MS Module 8 Difference in population proportions practice exam questions
(The attached PDF file has better formatting.)
A study on a treatment group vs a control group shows
treatment control
observations 60 89
successes 41 60
● The null hypothesis is H0: p1 = p2, where p1 and p2 are the true proportions of success for the two groups.
● The alternative hypothesis is Ha: p1 ≠ p2.
● The difference in population proportions is the proportion of successes for the treatment group minus the proportion of successes for the control group.
Question 1.2: Sample difference in the proportions of success
What is the sample difference in the proportions of success between the two groups?
Answer 1.2: 41 / 60 – 60 / 89 = 0.0092
Question 1.3: Proportion of success in combined sample
What is the sample proportion of success in the combination of the two groups?
Answer 1.3: (41 + 60) / (60 + 89) = 0.6779
(total successes / total observations)
Question 1.4: Variance of the sample difference
What is the variance of the sample difference in the proportions of success between the two groups if the null hypothesis is true?
Answer 1.4: (0.6779 × (1 – 0.6779) ) × (1 / 60 + 1 / 89) = 0.006093
(if null hypothesis is true, use the combined proportion of success for each group)
Question 1.5: Standard deviation of the sample difference
What is the standard deviation of the sample difference in the proportions of success between the two groups if the null hypothesis is true?
Answer 1.5: 0.0060930.5 = 0.0781
(standard deviation = square root of variance)
Question 1.6: Z statistic
What is the z statistic to test the null hypothesis H0: p1 = p2?
Answer 1.6: 0.0092 / 0.0781 = 0.1178
(z statistic = difference in proportions / standard deviation of this difference)
Question 1.7: p value
What is the p value for the two-tailed z test of the null hypothesis H0: p1 = p2?
Answer 1.7: 2 × Φ(-0.1178) = 0.9062
Interpolating in the statistical tables:
Φ(0.11) = 0.5438
Φ(0.12) = 0.5478
Φ(0.1178) = ( (0.1178 – 0.11) × 0.5478 + (0.12 – 0.1178) × 0.5438) / (0.12 – 0.11) = 0.5469
2 × Φ(-0.1178) = 2 × (1 – 0.5469) = 0.9062
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