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 two independent sample means of body temperatures. Here's our problem statement: A study was done on body temperatures of men and women. The results are shown in the table. Assume that the two samples are independent simple random samples selected from normally distributed populations and do not assume that the population standard deviations are equal. Complete Parts A and B below. Use a 1% significance level for both problems.
OK, Part A says, “Test the claim that men have a higher mean body temperature than women.” We're first asked to determine the null and alternative hypotheses. Remember that in order to determine the null and alternative hypotheses, we must first consider the claim the claim being made. Here it is that men have higher mean body temperature than women.
We notice from our sample statistics table listed here next to the problem statement that the men are being assigned Group 1. So all of the statistics and parameters that have a 1 subscript will be those for the men. The women are assigned a subscript 2, so all of the statistics and parameters that have a subscript 2 will be for the women. If the claim is that the men have a higher mean body temperature than the women, then that means that mu-1 will be greater than mu-2.
There's no semblance of equality with that statement, and so we can adopt that claim as our alternative hypothesis. The null hypothesis is by definition a statement of equality. So we're looking for the answer option where the null hypothesis has the two population parameters equal to each other and the alternative hypothesis has mu-1 greater than mu-2. Looking over my answer options, I see that is going to be here answer option C. I select that option and check my answer. Fantastic!
Now we're asked identify the test statistic, and to do that I'm going to run a hypothesis test inside StatCrunch. so here's StatCrunch. Inside StatCrunch, I’m going to go to Stat –> T Stats (because we're looking at comparing means without knowing the population standard deviation) –> Two Sample (because we're looking at two samples that are independent — very important to distinguish between dependent and independent samples — these samples here are independent; there's no real relationship between any one man in the first group and any one woman in the second group) –> With Summary (because we're not given actual data, just summary statistics).
Here in my options window, I'm going to put in the summary stats that were given there next to the problem statement. So Sample 1 is for the men, and Sample 2 is for the women. Notice this box next to “Pool variances” is unchecked by default. This is what we want, so we're going to leave that alone. The default radio button selection is also for the hypothesis test, so I'm going to leave that. Make sure that this alternative hypothesis matches what we selected previously, and away we go. Hit Compute!, and here we have our results window. The test statistic is always the second-to-last value in that table at the bottom of the results window. Nice work!
Now we're asked for the P-value. The P-value is always the last value in that same table in the results window. Excellent!
Now we are asked to state the conclusion for the test. The simplest way to do that is compare the P-value with the significance level alpha. We're asked to use a 1% significance level. Our P-value is 8.7%, and it’s easy to see that 8.7% is greater than 1%. So therefore, the area of the P-value is larger than the area for the significance level and cannot fit inside it because it's larger. Therefore, we are outside the region of rejection and we fail to reject the null hypothesis. Always when we fail to reject the null hypothesis, there is not sufficient evidence, so I select that answer option. Good job!
Now, Part B asks us to construct a confidence interval. I can go through the menu options in StatCrunch and input once again all of these summary stats, or I can take a shortcut by using the current results window and clicking on this button in the upper left-hand corner called Options, then in the drop-down that follows click Edit. Here I have all of the same summary stats, just switch this radio button to the confidence interval.
I need to make sure I have the appropriate confidence level here. I'm asked to use a 1% significance level. Normally in constructing the confidence interval, we would then select the complement of 1%, which is 99%, for our confidence interval level. However, in this case we have two independent samples, so the complement we're looking for is not 1 minus alpha but 1 minus 2 alpha. So we take twice the alpha to give us 2%, and we subtract that from 100% to give us the appropriate level for our confidence interval. We do this because we're looking at the difference between two independent samples and not just one sample. Hit Compute!, and now we have upper and lower limits for a confidence interval. Nice work!
Finally, we're asked, “Does the confidence intervals support the conclusion of the test?” Well, the conclusion we had from our hypothesis test was that we fail to reject the null hypothesis because there's not sufficient evidence to support the claim that men have a higher mean body temperature than women. If there's not sufficient evidence to support the claim that men have a higher mean body temperature than women, that means it's possible that men could have the same mean body temperature as women.
So what we need to look for is in our confidence interval could it be that the difference between these two is zero. If the difference between these two is zero, then that means that they could potentially be the same. So looking at this confidence interval for the difference between these two population parameters, is zero inside the confidence interval? The answer is yes, which means the difference could very well be zero. And if the difference is zero, then the two population parameters are the same. That means men and women have the same mean body temperature.
Here we see zero is included in the confidence interval. So that means it is possible they're the same. So the answer here would be yes, it does support the conclusion of our hypothesis test because the confidence interval contains zero. The confidence interval contains zero. Therefore, they could be the same. Therefore, we have to reject that claim; we don't have sufficient evidence to support the claim that there's a difference between the two groups and that the one group has a higher mean body temperature than the other. I check my answer. Excellent!
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