(Solved): In 2016 the Better Business Bureau settled \( 80 \% \) of complaints they received in the United St ...

In 2016 the Better Business Bureau settled \( 80 \% \) of complaints they received in the United States. Suppose you have been hired by the Better Business Bureau to investigate the complaints they received this year involving new car dealers. You plan to select a sample of nev car dealer complaints to estimate the proportion of complaints the Better Business Bureau is able to settle. Assume the population proportion of complaints settled for new car dealers is \( 0.80 \), the same as the overall proportion of complaints settled in 2016. Use the ztable. a. Suppose you select a sample of 200 complaints involving new car dealers. Show the sampling distribution of \( \bar{p} \). \( \begin{array}{ll}E(\bar{p})= & \text { (to } 2 \text { decimals) } \\ \sigma_{\bar{p}}= & \text { (to } 4 \text { decimals) }\end{array} \) b. Based upon a sample of 200 complaints, what is the probability that the sample proportion will be within \( 0.07 \) of the population proportion (to 4 decimals)? probability \( = \) c. Suppose you select a sample of 420 complaints involving new car dealers. Show the sampling distribution of \( \bar{p} \). \( E(\bar{p})=\quad \) (to 2 decimals) \( \sigma_{\pi}=\quad \) (to 4 decimals) d. Based upon the larger sample of 420 complaints, what is the probability that the sample proportion will be within \( 0.07 \) of the population proportion (to 4 decimals)? probability \( = \) e. As measured by the increase in probability, how much do you gain in precision by taking the larger somple in part (d)? Thelprobability of the sample proportion being within \( 0.07 \) of the population mean is increased by (to 3 decimals). There is a gain in precision by the sample size.