hypothesis 5 steps

Hypothesis Testing Framework

Now that we've seen an example and explored some of the themes for hypothesis testing, let's specify the procedure that we will follow.

Hypothesis Testing Steps

The formal framework and steps for hypothesis testing are as follows:

• Identify and define the parameter of interest
• Define the competing hypotheses to test
• Set the evidence threshold, formally called the significance level
• Generate or use theory to specify the sampling distribution and check conditions
• Calculate the test statistic and p-value
• Evaluate your results and write a conclusion in the context of the problem.

We'll discuss each of these steps below.

Identify Parameter of Interest

First, I like to specify and define the parameter of interest. What is the population that we are interested in? What characteristic are we measuring?

By defining our population of interest, we can confirm that we are truly using sample data. If we find that we actually have population data, our inference procedures are not needed. We could proceed by summarizing our population data.

By identifying and defining the parameter of interest, we can confirm that we use appropriate methods to summarize our variable of interest. We can also focus on the specific process needed for our parameter of interest.

In our example from the last page, the parameter of interest would be the population mean time that a host has been on Airbnb for the population of all Chicago listings on Airbnb in March 2023. We could represent this parameter with the symbol $\mu$. It is best practice to fully define $\mu$ both with words and symbol.

Define the Hypotheses

For hypothesis testing, we need to decide between two competing theories. These theories must be statements about the parameter. Although we won't have the population data to definitively select the correct theory, we will use our sample data to determine how reasonable our "skeptic's theory" is.

The first hypothesis is called the null hypothesis, $H_0$. This can be thought of as the "status quo", the "skeptic's theory", or that nothing is happening.

Examples of null hypotheses include that the population proportion is equal to 0.5 ($p = 0.5$), the population median is equal to 12 ($M = 12$), or the population mean is equal to 14.5 ($\mu = 14.5$).

The second hypothesis is called the alternative hypothesis, $H_a$ or $H_1$. This can be thought of as the "researcher's hypothesis" or that something is happening. This is what we'd like to convince the skeptic to believe. In most cases, the desired outcome of the researcher is to conclude that the alternative hypothesis is reasonable to use moving forward.

Examples of alternative hypotheses include that the population proportion is greater than 0.5 ($p > 0.5$), the population median is less than 12 ($M < 12$), or the population mean is not equal to 14.5 ($\mu \neq 14.5$).

There are a few requirements for the hypotheses:

• the hypotheses must be about the same population parameter,
• the hypotheses must have the same null value (provided number to compare to),
• the null hypothesis must have the equality (the equals sign must be in the null hypothesis),
• the alternative hypothesis must not have the equality (the equals sign cannot be in the alternative hypothesis),
• there must be no overlap between the null and alternative hypothesis.

You may have previously seen null hypotheses that include more than an equality (e.g. $p \le 0.5$). As long as there is an equality in the null hypothesis, this is allowed. For our purposes, we will simplify this statement to ($p = 0.5$).

To summarize from above, possible hypotheses statements are:

$H_0: p = 0.5$ vs. $H_a: p > 0.5$

$H_0: M = 12$ vs. $H_a: M < 12$

$H_0: \mu = 14.5$ vs. $H_a: \mu \neq 14.5$

In our second example about Airbnb hosts, our hypotheses would be:

$H_0: \mu = 2100$ vs. $H_a: \mu > 2100$.

Set Threshold (Significance Level)

There is one more step to complete before looking at the data. This is to set the threshold needed to convince the skeptic. This threshold is defined as an $\alpha$ significance level. We'll define exactly what the $\alpha$ significance level means later. For now, smaller $\alpha$s correspond to more evidence being required to convince the skeptic.

A few common $\alpha$ levels include 0.1, 0.05, and 0.01.

For our Airbnb hosts example, we'll set the threshold as 0.02.

Determine the Sampling Distribution of the Sample Statistic

The first step (as outlined above) is the identify the parameter of interest. What is the best estimate of the parameter of interest? Typically, it will be the sample statistic that corresponds to the parameter. This sample statistic, along with other features of the distribution will prove especially helpful as we continue the hypothesis testing procedure.

However, we do have a decision at this step. We can choose to use simulations with a resampling approach or we can choose to rely on theory if we are using proportions or means. We then also need to confirm that our results and conclusions will be valid based on the available data.

Required Condition

The one required assumption, regardless of approach (resampling or theory), is that the sample is random and representative of the population of interest. In other words, we need our sample to be a reasonable sample of data from the population.

Using Simulations and Resampling

If we'd like to use a resampling approach, we have no (or minimal) additional assumptions to check. This is because we are relying on the available data instead of assumptions.

We do need to adjust our data to be consistent with the null hypothesis (or skeptic's claim). We can then rely on our resampling approach to estimate a plausible sampling distribution for our sample statistic.

Recall that we took this approach on the last page. Before simulating our estimated sampling distribution, we adjusted the mean of the data so that it matched with our skeptic's claim, shown in the code below.

We'll see a few more examples on the next page.

Using Theory

On the other hand, we could rely on theory in order to estimate the sampling distribution of our desired statistic. Recall that we had a few different options to rely on:

• the CLT for the sampling distribution of a sample mean
• the binomial distribution for the sampling distribution of a proportion (or count)
• the Normal approximation of a binomial distribution (using the CLT) for the sampling distribution of a proportion

If relying on the CLT to specify the underlying sampling distribution, you also need to confirm:

• having a random sample and
• having a sample size that is less than 10% of the population size if the sampling is done without replacement
• having a Normally distributed population for a quantitative variable OR
• having a large enough sample size (usually at least 25) for a quantitative variable
• having a large enough sample size for a categorical variable (defined by $np$ and $n(1-p)$ being at least 10)

If relying on the binomial distribution to specify the underlying sampling distribution, you need to confirm:

• having a set number of trials, $n$
• having the same probability of success, $p$ for each observation

After determining the appropriate theory to use, we should check our conditions and then specify the sampling distribution for our statistic.

For the Airbnb hosts example, we have what we've assumed to be a random sample. It is not taken with replacement, so we also need to assume that our sample size (700) is less than 10% of our population size. In other words, we need to assume that the population of Chicago Airbnbs in March 2023 was at least 7000. Since we do have our (presumed) population data available, we can confirm that there were at least 7000 Chicago Airbnbs in the population in 2023.

Additionally, we can confirm that normality of the sampling distribution applies for the CLT to apply. Our sample size is more than 25 and the parameter of interest is a mean, so this meets our necessary criteria for the normality condition to be valid.

With the conditions now met, we can estimate our sampling distribution. From the CLT, we know that the distribution for the sample mean should be $\bar{X} \sim N(\mu, \frac{\sigma}{\sqrt{n}})$.

Now, we face our next challenge -- what to plug in as the mean and standard error for this distribution. Since we are adopting the skeptic's point of view for the purpose of this approach, we can plug in the value of $\mu_0 = 2100$. We also know that the sample size $n$ is 700. But what should we plug in for the population standard deviation $\sigma$?

When we don't know the value of a parameter, we will generally plug in our best estimate for the parameter. In this case, that corresponds to plugging in $\hat{\sigma}$, or our sample standard deviation.

Now, our estimated sampling distribution based on the CLT is: $\bar{X} \sim N(2100, 41.4045)$.

If we compare to our corresponding skeptic's sampling distribution on the last page, we can confirm that the theoretical sampling distribution is similar to the simulated sampling distribution based on resampling.

Assumptions not met

What do we do if the necessary conditions aren't met for the sampling distribution? Because the simulation-based resampling approach has minimal assumptions, we should be able to use this approach to produce valid results as long as the provided data is representative of the population.

The theory-based approach has more conditions, and we may not be able to meet all of the necessary conditions. For example, if our parameter is something other than a mean or proportion, we may not have appropriate theory. Additionally, we may not have a large enough sample size.

• First, we could consider changing approaches to the simulation-based one.
• Second, we might look at how we could meet the necessary conditions better. In some cases, we may be able to redefine groups or make adjustments so that the setup of the test is closer to what is needed.
• As a last resort, we may be able to continue following the hypothesis testing steps. In this case, your calculations may not be valid or exact; however, you might be able to use them as an estimate or an approximation. It would be crucial to specify the violation and approximation in any conclusions or discussion of the test.

Calculate the evidence with statistics and p-values

Now, it's time to calculate how much evidence the sample contains to convince the skeptic to change their mind. As we saw above, we can convince the skeptic to change their mind by demonstrating that our sample is unlikely to occur if their theory is correct.

How do we do this? We do this by calculating a probability associated with our observed value for the statistic.

For example, for our situation, we want to convince the skeptic that the population mean is actually greater than 2100 days. We do that by calculating the probability that a sample mean would be as large or larger than what we observed in our actual sample, which was 2188 days. Why do we need the larger portion? We use the larger portion because a sample mean of 2200 days also provides evidence that the population mean is larger than 2100 days; it isn't limited to exactly what we observed in our sample. We call this specific probability the p-value.

That is, the p-value is the probability of observing a test statistic as extreme or more extreme (as determined by the alternative hypothesis), assuming the null hypothesis is true.

Our observed p-value for the Airbnb host example demonstrates that the probability of getting a sample mean host time of 2188 days (the value from our sample) or more is 1.46%, assuming that the true population mean is 2100 days.

Test statistic

Notice that the formal definition of a p-value mentions a test statistic . In most cases, this word can be replaced with "statistic" or "sample" for an equivalent statement.

Oftentimes, we'll see that our sample statistic can be used directly as the test statistic, as it was above. We could equivalently adjust our statistic to calculate a test statistic. This test statistic is often calculated as:

$\text{test statistic} = \frac{\text{estimate} - \text{hypothesized value}}{\text{standard error of estimate}}$

P-value Calculation Options

Note also that the p-value definition includes a probability associated with a test statistic being as extreme or more extreme (as determined by the alternative hypothesis . How do we determine the area that we consider when calculating the probability. This decision is determined by the inequality in the alternative hypothesis.

For example, when we were trying to convince the skeptic that the population mean is greater than 2100 days, we only considered those sample means that we at least as large as what we observed -- 2188 days or more.

If instead we were trying to convince the skeptic that the population mean is less than 2100 days ($H_a: \mu < 2100$), we would consider all sample means that were at most what we observed - 2188 days or less. In this case, our p-value would be quite large; it would be around 99.5%. This large p-value demonstrates that our sample does not support the alternative hypothesis. In fact, our sample would encourage us to choose the null hypothesis instead of the alternative hypothesis of $\mu < 2100$, as our sample directly contradicts the statement in the alternative hypothesis.

If we wanted to convince the skeptic that they were wrong and that the population mean is anything other than 2100 days ($H_a: \mu \neq 2100$), then we would want to calculate the probability that a sample mean is at least 88 days away from 2100 days. That is, we would calculate the probability corresponding to 2188 days or more or 2012 days or less. In this case, our p-value would be roughly twice the previously calculated p-value.

We could calculate all of those probabilities using our sampling distributions, either simulated or theoretical, that we generated in the previous step. If we chose to calculate a test statistic as defined in the previous section, we could also rely on standard normal distributions to calculate our p-value.

Evaluate your results and write conclusion in context of problem

Once you've gathered your evidence, it's now time to make your final conclusions and determine how you might proceed.

In traditional hypothesis testing, you often make a decision. Recall that you have your threshold (significance level $\alpha$) and your level of evidence (p-value). We can compare the two to determine if your p-value is less than or equal to your threshold. If it is, you have enough evidence to persuade your skeptic to change their mind. If it is larger than the threshold, you don't have quite enough evidence to convince the skeptic.

Common formal conclusions (if given in context) would be:

• I have enough evidence to reject the null hypothesis (the skeptic's claim), and I have sufficient evidence to suggest that the alternative hypothesis is instead true.
• I do not have enough evidence to reject the null hypothesis (the skeptic's claim), and so I do not have sufficient evidence to suggest the alternative hypothesis is true.

The only decision that we can make is to either reject or fail to reject the null hypothesis (we cannot "accept" the null hypothesis). Because we aren't actively evaluating the alternative hypothesis, we don't want to make definitive decisions based on that hypothesis. However, when it comes to making our conclusion for what to use going forward, we frame this on whether we could successfully convince someone of the alternative hypothesis.

A less formal conclusion might look something like:

Based on our sample of Chicago Airbnb listings, it seems as if the mean time since a host has been on Airbnb (for all Chicago Airbnb listings) is more than 5.75 years.

Significance Level Interpretation

We've now seen how the significance level $\alpha$ is used as a threshold for hypothesis testing. What exactly is the significance level?

The significance level $\alpha$ has two primary definitions. One is that the significance level is the maximum probability required to reject the null hypothesis; this is based on how the significance level functions within the hypothesis testing framework. The second definition is that this is the probability of rejecting the null hypothesis when the null hypothesis is true; in other words, this is the probability of making a specific type of error called a Type I error.

Why do we have to be comfortable making a Type I error? There is always a chance that the skeptic was originally correct and we obtained a very unusual sample. We don't want to the skeptic to be so convinced of their theory that no evidence can convince them. In this case, we need the skeptic to be convinced as long as the evidence is strong enough . Typically, the probability threshold will be low, to reduce the number of errors made. This also means that a decent amount of evidence will be needed to convince the skeptic to abandon their position in favor of the alternative theory.

p-value Limitations and Misconceptions

In comparison to the $\alpha$ significance level, we also need to calculate the evidence against the null hypothesis with the p-value.

The p-value is the probability of getting a test statistic as extreme or more extreme (in the direction of the alternative hypothesis), assuming the null hypothesis is true.

Recently, p-values have gotten some bad press in terms of how they are used. However, that doesn't mean that p-values should be abandoned, as they still provide some helpful information. Below, we'll describe what p-values don't mean, and how they should or shouldn't be used to make decisions.

Factors that affect a p-value

What features affect the size of a p-value?

• the null value, or the value assumed under the null hypothesis
• the effect size (the difference between the null value under the null hypothesis and the true value of the parameter)
• the sample size

More evidence against the null hypothesis will be obtained if the effect size is larger and if the sample size is larger.

Misconceptions

We gave a definition for p-values above. What are some examples that p-values don't mean?

• A p-value is not the probability that the null hypothesis is correct
• A p-value is not the probability that the null hypothesis is incorrect
• A p-value is not the probability of getting your specific sample
• A p-value is not the probability that the alternative hypothesis is correct
• A p-value is not the probability that the alternative hypothesis is incorrect
• A p-value does not indicate the size of the effect

Our p-value is a way of measuring the evidence that your sample provides against the null hypothesis, assuming the null hypothesis is in fact correct.

Using the p-value to make a decision

Why is there bad press for a p-value? You may have heard about the standard $\alpha$ level of 0.05. That is, we would be comfortable with rejecting the null hypothesis once in 20 attempts when the null hypothesis is really true. Recall that we reject the null hypothesis when the p-value is less than or equal to the significance level.

Consider what would happen if you have two different p-values: 0.049 and 0.051.

In essence, these two p-values represent two very similar probabilities (4.9% vs. 5.1%) and very similar levels of evidence against the null hypothesis. However, when we make our decision based on our threshold, we would make two different decisions (reject and fail to reject, respectively). Should this decision really be so simplistic? I would argue that the difference shouldn't be so severe when the sample statistics are likely very similar. For this reason, I (and many other experts) strongly recommend using the p-value as a measure of evidence and including it with your conclusion.

Putting too much emphasis on the decision (and having a significant result) has created a culture of misusing p-values. For this reason, understanding your p-value itself is crucial.

Searching for p-values

The other concern with setting a definitive threshold of 0.05 is that some researchers will begin performing multiple tests until finding a p-value that is small enough. However, with a p-value of 0.05, we know that we will have a p-value less than 0.05 1 time out of every 20 times, even when the null hypothesis is true.

This means that if researchers start hunting for p-values that are small (sometimes called p-hacking), then they are likely to identify a small p-value every once in a while by chance alone. Researchers might then publish that result, even though the result is actually not informative. For this reason, it is recommended that researchers write a definitive analysis plan to prevent performing multiple tests in search of a result that occurs by chance alone.

Best Practices

With all of this in mind, what should we do when we have our p-value? How can we prevent or reduce misuse of a p-value?

• Report the p-value along with the conclusion
• Specify the effect size (the value of the statistic)
• Define an analysis plan before looking at the data
• Interpret the p-value clearly to specify what it indicates
• Consider using an alternate statistical approach, the confidence interval, discussed next, when appropriate

11.2.1 - Five Step Hypothesis Testing Procedure

The examples on the following pages use the five step hypothesis testing procedure outlined below. This is the same procedure that we used to conduct a hypothesis test for a single mean, single proportion, difference in two means, and difference in two proportions.

When conducting a chi-square goodness-of-fit test, it makes the most sense to write the hypotheses first. The hypotheses will depend on the research question. The null hypothesis will always contain the equalities and the alternative hypothesis will be that at least one population proportion is not as specified in the null.

In order to use the chi-square distribution to approximate the sampling distribution, all expected counts must be at least five.

Expected Count

Where \(n\) is the total sample size and \(p_i\) is the hypothesized population proportion in the "ith" group.

To check this assumption, compute all expected counts and confirm that each is at least five.

In Step 1 you already computed the expected counts. Use this formula to compute the chi-square test statistic:

Chi-Square Test Statistic

\(\chi^2=\sum \dfrac{(O-E)^2}{E}\)

Where \(O\) is the observed count for each cell and \(E\) is the expected count for each cell.

Construct a chi-square distribution with degrees of freedom equal to the number of groups minus one. The p-value is the area under that distribution to the right of the test statistic that was computed in Step 2. You can find this area by constructing a probability distribution plot in Minitab. 

Unless otherwise stated, use the standard 0.05 alpha level.

\(p \leq \alpha\) reject the null hypothesis.

\(p > \alpha\) fail to reject the null hypothesis.

Go back to the original research question and address it directly. If you rejected the null hypothesis, then there is convincing evidence that at least one of the population proportions is not as stated in the null hypothesis. If you failed to reject the null hypothesis, then there is not enough evidence that any of the population proportions are different from what is stated in the null hypothesis. 

11.2.1.1 - Video: Cupcakes (Equal Proportions)

11.2.1.2- Cards (Equal Proportions)

Example: cards.

Research question : When randomly selecting a card from a deck with replacement, are we equally likely to select a heart, diamond, spade, and club?

I randomly selected a card from a standard deck 40 times with replacement. I pulled 13 hearts, 8 diamonds, 8 spades, and 11 clubs.

Let's use the five-step hypothesis testing procedure:

\(H_0: p_h=p_d=p_s=p_c=0.25\) \(H_a:\) at least one \(p_i\) is not as specified in the null

We can use the null hypothesis to check the assumption that all expected counts are at least 5.

\(Expected\;count=n (p_i)\)

All \(p_i\) are 0.25. \(40(0.25)=10\), thus this assumption is met and we can approximate the sampling distribution using the chi-square distribution.

 \(\chi^2=\sum \dfrac{(Observed-Expected)^2}{Expected} \)

All expected values are 10. Our observed values were 13, 8, 8, and 11.

\(\chi^2=\dfrac{(13-10)^2}{10}+\dfrac{(8-10)^2}{10}+\dfrac{(8-10)^2}{10}+\dfrac{(11-10)^2}{10}\) \(\chi^2=\dfrac{9}{10}+\dfrac{4}{10}+\dfrac{4}{10}+\dfrac{1}{10}\) \(\chi^2=1.8\)

Our sampling distribution will be a chi-square distribution.

\(df=k-1=4-1=3\)

We can find the p-value by constructing a chi-square distribution with 3 degrees of freedom to find the area to the right of \(\chi^2=1.8\)

The p-value is 0.614935

\(p>0.05\) therefore we fail to reject the null hypothesis.

There is not enough evidence to state that the proportion of hearts, diamonds, spades, and clubs that are randomly drawn from this deck are different.

11.2.1.3 - Roulette Wheel (Different Proportions)

Example: roulette wheel.

Research Question : An American roulette wheel contains 38 slots: 18 red, 18 black, and 2 green.  A casino has purchased a new wheel and they want to know if there is convincing evidence that the wheel is unfair. They spin the wheel 100 times and it lands on red 44 times, black 49 times, and green 7 times.

If the wheel is fair then \(p_{red}=\dfrac{18}{38}\), \(p_{black}=\dfrac{18}{38}\), and \(p_{green}=\dfrac{2}{38}\).

All of these proportions combined equal 1.

\(H_0: p_{red}=\dfrac{18}{38},\;p_{black}=\dfrac{18}{38}\;and\;p_{green}=\dfrac{2}{38}\)

\(H_a: At\;least\;one\;p_i\;is \;not\;as\;specified\;in\;the\;null\)

In order to conduct a chi-square goodness of fit test all expected values must be at least 5. 

For both red and black: \(Expected \;count=100(\dfrac{18}{38})=47.368\)

For green: \(Expected\;count=100(\dfrac{2}{38})=5.263\)

All expected counts are at least 5 so we can conduct a chi-square goodness of fit test. 

In the first step we computed the expected values for red and black to be 47.368 and for green to be 5.263.

 \(\chi^2= \dfrac{(44-47.368)^2}{47.368}+\dfrac{(49-47.368)^2}{47.368}+\dfrac{(7-5.263)^2}{5.263} \)

 \(\chi^2=0.239+0.056+0.573=0.868\)

\(df=k-1=3-1=2\)

We can find the p-value by constructing a chi-square distribution with 2 degrees of freedom to find the area to the right of \(\chi^2=0.868\)

The p-value is 0.647912

\(p>0.05\) therefore we should fail to reject the null hypothesis. 

There is not enough evidence that this roulette wheel is unfair.

Step-by-step guide to hypothesis testing in statistics

Hypothesis testing in statistics helps us use data to make informed decisions. It starts with an assumption or guess about a group or population—something we believe might be true. We then collect sample data to check if there is enough evidence to support or reject that guess. This method is useful in many fields, like science, business, and healthcare, where decisions need to be based on facts.

Learning how to do hypothesis testing in statistics step-by-step can help you better understand data and make smarter choices, even when things are uncertain. This guide will take you through each step, from creating your hypothesis to making sense of the results, so you can see how it works in practical situations.

What is Hypothesis Testing?

Table of Contents

Hypothesis testing is a method for determining whether data supports a certain idea or assumption about a larger group. It starts by making a guess, like an average or a proportion, and then uses a small sample of data to see if that guess seems true or not.

For example, if a company wants to know if its new product is more popular than its old one, it can use hypothesis testing. They start with a statement like “The new product is not more popular than the old one” (this is the null hypothesis) and compare it with “The new product is more popular” (this is the alternative hypothesis). Then, they look at customer feedback to see if there’s enough evidence to reject the first statement and support the second one.

Simply put, hypothesis testing is a way to use data to help make decisions and understand what the data is really telling us, even when we don’t have all the answers.

Importance Of Hypothesis Testing In Decision-Making And Data Analysis

Hypothesis testing is important because it helps us make smart choices and understand data better. Here’s why it’s useful:

• Reduces Guesswork : It helps us see if our guesses or ideas are likely correct, even when we don’t have all the details.
• Uses Real Data : Instead of just guessing, it checks if our ideas match up with real data, which makes our decisions more reliable.
• Avoids Errors : It helps us avoid mistakes by carefully checking if our ideas are right so we don’t make costly errors.
• Shows What to Do Next : It tells us if our ideas work or not, helping us decide whether to keep, change, or drop something. For example, a company might test a new ad and decide what to do based on the results.
• Confirms Research Findings : It makes sure that research results are accurate and not just random chance so that we can trust the findings.

Here’s a simple guide to understanding hypothesis testing, with an example:

1. Set Up Your Hypotheses

Explanation: Start by defining two statements:

• Null Hypothesis (H0): This is the idea that there is no change or effect. It’s what you assume is true.
• Alternative Hypothesis (H1): This is what you want to test. It suggests there is a change or effect.

Example: Suppose a company says their new batteries last an average of 500 hours. To check this:

• Null Hypothesis (H0): The average battery life is 500 hours.
• Alternative Hypothesis (H1): The average battery life is not 500 hours.

2. Choose the Test

Explanation: Pick a statistical test that fits your data and your hypotheses. Different tests are used for various kinds of data.

Example: Since you’re comparing the average battery life, you use a one-sample t-test .

3. Set the Significance Level

Explanation: Decide how much risk you’re willing to take if you make a wrong decision. This is called the significance level, often set at 0.05 or 5%.

Example: You choose a significance level of 0.05, meaning you’re okay with a 5% chance of being wrong.

4. Gather and Analyze Data

Explanation: Collect your data and perform the test. Calculate the test statistic to see how far your sample result is from what you assumed.

Example: You test 30 batteries and find they last an average of 485 hours. You then calculate how this average compares to the claimed 500 hours using the t-test.

5. Find the p-Value

Explanation: The p-value tells you the probability of getting a result as extreme as yours if the null hypothesis is true.

Example: You find a p-value of 0.0001. This means there’s a very small chance (0.01%) of getting an average battery life of 485 hours or less if the true average is 500 hours.

6. Make Your Decision

Explanation: Compare the p-value to your significance level. If the p-value is smaller, you reject the null hypothesis. If it’s larger, you do not reject it.

Example: Since 0.0001 is much less than 0.05, you reject the null hypothesis. This means the data suggests the average battery life is different from 500 hours.

7. Report Your Findings

Explanation: Summarize what the results mean. State whether you rejected the null hypothesis and what that implies.

Example: You conclude that the average battery life is likely different from 500 hours. This suggests the company’s claim might not be accurate.

Hypothesis testing is a way to use data to check if your guesses or assumptions are likely true. By following these steps—setting up your hypotheses, choosing the right test, deciding on a significance level, analyzing your data, finding the p-value, making a decision, and reporting results—you can determine if your data supports or challenges your initial idea.

Understanding Hypothesis Testing: A Simple Explanation

Hypothesis testing is a way to use data to make decisions. Here’s a straightforward guide:

1. What is the Null and Alternative Hypotheses?

• Null Hypothesis (H0): This is your starting assumption. It says that nothing has changed or that there is no effect. It’s what you assume to be true until your data shows otherwise. Example: If a company says their batteries last 500 hours, the null hypothesis is: “The average battery life is 500 hours.” This means you think the claim is correct unless you find evidence to prove otherwise.
• Alternative Hypothesis (H1): This is what you want to find out. It suggests that there is an effect or a difference. It’s what you are testing to see if it might be true. Example: To test the company’s claim, you might say: “The average battery life is not 500 hours.” This means you think the average battery life might be different from what the company says.

2. One-Tailed vs. Two-Tailed Tests

• One-Tailed Test: This test checks for an effect in only one direction. You use it when you’re only interested in finding out if something is either more or less than a specific value. Example: If you think the battery lasts longer than 500 hours, you would use a one-tailed test to see if the battery life is significantly more than 500 hours.
• Two-Tailed Test: This test checks for an effect in both directions. Use this when you want to see if something is different from a specific value, whether it’s more or less. Example: If you want to see if the battery life is different from 500 hours, whether it’s more or less, you would use a two-tailed test. This checks for any significant difference, regardless of the direction.

3. Common Misunderstandings

• Clarification: Hypothesis testing doesn’t prove that the null hypothesis is true. It just helps you decide if you should reject it. If there isn’t enough evidence against it, you don’t reject it, but that doesn’t mean it’s definitely true.
• Clarification: A small p-value shows that your data is unlikely if the null hypothesis is true. It suggests that the alternative hypothesis might be right, but it doesn’t prove the null hypothesis is false.
• Clarification: The significance level (alpha) is a set threshold, like 0.05, that helps you decide how much risk you’re willing to take for making a wrong decision. It should be chosen carefully, not randomly.
• Clarification: Hypothesis testing helps you make decisions based on data, but it doesn’t guarantee your results are correct. The quality of your data and the right choice of test affect how reliable your results are.

Benefits and Limitations of Hypothesis Testing

• Clear Decisions: Hypothesis testing helps you make clear decisions based on data. It shows whether the evidence supports or goes against your initial idea.
• Objective Analysis: It relies on data rather than personal opinions, so your decisions are based on facts rather than feelings.
• Concrete Numbers: You get specific numbers, like p-values, to understand how strong the evidence is against your idea.
• Control Risk: You can set a risk level (alpha level) to manage the chance of making an error, which helps avoid incorrect conclusions.
• Widely Used: It can be used in many areas, from science and business to social studies and engineering, making it a versatile tool.

Limitations

• Sample Size Matters: The results can be affected by the size of the sample. Small samples might give unreliable results, while large samples might find differences that aren’t meaningful in real life.
• Risk of Misinterpretation: A small p-value means the results are unlikely if the null hypothesis is true, but it doesn’t show how important the effect is.
• Needs Assumptions: Hypothesis testing requires certain conditions, like data being normally distributed . If these aren’t met, the results might not be accurate.
• Simple Decisions: It often results in a basic yes or no decision without giving detailed information about the size or impact of the effect.
• Can Be Misused: Sometimes, people misuse hypothesis testing, tweaking data to get a desired result or focusing only on whether the result is statistically significant.
• No Absolute Proof: Hypothesis testing doesn’t prove that your hypothesis is true. It only helps you decide if there’s enough evidence to reject the null hypothesis, so the conclusions are based on likelihood, not certainty.

Final Thoughts 

Hypothesis testing helps you make decisions based on data. It involves setting up your initial idea, picking a significance level, doing the test, and looking at the results. By following these steps, you can make sure your conclusions are based on solid information, not just guesses.

This approach lets you see if the evidence supports or contradicts your initial idea, helping you make better decisions. But remember that hypothesis testing isn’t perfect. Things like sample size and assumptions can affect the results, so it’s important to be aware of these limitations.

In simple terms, using a step-by-step guide for hypothesis testing is a great way to better understand your data. Follow the steps carefully and keep in mind the method’s limits.

What is the difference between one-tailed and two-tailed tests?

 A one-tailed test assesses the probability of the observed data in one direction (either greater than or less than a certain value). In contrast, a two-tailed test looks at both directions (greater than and less than) to detect any significant deviation from the null hypothesis.

How do you choose the appropriate test for hypothesis testing?

The choice of test depends on the type of data you have and the hypotheses you are testing. Common tests include t-tests, chi-square tests, and ANOVA. You get more details about ANOVA, you may read Complete Details on What is ANOVA in Statistics ?  It’s important to match the test to the data characteristics and the research question.

What is the role of sample size in hypothesis testing?  

Sample size affects the reliability of hypothesis testing. Larger samples provide more reliable estimates and can detect smaller effects, while smaller samples may lead to less accurate results and reduced power.

Can hypothesis testing prove that a hypothesis is true?  

Hypothesis testing cannot prove that a hypothesis is true. It can only provide evidence to support or reject the null hypothesis. A result can indicate whether the data is consistent with the null hypothesis or not, but it does not prove the alternative hypothesis with certainty.

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