Question

In a survey of 500 residents in a certain town, 312 said they were opposed to constructing a new shopping mall. Can you conclude that at least 60% of the residents in this town are opposed to constructing a new shopping mall? Use ? = 0.05.

a. Find 95% confidence interval and use it to test the hypothesis.

b. Clearly write your conclusion.

c. If the true percentage of town residents opposing the new shopping mall is 65%, what is the type II error in our hypothesis?

d. If the true percentage of town residents opposing the new shopping mall is 65%, what is the sample size needed to recognize it with 99% certainty.

Answer 1

(a) The 95% confidence interval is between 0.5815 and 0.6665.

Observed
0.624	p (as decimal)
312/500	p (as fraction)
312.	X
500	n
0.5815	confidence interval 95.% lower
0.6665	confidence interval 95.% upper
0.0425	margin of error

(b) Since the confidence interval's lower limit is less than 60%, we cannot conclude that at least 60% of the residents in this town are opposed to constructing a new shopping mall.

(c) Power, 1−β = 0.7441

Type II error = β = 1 - 0.7441 = 0.2559

(d) n = 1463