Eric wants to estimate the percentage of elementary school children who have a social media account. He surveys 450 elementary school children and finds that 280 have a social media account. Identify the values needed to calculate a confidence interval at the 99% confidence level. Then find the confidence interval.
Accepted Solution
A:
Answer with explanation:The confidence interval for population mean is given by :-[tex]\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex], where [tex]\hat{p}[/tex] is the sample proportion, n is the sample size , [tex]z_{\alpha/2}[/tex] is the critical z-value.The values needed to calculate a confidence interval at the 99% confidence level are :Given : Significance level : [tex]\alpha:1-0.99=0.01[/tex]Sample size : n=450Critical value : [tex]z_{\alpha/2}=2.576[/tex]Sample proportion: [tex]\hat{p}=\dfrac{280}{450}\approx0.62[/tex]Now, the 99% confidence level will be :[tex]\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\\\\=0.62\pm(2.576)\sqrt{\dfrac{0.62(1-0.62)}{450}}\\\\\approx0.62\pm0.023\\\\=(0.62-0.023,\ 0.62+0.023)=(0.597,\ 0.643)[/tex]Hence, the 99% confidence interval is [tex](0.597,\ 0.643)[/tex]