Question

Two hundred randomly selected residents in Hoptown were surveyed and asked, "Do you support the city's decision to build a public health clinic in your area?" A total of 108 said no, 8 said yes, and 84 had no opinion. A large sample z-confidence interval, p̂ ± z*, was constructed from these data to estimate the proportion that support building the clinic. Which statement is true for the confidence interval?

A. A sample size of more than 30 was used; therefore, the confidence interval is valid.
B. Each resident was asked the same question; therefore, the confidence interval is valid.
C. Because there are no conditions required for constructing a large sample confidence interval for a proportion, the confidence interval is valid.
D. This confidence interval is not valid because the quantity np̂ is too small.
E. "No opinion" was included as a response category for the question; therefore, this confidence interval is not valid.

Answer 1

The confidence interval from the Hoptown survey is invalid due to a small value of the quantity np™, which is required to be larger for a valid large sample z-confidence interval. The correct answer is option D. This confidence interval is not valid because the quantity np̂ is too small.

Step-by-step explanation:

The true statement about the confidence interval for the survey in Hoptown is D. This confidence interval is not valid because the quantity npâ„¢ is too small.

A confidence interval is valid if the sampled proportions meet the requirements of the large sample size condition for a proportion, which typically involve npâ„¢ and n(1 - pâ„¢) both being greater than 5 when constructing a binomial confidence interval. However, in the case of this survey, there are 200 residents and only 8 said yes.

Given that 8 is less than 10, this means that npâ„¢ (n being the sample size and pâ„¢ the sample proportion) would indeed be too small, rendering the confidence interval invalid.