Question

The CBS’ television show 60 Minutes has been successful for many years. That show recently had a share of 20, which means that among the TV sets in use at the time the show aired, 20% were tuned to 60 Minutes. Assume that this is based on a sample size of 5000 – which is a typical sample size for this kind of experiments. Construct a 95% confidence interval for the true proportion of TV sets that are tuned to 60 Minutes.

Answer 1

The formula for the 95% confidence interval for true population proportion is :

$\left [ \hat{p} - Z_{\frac{1-0.95}{2}}\times\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}, \hat{p} + Z_{\frac{1-0.95}{2}}\times\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \ \right ]$

Here,

$\hat{p}=20\%=0.2\\\\ (1-\hat{p})=(1-0.2)=0.8\\\\n=5000\\\\Z_{\frac{1-0.95}{2}}=Z_{0.025}=1.96$

So,

0.2 - 1.96 x 0.2 x 0.8 5000 0.2 +1.96 x 0.2 x 0.8 5000 [0.1889, 0.2111]

Hence ,

the 95% confidence interval for the true proportion of TV sets that are tuned to 60 Minutes , is = [ 0.1889 , 0.2111 ]