Ap Stats Question Pls Help

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? (5 points)

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

Answer 1

One of the conditions for 1 prop Z-interval is np̂≥10, and for this problem np̂=8.

Therefore that condition is not satisfied and the answer should be the 4th option "This confidence interval is not valid because the quantity np̂ is too small."

First option would have been correct if we were working with means not proportions, this problem is about proportions.

Answer 2

So, I believe that this question looks trickier than it is. All it's asking for is whether the confidence intervals are valid, which worded differently could mean, "are the assumptions of this test met?" In this case the assumption is that the sample size is large, and normally distributed. We can't no the normal distribution, but we do know the sample size is large enough (>30), so I believe the first answer should be correct.