Question

6. The manufacturer of the ColourSmart television claims that 85% of its sets last at least five years without needing a single repair. In order to test this claim, a consumer group randomly selected 400 consumers who have owned a ColourSmart television set for five years. Of these 400 consumers, 65 say that their ColourSmart television sets did need at least one repair.
a. Obtain the point estimate for the proportion of all ColourSmart television sets that have lasted at least five years without needing a single repair. *
b. Determine whether the condition is satisfied to use the large sample formula of the confidence interval of the population proportion. (Check the assumption to use Z distribution) *
c. Find a 99% confidence interval for the true proportion of all ColourSmart television sets that have lasted at least five years without needing a single repair. (Type your answer in this format: Margin error =___x ____ =____; 99% CI = __________ <___<_____ ) *

Answer 1

Here we have

$n =400$

$p_0=0.85$

The point estimate of population proportion is $\hat p=\frac{65}{400}=0.1625$

At 99% confidence level margin of error is

$ME=\sqrt{\frac{\hat p(1-\hat p)}{n}}\cdot Z_{\frac{1-0.99}{2}}$

$ME=\sqrt{\frac{0.1625\cdot(1-0.1625)}{400}}\cdot Z_{0.005}=\sqrt{\frac{0.1625\cdot0.8375}{400}}\cdot 2.58=0.0475$

$n\cdot p_0= 400\cdot 0.85 = 340 >10$ and $n\cdot (1 - p_0)= 400\cdot (1-0.85) = 60 >10$

So, the condition is satisfied to use the large sample formula.

$\hat p\pm ME=0.1625\pm 0.0475$

A 99% confidence interval for the population proportion is $(0.115, 0.210)$

Answer:

a) 0.1625

b) yes

c) 0.0475, (0.115, 0.210)