Question

5. A telephone sales solicitor, trying to decide between two alternative sales pitches, randomly alternated between them during a day of calls. Using approach A, 20% of 100 calls led to requests for the mailing of additional product information. For approach B in another 100 calls, only 14 % led to requests for the product information mailing.

.At the 0.05 significance level, can we conclude that the difference in results was due to chance?

- Construct the 95% confidence level for the difference between population proportions (π1 - π2).

- Identify and interpret the p-value for the test.

ANSWER MUST FOLLOW THIS FORMAT:

Define H0 :

Define H1 :

Test statistic

Critical value of test statistic

Decision rule

Calculated value of test statistic

Reject or fail to reject H0?

Conclusion about differences in sales pitches

95% confidence interval for difference between proportions

Find the p-value

Interpret p-value

Answer 1

Define H₀ :

Define H₁ :

Test statistic: For difference in proportions, we will calculate z test statistic, to conduct two-proportion z-test.

Critical value of test statistic: 1.96 (for two-tail tests and 0.05 significance level)

Decision rule: Reject H0 is the test statistic is less than -1.96 or greater than 1.96

Calculated value of test statistic

Sample proportions are,

p1 = 0.2 and p2 = 0.14

Pooled sample proportion, p = (n1 * p1 + n2 * p2) / (n1 + n2)

= (100 * 0.2 + 100 * 0.14) / (100 + 100) = 0.17

Standard error (SE) of sampling distribution difference between two proportions =

SE = sqrt{ p * ( 1 - p ) * [ (1/n₁) + (1/n₂) ] }

= sqrt{ 0.17 * ( 1 - 0.17 ) * [ (1/100) + (1/100) ] }

= 0.0531225

Test statistic: z = (p1 - p2) / SE = (0.2 - 0.14) / 0.0531225 = 1.13

Reject or fail to reject H₀ - Since, calculated test statistic lies between -1.96 and +1.96, we fail to reject null hypothesis H0.

Conclusion about differences in sales pitches - Since, we fail to reject H0, there is no significant evidence of difference between population proportions of  sales pitches.

95% confidence interval for difference between proportions -

Z value for 95% confidence interval is 1.96

Sample difference in proportions = p1 - p2 = 0.2 - 0.14 = 0.06

95% confidence interval for difference in proportions = (0.06 - 1.96 * 0.0531225, 0.06 + 1.96 * 0.0531225)

= (-0.0441201, 0.1641201)

Find the p-value -

P(Z > 1.13) = 0.1292

For two-tail tests, P-value = 2 * 0.1292 = 0.2584

Interpret p-value - The probability of finding the observed, or more extreme, results when the null hypothesis (H₀) is true is 0.2584.