Question

Vesco, Inc. is the manufacturer of a popular television set brand, ColorCrisp-510. Vesco claims that 95 percent of its set last at least five years without needing a single repair. In order to test this claim, a consumer group randomly selects 400 customers who have owned a ColorCrisp-510 television set for five years. Of these 400 customers, 316 say their ColorCrisp-510 television sets did not need a repair, whereas 84 say their ColorCrisp-510 television sets did need at least one repair.

1. Find a 99 percent confidence interval (C.I.) for the proportion of all ColorCrisp-510 television sets that have lasted at least five years without needing a single repair.
2. Does this C.I. provide strong evidence that the percentage of all ColorCrisp-510 television sets that last at least five years without a single repair is less than the 95 percent claimed by Vesco, Inc.?

Answer 1

a) = 316/400 = 0.79

At 99% confidence level, the critical value is z_0.005 = 2.58

The 99% confidence interval is

+/- z_0.005 * sqrt((1 - )/n)

= 0.79 +/- 2.58 * sqrt(0.79(1 - 0.79)/400)

= 0.79 +/- 0.0525

= 0.7375, 0.8425

b) The hypothesized value 0.95 doesn't lie in confidence interval.

Yes, the confidence interval provide strong evidence that the percentage of all ColorCrisp -510 television sets that last at least five years without a single repair is less than the 95% claimed by Vesco, Inc.