The manufacturer of the ColorSmart-5000 television set claims 95 percent of its sets last at least five years without needing a single repair. In order to test this claim, a consumer group randomly selects 420 consumers who have owned a ColorSmart-5000 television set for five years. Of these 420 consumers, 303 say their ColorSmart-5000 television sets did not need a repair, whereas 117 say their ColorSmart-5000 television sets did need at least one repair. (a) Find a 99 percent confidence interval for the proportion of all ColorSmart-5000 television sets that have lasted at least five years without needing a single repair.

Answer: (0.665 , 0.777)Step-by-step explanation:Given : Sample size of consumers: n=420  ;Number of consumers say their ColorSmart-5000 television sets did not need a repair=303 The the sample proportion for consumers say their ColorSmart-5000 television sets did not need a repair : [tex]\hat{p}=\dfrac{303}{420}\approx0.721[/tex]Significance level : [tex]\alpha:1-0.99=0.01[/tex]Critical value = [tex]z_{\alpha/2}=\pm2.576[/tex]The confidence interval for population proportion is given by :_ [tex]\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{p(1-p)}{n}}[/tex]i.e.  [tex]0.721\pm (2.576)\sqrt{\dfrac{0.721(1-0.721)}{420}}[/tex][tex]\approx0.721\pm 0.056=(0.721-0.056,0.721+0.056)\\\\=(0.665\ ,\ 0.777)[/tex]Hence,  a 99 percent confidence interval for the proportion of all ColorSmart-5000 television sets that have lasted at least five years without needing a single repair = (0.665 , 0.777)