Question

A team of researchers used the Survey of Consumer Attitudes to investigate whether incentives would improve response rates on telephone surveys. A national sample of 735 households were selected for the survey. 368 households were randomly assigned to receive a monetary incentive and of these 286 responded to the survey. The other 367 households were assigned to only receive an advance letter and of these 245 responded to the survey.

Compute the conditional proportions to evaluate if monetary incentives improve the response rate.

Answer 1

p₁ = 286/368 = 0.777

p₂ = 245/367 = 0.668

The Pooled sample proportion (P) = (p1 * n1 + p2 * n2)/(n1 + n2)

                                                       = (0.777 * 368 + 0.668 * 367)/(368 + 367) = 0.723

SE = sqrt(P * (1 - P) * (1/n1 + 1/n2))

      = sqrt(0.723 * (1 - 0.723) * (1/368 + 1/367))

      = 0.033

The test statstic z = (p1 - p2)/SE

                             = (0.777 - 0.668)/0.033 = 3.30

P-value = P(Z > 3.30)

             = 1 - P(Z < 3.30)

             = 1 - 0.9995

             = 0.0005

As the P-value is less than 0.05, so the null hypothesis is rejected.

We have very strong evidence against the null hypothesis.

The test statistic z = (p1 - p2)/SE

                             = (0.777 - 0.668)/0.033 = 3.30

At 5% significance level the critical value is z^* = 1.96

As the test statistic value is greater than the critical value (3.3 > 1.96), so the null hypothesis is rejected.