Question

A researcher wishes to be 95% confident that her estimate of the true proportion of individuals who travel overseas is within 3% of the true proportion. a) Find the sample necessary if, in a prior study, a sample of 200 people showed that 40 traveled overseas last year. b) If no estimate of the sample proportion is available, how large should the sample be?

Answer 1

Answer:  a) 683   b) 1067

Step-by-step explanation:

The confidence interval for population proportion is given by :-

$p\pm z_{\alpha/2}\sqrt{\dfrac{p(1-p)}{n}}$

a) Given : Significance level :$\alpha=1-0.95=0.05$

Critical value : $z_{\alpha/2}}=\pm1.96$

Margin of error : $E=0.03$

Formula to calculate the sample size needed for interval estimate of population proportion :-

$n=p(1-p)(\dfrac{z_{\alpha/2}}{E})^2\\\\=0.2(0.8)(\dfrac{1.96}{0.03})^2=682.951111111\approx683$

Hence, the required sample size would be 683 .

b) If no estimate of the sample proportion is available then the formula to calculate sample size will be :-

$n=0.25(\dfrac{z_{\alpha/2}}{E})^2\\\\=0.25(\dfrac{1.96}{0.03})^2=1067.11111111\approx1067$

Hence, the required sample size would be 1067 .