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Conducting Hypothesis Testing for a Mean: Process & Examples

Lesson Transcript
Instructor: Natalie Boyd

Natalie is a teacher and holds an MA in English Education and is in progress on her PhD in psychology.

Read this lesson to learn how you can use hypothesis testing to test for a mean. Learn what conditions need to be met before you can use hypothesis testing to find the average for the test subject. Updated: 10/21/2020

Hypothesis Testing

Karen teaches at an all-girls school, and she believes her students are smarter than most other students. She knows that the average IQ score for the general population is 100 and she's given IQ tests to 30 girls from her school. Now, she has the IQ scores from the test, but she's not sure what to do with them. How can she take these IQs and say whether her subjects are smarter than other students?

Karen is facing the question of hypothesis testing, which means designing a study and analyzing the data in order to see if your scientific prediction is correct. If Karen believes that her students are smarter than the average student, then she'll expect that her girls will have an average IQ higher than 100, which is the average IQ score. Hypothesis testing can help her find out if she's right. Let's work with Karen to test her hypothesis.

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  • 0:04 Hypothesis Testing
  • 0:56 Setting Up the Problem
  • 2:53 Analyzing the Data
  • 7:14 Lesson Summary
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Setting Up the Problem

The first thing Karen will want to do when she is testing her hypothesis is to set up the problem. To do this, she needs to fulfill several steps. First, she needs to identify the null and alternative hypotheses. It's easiest to start with Karen's hypothesis, which is that her students are smarter than students on average. The null hypothesis is the hypothesis that a person starts with. Thus, the null hypothesis is that the girls who attend the school where Karen teaches are smarter than students in general. The alternative hypothesis is the opposite of the null hypothesis. In Karen's case, the alternative hypothesis is that girls at Karen's school are not smarter than other students.

Second, she needs to operationalize the variables. Once Karen has identified the null and alternative hypotheses, she needs to make sure that all of her variables are measurable. She can measure whether someone goes to the school where she teaches pretty easily, but what about the variable of 'smarter'? How can you measure smarts? Does it mean that the person is good at math? Well read? Good at puzzles? There are many ways to think about 'smart.' Karen will need to operationalize the variable 'smart.' In her case, she might say that IQ is a measure of intelligence and, thus, a high IQ means a person is smart. Other scientists might operationalize smart based on how big a person's vocabulary is or how good they are at critical thinking.

Third, she needs to gather data. Once she has identified her null hypothesis and operationalized her variables, Karen is ready to gather data. This is the part where she goes out and does the study. For example, she gave an IQ test to 30 girls at her school. That's when she gathered the data. Now that she has her null and alternative hypotheses and her data, she's ready to analyze that data. She will be testing whether the mean, which is the average of the two groups, is significantly different. To do that, she'll use probabilities.

Analyzing the Data

Karen's prepared the data and now she's ready to analyze it. She's going to use the p-value approach. The p in p-value stands for probability and that's what Karen is going to calculate. The probability of finding a more extreme case towards the alternative hypothesis if the null hypothesis was true. In other words, what she's asking is, 'If girls at my school are smarter than students in general, what's the likelihood that my subjects will score closer to or lower than the average IQ for students in the general population?' This might seem weird but, essentially, Karen wants to know if the alternative hypothesis could be true instead of the null hypothesis.

If Karen calculates the p-value and realizes that it's very low, then it's not very likely that the null hypothesis is true. In contrast, if it's very high, the null hypothesis that girls at her school are smarter is likely to be true. So how does Karen go about calculating the p-value and determining if it's high or low? She'll want to first calculate the t statistic, which can be expressed like this:

t-statistic equation

In this equation, x bar stands for the mean in the study. For example, if Karen's subjects had an average IQ of 110, that would be her x bar. In the equation, μ stands for the mean of the entire population. For Karen's study, this would be the average IQ of every student in the United States, which she knows is 100. Essentially, this is what Karen would expect the average to be. She's testing whether her sample's mean is significantly different from the expected population mean. The denominator in this case is s / square root of n. Here, s is the standard deviation of the sample in the study. For example, Karen's students may fall on a distribution with an s of 10. This number will be divided by the square root of n or the number of students in Karen's study. So when Karen plugs in the numbers, she gets:

t-statistic problem

Thus, for Karen, t = 5.476.

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