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 the proportion of polygraph results.
Here's our problem statement: Trials in an experiment with a polygraph include 98 results that include 23 cases of wrong results and 75 cases of correct results. Use a 1% significance level to test the claim that such polygraph results are correct less than 80 percent of the time. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. Use the P-value method. Use the normal distribution as approximation of the binomial distribution.
OK, the first part of our problem says, “Let p be the population proportion of correct polygraph results. Identify the null and alternative hypotheses. Choose the correct answer below.” To form our null and alternative hypotheses, we need to go back to our problem statement and look for the claim. The first step is to identify the claim.
So here we find in our problem statement, it says the claim is that “polygraph results are correct less than 80% of the time.” So that means our proportion is going to be less than 80% of the time; that's the claim.
Now the claim has no semblance of equality in it. Therefore, it will become our alternative hypothesis. The null hypothesis will then be a statement of equality because that's what it is by definition. So we look at our answer options here, and answer options A, E, and F are obviously wrong because they use 20% for their values and the value that we're looking for for our claim is 80%. So we want to choose between answer options B, C, and D.
Look at the null hypotheses. In each case, is a statement of equality to be found? In each instance, the answer is yes, so we can't use that to differentiate our answer. Look at the alternative hypotheses. Which one matches the claim? Well, the one that matches the claim is here in answer option D. So I select that and check my answer. Excellent!
Now the next part of our question asks us to identify the test statistic. To do this, we're going to go into StatCrunch. So I’m gonna pull up StatCrunch here, and inside StatCrunch, I'm going to go to Stat –> Proportion Stats (because I'm looking for proportions) –> One Sample (because I'm only given one sample), and With Summary (because I don't have actual data but I do have some summary statistics in the problem statement).
In the options window that appears, the number of successes — remember that the population proportion is identified as being those with the correct polygraph results. So looking at the problem statement here, how many of the sample results are correct results. That's going to be the 75. So that's what I put here in the first field. The number of observations is the total number in the sample; that's 98. I want hypothesis testing. Here we need to match these fields with the correct null and alternative hypothesis that we identified earlier. So I change these fields to match. And once I've done that, I'm ready to hit Compute! and get my results.
Here in the results window towards the end, we see the second to last value in that table is my Z-statistic. So that's the number I'm going to slip in right here rounded to two places. I check my answer. Well done!
The P-value — I get that from the same results window here in StatCrunch. Well done!
And now the last part of our problem asks us to “identify the conclusion about the null hypothesis and the final conclusion that addresses the original claim.” To do this we're going to use the P-value method as we were instructed earlier. This means we compare our P-value with our significance level, alpha, from our problem state.
And we see that alpha equals 1%. Our P-value is almost 20%, and since that is greater than 1%, our P-value is greater than alpha. And that means we are outside the rejection region. Therefore we're going to fail to reject the null hypothesis. And because we fail to reject there is not sufficient evidence to support the claim. I check my answer. Nice work!
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