According to USA​ Today, customers are not settling for automobiles straight off the production lines. As an​ example, those who purchase a​ $355,000 Rolls-Royce typically add​ $25,000 in accessories. One of the affordable automobiles to receive additions is​ BMW's Mini Cooper. A sample of 179 recent Mini purchasers yielded a sample mean of​ $5,000 above the​ $20,200 base sticker price. Suppose the cost of accessories purchased for all Mini Coopers has a standard deviation of​ $1,500. Calculate a​ 95% confidence interval for the average cost of accessories on Mini Coopers.

Accepted Solution

Answer:  [tex](24980.25,\ 25419.75)[/tex]Step-by-step explanation:The confidence interval for population mean is given by :-[tex]\overline{x}\pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}[/tex]Given : Sample size : [tex]n= 179 [/tex] , which is a large sample , so we apply z-test .Sample mean : [tex]\overline{x}=20200+5000=25200 [/tex]Standard deviation : [tex]\sigma= 1500[/tex]Significance level : [tex]\alpha=1-0.95=0.05[/tex]Critical value : [tex]z_{\alpha/2}=1.96[/tex]Now, a confidence interval at the 95% level of confidence will be :-[tex]25200\pm(1.96)\dfrac{1500}{\sqrt{179}}\\\\\approx25200\pm219.75\\\\=(25200-219.75,\ 25200+219.75)\\\\=(24980.25,\ 25419.75)[/tex]