Question

A poll asked adults in a certain country whether immigration was a good thing or a bad thing for the country. Out of 10781078 ​respondents, 659659 said it was a good thing. a. What percentage of those taking the poll said immigration was a good​ thing? b. Check the conditions to determine whether you can apply the CLT to find a confidence interval. c. Find a​ 95% confidence interval for the proportion who believe immigration is a good thing.

Answer 1

Answer with explanation:

a. Given : Sample size : n= 1078

The number of adults said it was a good thing= 659

Then , the proportion of success : $p=\dfrac{659}{1078}\approx0.611$

b. The conditions to determine whether you can apply the CLT to find a confidence interval are :-

• The data must be randomly sampled.
• The sample size must be large (n>30).
• The sample values should be independent of each other.

Since the given data satisfies all the above  conditions , so we can apply the CLT to find a confidence interval.

c. Given : Significance level : $1-0.95=0.05$

Critical value : $z_{\alpha/2}=1.96$

The confidence interval for population proportion is given by :-

$p\ \pm\ z_{\alpha/2}\sqrt{\dfrac{p(1-p)}{n}}\\\\=0.611\pm (1.96)\sqrt{\dfrac{(0.611)(1-0.611)}{1078}}\approx0.611\pm0.029\\\\=(0.611-0.029,0.611+0.029)\\\\=(0.582, 0.64)$

Hence, a​ 95% confidence interval for the proportion who believe immigration is a good thing is (0.582, 0.64).