Question

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 1

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 : $\hat{p}=\dfrac{303}{420}\approx0.721$

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

Critical value = $z_{\alpha/2}=\pm2.576$

The confidence interval for population proportion is given by :_

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

i.e.  $0.721\pm (2.576)\sqrt{\dfrac{0.721(1-0.721)}{420}}$

$\approx0.721\pm 0.056=(0.721-0.056,0.721+0.056)\\\\=(0.665\ ,\ 0.777)$

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)