Question

A college student organization wants to start a nightclub for students under the age of 21. To assess support for this proposal, they will select an SRS of students and ask each respondent if he or she would patronize this type of establishment. They expect that about 69% of the student body would respond favorably. (a) What sample size is required to obtain a 95% confidence interval with an approximate margin of error of 0.031? (b) Suppose that 49% of the sample responds favorably. Calculate the margin of error for the 95% confidence interval.

Answer 1

a)Sample size should be atleast 855

b) (0.4565, 0.5235)

Explanation:

given that a college student organization wants to start a nightclub for students under the age of 21.

`H_0: p=0.69\\H_a: p \\eq 0.69`

(Two tailed test)

a) for margin of error to be less than 0.031

we have

`\sqrt{(pq)/(n) } *1/96<0.031\\\sqrt{(pq)/(n) }<0.015816\\(0.4624)/(√(n) ) <0.01582\\√(n) >29.235\\n>855`

b) given that sample P = 0.49

Sample q = 0.51

Std error =
`√(0.49*0.51/855) =0.0171`

Margin of error = 1.96*0.0171 = 0.0335

Confidence interval =
`(0.49-0.0335, 0.49+0.0335)\\=(0.4565, 0.5235)`