Question

1. A university wants to estimate the proportion of its students who smoke regularly. Suppose that this university has N = 30, 000 students in total and a SRSWOR of size n will be taken from the population. a) If the university wants the proportion estimator ˆp to achieve the precision with tolerance level e = 0.03 and risk α = 0.1. Estimate the minimal sample size needed for this estimation. (z0.1 = 1.28, z0.05 = 1.65) b) Suppose that, in a pilot study, the university takes a SRSWOR of size 20 students, among which only 2 students are found to be smokers. Based on this preliminary result, re-compute the minimal sample size for part (a). c) Suppose the university chooses the sample size estimated from (b), and finds that the sample proportion is ˆp = 0.15. Construct a 90% confidence interval for the population proportion p

Answer 1

ANSWER:

Minimal Sample size needed for estimation population proportion

(a) Formula for Computing n

Given,

e= 0.03;

=0.1 ; /2= 0.05;

=1.65

Substituting these values in forrmula

Sample Size needed for this estimation = 756

(b)

In pilot study, 2 students for found to be smokers in a sample of 20 students.

Guess value for proportion of students who smoke : = 2/20 = 0.1

Formula for sample size : n when we have a guess value for proportion

e= 0.03;

=0.1 ; /2= 0.05;

=1.65

Substituting these values in forrmula

Sample Size needed for this estimation = 272

c)

Given,

Sample proportion : = 0.15

Formula for confidence interval for population proportions : p

Given
n : Sample Size of Sample	272
: Sample Propotion of Sample : x/n	0.15
Confidence Level	90%
(= 100-90/100=10/100 ) = 0.1	0.1
/2 (=0.1/2=0.05)	0.05
	1.65

Confidence Interval for Population Proportion

Confidence interval for the population proportion p =(0.114276,0.185724)

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