Natalie is a teacher and holds an MA in English Education and is in progress on her PhD in psychology.
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.
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:
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:
Thus, for Karen, t = 5.476.
There are two more things that Karen needs to know to see if her result is significant and the students at her school really do have a higher mean IQ than the population. The first is the degrees of freedom, which for a t statistic is n - 1, or 29 for Karen. The second is the alpha value, or the significance level. This should be 0.10, 0.05, or 0.01. The smaller Karen sets the alpha value, the more stringent the test. That is, a smaller significance level means that it's more likely she'll have to reject her null hypothesis that girls in her school are smarter for the alternative hypothesis that they aren't. Let's say that she chooses alpha value = 0.05.
Finally, Karen will want to check a t-table. This table, which can often be found in the back of a statistics textbook, has the degrees of freedom down the columns and the significance levels along the top. She'll look down until she finds 29 degrees of freedom and then across until she finds the alpha value of 0.05. The number at that point, the t-critical value, is 1.699. Since Karen's t-statistic value of 5.447 is higher than 1.699, she will reject the null hypothesis. She cannot say that her students have demonstrated a significantly higher IQ than other students. If, on the other hand, her t-statistic value was lower than the t-critical value, she would accept the null hypothesis and reject the alternative hypothesis and be able to say that her students do have a higher average IQ than the rest of the population. This data can be shown on a bell curve.
To sum up, Karen will calculate the t-statistic using her data and our prior formula, then calculate the t-critical value by using the degrees of freedom and setting the significance level (alpha value) then finding that value on a t-table. If the t-statistic value is larger than the t-critical value, the null hypothesis is rejected. If the t-statistic value is smaller than the t-critical value, the null hypothesis is accepted.
Hypothesis testing involves designing a study and analyzing the data in order to see if the mean of the study significantly differs from the population mean. Designing a study can be done in three steps. First, identify the null hypothesis, or what the researcher believes to be true, and the alternative hypothesis, which is the opposite of the null hypothesis. Second, operationalize the variables, or define them in such a way so that they are measurable. Third, gather data by conducting the experiment. Once the mean, or the average of the data, has been identified, use the p-value approach to test the probability that the alternative hypothesis could be true. This too requires several steps. It involves calculating the t-statistic using the following formula:
It also involves calculating the degrees of freedom, which is n - 1, or one less the number of subjects. Finally, it involves setting the significance level at 0.10, 0.05, or 0.01. Lastly, it involves setting a t-critical value for the number at the significance level and degrees of freedom. If the t-statistic is lower than the t-critical value, the null hypothesis is accepted. If the t-statistic is higher than the t-critical value, the null hypothesis is rejected.
Vocabulary & Definitions
Hypothesis testing: Hypothesis testing is designing a study to determine if the researcher's hypothesis is true.
Null hypothesis: A null hypothesis is a person's hypothesis in the beginning.
Alternative hypothesis: An alternative hypothesis is used in hypothesis testing to contradict the null hypothesis in an attempt to reject the null hypothesis.
Mean: The mean is the average of the data.
After viewing this lesson, you should be able to:
- Identify the steps in conducting hypothesis testing for a mean
- List the conditions that must be met before being able to test the hypothesis
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