Question

The manufacturer of the ColorSmart-5000 television set claims that 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 399 consumers who have owned a ColorSmart-5000 television set for five years. Of these 399 consumers, 310 say that their ColorSmart-5000 television sets did not need repair, while 89 say that their ColorSmart-5000 television sets did need at least one repair.

(a) Letting p be the proportion of ColorSmart-5000 television sets that last five years without a single repair, set up the null and alternative hypotheses that the consumer group should use to attempt to show that the manufacturer’s claim is false.

H₀ :p  (Click to select)  ≥  >  ≠  <  .95 versus H_a : p  (Click to select)  <  ≥  >  ≠  .95.

(b) Use critical values and the previously given sample information to test the hypotheses you set up in part a by setting α equal to .10, .05, .01, and .001. How much evidence is there that the manufacturer’s claim is false?

(Click to select)  Reject  Do not reject  H₀ at each value of α;   (Click to select)  none  very strong  extremely strong  strong  some  evidence.

Answer 1

a)
Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p >= 0.95
Alternative Hypothesis, Ha: p < 0.95

b)

Test statistic,
z = (pcap - p)/sqrt(p*(1-p)/n)
z = (0.7769 - 0.95)/sqrt(0.95*(1-0.95)/399)
z = -15.865

This is left tailed test, for α = 0.1
Critical value of z is -1.28.
Hence reject H0 if z < -1.28

This is left tailed test, for α = 0.001
Critical value of z is -3.09.
Hence reject H0 if z < -3.09

Reject H0 at each value of α; extremely strong evidence