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 proportions of jury selections. Here's our problem statement: In a recent court case, it was found that, during a period of 11 years, 867 people were selected for grand jury duty, and 37% of them were from the same ethnicity. Among the people eligible for grand jury duty, 80.7% were of this ethnicity. Use a 0.01 significance level to test the claim that the selection process is biased against allowing this ethnicity to sit on the grand jury. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that address the original claim. Use the P-value method and the Normal distribution as an approximation to the binomial distribution.
OK, so this first part of our problem is asking us to identify the null and alternative hypothesis. The null hypothesis, of course, by definition is a statement of equality. So all these answer options where the null hypothesis is not a statement of equality (answer options B, for instance) will not be the correct answer. So of the answer options where we do have the null hypothesis as a statement of equality, we need to look at the alternative hypothesis to select the correct answer.
Here we go back to the problem statement. Typically, the alternative hypothesis reflects the claim. And here we see the claim is that the selection process is biased against allowing this ethnicity to set on the grand jury. If you’re biased against an ethnicity, that means you have fewer of that ethnicity than you should have. So were going to look for the option that has the alternative hypothesis that says, “less than.” And that’s going to be answer option A here. Good job!
Now the second part asks us to find the test statistic. And to do that, we’re going to look in StatCrunch. So here I have StatCrunch open, and inside StatCrunch, to do my hypothesis test, I’m going to go to Stat –> Proportion Stats (because we’re dealing with proportions) –> One Sample (because we have only the one sample) and With Summary (because we don't have any data).
Here we need to supply the number of successes and the number of observations. The number of successes is going to be the ethnicity that we find in the general populace. And so that’s 37%. But here StatCrunch is organized so we have to actually put in an actual number. So I’m going to get my calculator out, and I'm going to calculate that — 37% of the total [is] 320.79, which rounds up to 321. If the number had been such that I would round down, I’d still round up because I want to catch that partial member or person.
Of course, the total number we put here. We want a hypothesis test, and here I need to make sure this area matches the answer option that I got from the previous part. I hit Compute! and here's the results from my hypothesis test. The z-score is the next-to-last number there listed in the table. So I’ll stick that in here. Fantastic!
The third part of the problem asks us for the P-value, which is right here, the last number on the edge of this table. It says “less than .0001,” which is the same thing as zero for all practical purposes. It’s technically not zero, but it's a number so low that it might as well just be zero. So I can stick that here in my answer field. Well done!
Now the next part asks, “What is the conclusion of the null hypothesis?” Well, if I compare my P-value to my significance level — here in the problem we’re comparing with 1% — zero is definitely less than 1%. Therefore, we’re in the region of rejection, and we’re going to reject the null hypothesis. So I select that answer option here. Well done!
And now the last part of the problem asks, “Does the jury selection system appear to be fair?” Well, I go to the previous portion, right, where it says I’m going to reject the null hypothesis. We reject the null hypothesis, so there is sufficient evidence to support the claim. So I'm going to look to my answer options and select that option — “There is sufficient evidence to support the claim.” Excellent!
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