Question

Show your work. Carry out all calculations to at least 3 significant digits.

A manufacturer of TV sets claims that at least98% of its TV sets can last more than10years without needing a single repair. In order to verify and challenge this claim, a consumer group randomly selected 800 consumers who had owned a TV set made by this manufacturer for 10 years. Of these 800 consumers, 60 said that their TV sets needed some repair at least once.

a. Is there significant evidence showing that the manufacturer’s claim is false? Test using ? = 0.01.

b. Do the data support that the manufacturer’s actual no-repair rate does not even reach 94%? Use ? = 0.01.

Answer 1

The following information is provided: The sample size is N = 800, the number of favorable cases is X = 800 - 60 = 740, and the sample proportion is , and the significance level is α = 0.01

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p ≤ 0.98

Ha: p > 0.98

This corresponds to a right-tailed test, for which a z-test for one population proportion needs to be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.01, and the critical value for a right-tailed test is z_c = 2.33.

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that z =−11.112 ≤ zc​=2.33, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p = 1, and since p = 1 ≥ 0.01, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion p is greater than 0.98​, at the α = 0.01 significance level.

a. Is there significant evidence showing that the manufacturer’s claim is false?

His claim is False at α = 0.01

Confidence Interval

The 99% confidence interval for p is: 0.901 < p < 0.949.

b. Do the data support that the manufacturer’s actual no-repair rate does not even reach 94%?

His maximum value of p during the interval is 0.949 hence, > 0.94

He reaches at 94% during the interval