Andre read a report stating that 35% of people in his country approved of the job their current Prime Minister was doing, 35% disapproved, and 30% neither approved nor disapproved. He wondered if these percentages held true in his city, so he obtained a random sample of 16 responses from people in his city. Here are the results:

| Opinion | Approve | Disapprove | Neither |
|--------------|---------|------------|---------|
| Observed counts | 5 | 8 | 3 |

Andre wanted to use these results to carry out a [tex]$\chi^2$[/tex] goodness-of-fit test to determine if the sample disagreed with the reported distribution. Which count(s) make this sample fail the large counts condition for this test?

A. The observed count of people who approve of the Prime Minister's job.
B. The observed count of people who disapprove of the Prime Minister's job.
C. The observed count of people who neither approve nor disapprove of the Prime Minister's job.
D. The expected count of people who approve of the Prime Minister's job.
E. The expected count of people who neither approve nor disapprove of the Prime Minister's job.

Andre's sample fails the large counts condition for a χ^2 goodness-of-fit test due to the expected count of people who neither approve nor disapprove of the Prime Minister's job, which is less than 5.

Explanation:

Andre is interested in whether the percentages reported for national approval of the Prime Minister apply to his city. He collected a sample of 16 responses to perform a χ2 goodness-of-fit test, but before carrying out the test, he needs to check the large counts condition. This condition requires that all expected counts have to be at least 5 for the test to be valid. To determine if the sample fails this condition, we must calculate the expected counts:

Expected count for 'Approve': 16 * 35% = 5.6
Expected count for 'Disapprove': 16 * 35% = 5.6
Expected count for 'Neither': 16 * 30% = 4.8

From the calculations above, it is clear that the observed counts for 'Approve' and 'Disapprove' meet the condition (since both are higher than 5), whereas the expected count for 'Neither' does not meet the condition, being less than the minimum required 5. Thus, the correct answer would be:

E. The expected count of people who neither approve nor disapprove of the Prime Minister's job.