Question

A newspaper publisher is considering launching a new "national" newspaper in Anytown. It is believed that the newspaper would have to capture over 12% of the market in order to be financially viable. During the planning stages of this newspaper, a market survey was conducted of a sample of 400 readers. After providing a brief description of the proposed newspaper, one question asked if the survey participant would subscribe to the newspaper if the cost did not exceed $20 per month. Suppose that 58 participants said they would subscribe. a. Can the publisher conclude that the proposed newspaper will be financially viable? Perform the appropriate test at a 1% level of significance. b. Suppose the actual value of the overall proportion of readers who would subscribe to this newspaper is 0.13. Was the decision made in part (a) correct? If not, what type of error was made? c. State the meaning of a Type I and Type II error in the context of this scenario. And what would be the repercussions of making these errors to the publisher?

Answer 1

Sample

Proportion = 58/500 = 0.145

a)

alpha = 0.01

Test Statistic

H₀: µ = 0.12

H_a: µ > 0.12

Test Statistic

t = (p - µ₀)/ {p*(1-p)/n}^1/2 = (0.145 – 0.12)/ {0.145*(1-0.145)/n}^1/2 = 1.42

p-value = TDIST(1.42,400-1,1) = 0.08

Result

Since the p-value is greater than 0.01, we fail to reject the null hypothesis.

Conclusion

Proposed newspaper will not be financially viable

b)

Suppose, p = 0.13

Then,

Test Statistic

t = (p - µ₀)/ {p*(1-p)/n}^1/2 = (0.13 – 0.12)/ {0.13*(1-0.13)/n}^1/2 = 0.59

p-value = TDIST(0.59,400-1,1) = 0.28

Still the result was correct in part a as p-value is greater than 0.01.

c)

type I error is the rejection of a true null hypothesis while type II error is the non-rejection of a false null hypothesis

Context of Problem

Type I error ie When in actual Proposed newspaper was financially viable but we concluded Proposed newspaper will not be financially viable

Type II error ie when in actual Proposed newspaper was not financially viable but we concluded Proposed newspaper was financially viable