Question

In spite of potential safety hazards, some people would like to have internet connection in
their car. A preliminary survey of working adults has estimated this proportion to be
somewhere around 0.3.
(a) Determine the sample size required to estimate the proportion of working adults who
would like internet connection in their car within 0.02 with 95% confidence.
(b) What is the 97% confidence interval if the sample size now is 100?
(c) Interpret the confidence interval obtained in (b).

Answer 1

(a)

$ME=z_{0.025}\sqrt{\frac{p(1-p)}{n}}\to\; n=(\frac{z_{0.025}}{ME})^2p(1-p)=\\ =(\frac{1.96}{0.02})^2*0.3(1-0.3)=2017.$

(b)

$97\%CI=(p-z_{0.015}\sqrt{\frac{p(1-p)}{n}},\;p+z_{0.015}\sqrt{\frac{p(1-p)}{n}})=\\ =(0.3-2.17\sqrt{\frac{0.3(1-0.3)}{100}},\;0.3+2.17\sqrt{\frac{0.3(1-0.3)}{100}})=\\ =(0.20,\;0.40).$

(c)

We are 97% confident that the true proportion of people who would like to have internet connection in their car lies between 0.20 and 0.40.