesis testing on graphs is gaining importance. In this work, we formalize node, edge, and path hypotheses on attributed graphs.
Exclusive hypothesis testing is a new and special class of hypothesis testing. This kind of testing can be applied in survival analysis to understand the association between genomics information and clinical information about the survival time. Besides, it is well known that Cox's proportional hazards model is the most commonly used model for regression analysis of failure time. In this ...