statistics question

Question

Cable TV choices: A telecommunications company provided its cable TV subscribers with free access to a new sports channel for a period of one month. It then chose a sample of 400 television viewers and asked them whether they would be willing to pay an extra $10 per month to continue to access the channel. A total of 25 of the 400 replied that they would be willing to pay. The marketing director of the company claims that more than 5% of all its subscribers would pay for the channel. Can you conclude that the director’s claim is true? Use the α = 0.01 level of significance.

Answer 1

The null hypothesis H₀ = p = 0.05. The alternative is not specified so assume H₁ = p ^= 0.05 (two-sided test)

When alpha = 0.01 z_.005 = 2.575

25/400 = 0.625

calculate Z = (0.0625 - 0.05) / sqrt((0.0625*0.9375)/400)

=0.0125/0.0121= 1.0328

Z < z_0.005 so H₀ cannot be rejected and the marketing director's claim has validity.