Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to use StatCrunch to perform hypothesis testing on means of course evaluation scores. Here's our problem statement: A data set includes data from student evaluations of courses. The summary statistics are sample size n = 94, sample mean x-bar = 3.57, and sample standard deviation = 0.55. Use a 5% significance level to test the claim that the population of student course evaluations has a mean equal to 3.75. Assume that a simple random sample has been selected. Identify the null and alternative hypotheses, test statistic, P-value, and state the final conclusion that addresses the original claim.
OK, so the first part of the problem asks us to identify the null and alternative hypotheses. To do this, we first need to think about the claim that's being made. If we go back to the problem statement, we can see that the claim is that the population of student course evaluations has a mean equal to 3.75.
So down here in our answer options, we're going to first form our null hypothesis the null hypothesis. By definition, this is a statement of equality. So right off the bat, we can eliminate answer option B, because the null hypothesis here says “not equal to.”
Now we need to choose between answer options A, C, and D. To do that, we're going to have to select the correct alternative hypothesis. Generally, the alternative hypothesis reflects the claim being made. However, in this case, because the claim has a semblance of equality to it — it says the mean is equal to 3.75, and that equality by definition belongs with the null hypothesis — we can't take the claim and turn it into our alternative hypothesis.
So what we have to do is take the complement of the claim as our alternative hypothesis. Here we say the mean is equal to 3.75. The complement of being equal to is being not equal to, so that's going to be our alternative hypothesis: Mu is not equal to 3.75. This is answer option C. I check my answer. Well done!
Now the second part of the problem asks us to determine the test statistic. To do that, I'm gonna pull up StatCrunch, because statistical software like StatCrunch makes hypothesis testing really easy. To get the test statistic, I'm first going to go into Stat –> T Stats (because I don't know what the population standard deviation is — I do have a sample standard deviation but not the population standard deviation — so that means I'm using a Student-t distribution) –> One Sample (because I'm only given one sample) –> With Summary (because I don't have actual data just summary statistics).
In the options window, I'm going to put the summary statistics that were given to me in the problem statement. Then in the field for hypothesis testing, I'm going to make sure that this matches the null and alternative hypothesis that I previously selected. I press Compute!, and out comes my results window with my results. Here in the table, the test statistic is always the second-to-last item in that table. So I'm just going to put that here in my answer field. Nice work!
The third part of our problem asks us to determine the P-value. Again, I go to my results window, and in that table the P-value is always the last value listed in that table. Well done!
Now the last part of our problem asks us to “state the final conclusion that addresses the original claim.” To do that, we can either use the test statistic or the P-value. It's easier to use the P-value, so that's the route I'm going to take.
To use the P-value to state our final conclusion from a hypothesis test, we need to compare the P-value with the significance level for the hypothesis test. In the problem statement, we were instructed to use a 5% significance level. So I compare the P-value, 0.2% with 5%. Because 0.002 is less than 0.05, I'm inside the rejection region, and therefore I'm going to reject the null hypothesis. Because I reject the null hypothesis, there is sufficient evidence.
This final field needs to match the alternative hypothesis that we selected earlier. Here the alternative hypothesis says that the mean for the population is not equal to 3.75, so I select that here. Now I check my answer. Excellent!
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Frustrated with a particular MyStatLab/MyMathLab homework problem? No worries! I'm Professor Curtis, and I'm here to help.