Question

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.

Answer 1

Answer with explanation:

The confidence interval for population mean is given by :-

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

, where $\hat{p}$ is the sample proportion, n is the sample size , $z_{\alpha/2}$ is the critical z-value.

The  values needed to calculate a confidence interval at the 99% confidence level are :

Given : Significance level : $\alpha:1-0.99=0.01$

Sample size : n=450

Critical value : $z_{\alpha/2}=2.576$

Sample proportion: $\hat{p}=\dfrac{280}{450}\approx0.62$

Now, the  99% confidence level will be :

$\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\\\\=0.62\pm(2.576)\sqrt{\dfrac{0.62(1-0.62)}{450}}\\\\\approx0.62\pm0.023\\\\=(0.62-0.023,\ 0.62+0.023)=(0.597,\ 0.643)$

Hence, the  99% confidence interval is $(0.597,\ 0.643)$