3.9 Differences Between Sample Proportions — Answers and Rubrics
These solutions explain the new practice tasks in this lesson. For simulation parts, a valid recorded run may differ from another learner’s run; the rubric checks the model, recorded work and interpretation.
Original generated practice for the Fall 2026 course. Return to the lesson and independent checks.
Two Towns, Two Sampling Designs: worked answer
Town A has 10,000 households and a known population recycling proportion p_A=0.55. Town B has 15,000 households and known p_B=0.45. Take independent SRSs of 200 distinct A households and 300 distinct B households without replacement. Let D=sample proportion in A minus sample proportion in B. Use the usual 10% approximation and a normal approximation without a continuity correction where justified.
Part a
Calculate the mean and SD of D. Show both variance contributions and explain why the subtraction order affects the mean but not which contributions are added under the square root.
. . Reversing the difference would give mean −0.10, while the independent variances still add and the SD stays 0.0454148. This is an approximation for the small without-replacement sampling fractions.
| Evidence for credit | Points |
|---|---|
| Obtains mean 0.10 in the A-minus-B order. | 1 |
| Shows both indexed variance contributions and obtains SD 0.0454148. | 1 |
| Explains mean sign reversal versus unchanged sum-of-variances spread. | 1 |
Part b
Justify the original difference sampling model. Verify two independent random samples, both numerical 10% checks and all four expected-success/failure counts, then state the normal-model conclusion.
The problem provides independent SRSs. The two sampling checks are and . Expected A counts are 110 recycling and 90 not recycling; expected B counts are 135 and 165. Every count is at least 10. Thus an approximately normal sampling distribution for D is appropriate, using the usual small-fraction independence approximation.
| Evidence for credit | Points |
|---|---|
| Identifies the independent SRS design. | 1 |
| Checks 200≤1,000 and 300≤1,500 using the corresponding populations. | 1 |
| Calculates all four counts 110,90,135,165 and checks each against 10. | 1 |
| Justifies the approximate normal difference model. | 1 |
Part c
If Town B instead had known p_B=0.01 with the same n_B=300, which normality condition would fail? Would increasing only Town A’s sample fix that failure? Separately, explain why asking the same 200 households about two different months would not be two independent population samples.
Town B would have expected successes and failures 297. Its expected-success count fails; increasing only A does not increase B’s count. Repeated responses from the same households are paired and can be related, so the independent two-sample variance formula is not justified merely by having two month labels.
| Evidence for credit | Points |
|---|---|
| Calculates the failing expected count 3 and the passing count 297. | 1 |
| Explains why changing A cannot fix B’s count. | 1 |
| Identifies the same-household measurements as paired rather than independent samples. | 1 |
Part d
A separate experiment has 200 volunteer households randomly assigned, 100 to message A and 100 to message B. The supplied independent-response model has recycling probabilities 0.55 under message A and 0.45 under message B. Explain which randomization requirement is satisfied and whether two population SRSs or a 10% population-size check are required for this assignment comparison. Calculate all four expected counts and evaluate the normal approximation. Explain the limit on generalizing to all town households.
Random treatment assignment satisfies the experimental randomization condition. This experiment does not require two population SRSs or the sampling-without-replacement 10% checks merely to compare assigned treatments. Expected counts are 55 and 45 in A, 45 and 55 in B, so the supplied response model satisfies all four normality count conditions. The independent-response assumption is explicitly supplied; assignment alone should not be used to invent it. Volunteers are not a random population sample, so assignment does not establish representativeness of all town households.
| Evidence for credit | Points |
|---|---|
| Identifies random assignment to treatments as the experimental randomization condition. | 1 |
| Correctly separates assignment from random population sampling and its 10% checks. | 1 |
| Calculates and verifies 55,45,45,55. | 1 |
| States the supplied independent-response assumption and the volunteer-population generalization limit. | 1 |
Part e
For the original town SRSs, estimate . Show the standardized cutoff and area direction. Interpret D’s mean, SD and this probability in terms of the two populations and their specified repeated sample sizes.
; the upper-tail area is about 0.13546. Across repeated independent SRSs of 200 A households and 300 B households, the average A-minus-B sample-proportion difference is 0.10, or 10 percentage points; its typical deviation from 0.10 is about 0.04541, or 4.54 percentage points. About 13.55% of those sample pairs would show A’s sample recycling proportion more than 15 percentage points above B’s under this approximate model.
| Evidence for credit | Points |
|---|---|
| Uses the upper tail at z≈1.10096 and obtains about 0.13546. | 1 |
| Interprets mean 0.10 and SD 0.04541 in percentage-point difference units. | 1 |
| Interprets the probability for repeated 200/300-household sample pairs with the correct A-minus-B direction. | 1 |