Question

A recent poll found that ""433 of the 1548 randomly-selected adults questioned felt that unemployment compensation should be extended an additional six months while the country is in it current economic downturn."" Construct a 95% confidence interval to estimate the proportion of the U.S. adults who feel this way.

Answer 1

Answer: (0.257633, 0.302367)

Step-by-step explanation:

We know that , the confidence interval for population proportion is given by :-

$\hat{p}\pm z\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$ , where n= sample size , z= critical value , $\hat{p}$= sample proportion.

Let p be the proportion of the U.S. adults who feel  that unemployment compensation should be extended an additional six months while the country is in it current economic downturn.

Here , n= 1548 , $\hat{p}=\dfrac{433}{1548}\approx0.28$

For 95% confidence interval , z= 1.96

Now , a 95% confidence interval to estimate the proportion of the U.S. adults who feel this way would be :

$0.28\pm (1.96)\sqrt{\dfrac{0.28(1-0.28)}{1548}} \\\\=0.28\pm (1.96)\sqrt{\dfrac{0.28\times0.72}{1548}}\\\\=0.28\pm(1.96)(0.011412)\\\\=0.28\pm0.022367\\\\=(0.28-0.022367,\ 0.28+0.022367)\\\\=(0.257633,\ 0.302367)$

Hence, the 95% confidence interval to estimate the proportion of the U.S. adults who feel this way. would be : (0.257633, 0.302367)