4.6 Differences Between Two Sample Means — 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.
Compare Average Battery Lifetimes: worked answer
Brand A’s defined production population has N_A=100,000 batteries with known mean lifetime 10 hours and known SD 2 hours. Brand B has N_B=200,000 batteries with known mean lifetime 9 hours and known SD 3 hours. Both lifetime distributions are moderately skewed with finite variance and no extreme outliers. Take independent SRSs without replacement of 64 A batteries and 100 B batteries. Define D=mean lifetime in the A sample minus mean lifetime in the B sample. Use the usual 10% approximation and a justified normal approximation.
Part a
Calculate D’s sampling-distribution mean and SD with complete substitutions. Explain the squared population SDs, separate sample sizes and chosen subtraction order. State what would change if D were reversed.
hour. hour. The squared SDs give the two variance contributions; each is divided by its own n and the contributions add because the samples are independent. Reversing D changes the mean to −1 hour but leaves the SD unchanged.
| Evidence for credit | Points |
|---|---|
| Obtains mean 1 hour in the A-minus-B order. | 1 |
| Uses 2²/64+3²/100 under the square root and obtains 0.390512 hour. | 1 |
| Explains the separate variance contributions and sign reversal without changing SD. | 1 |
Part b
Justify the original sampling conditions and approximate normal shape for D. Include independent random samples, both numerical 10% checks, and both sample sizes and supplied population descriptions.
The samples are independent SRSs. The checks are and , supporting the usual independence approximation. Both populations are non-normal but moderately skewed without extreme outliers, and both n_A=64 and n_B=100 are at least 30. An approximately normal difference-mean model is therefore reasonable; it does not assert that individual battery lifetimes are normal.
| Evidence for credit | Points |
|---|---|
| Identifies independent SRSs. | 1 |
| Checks both sample fractions against their correct populations. | 1 |
| Checks both n≥30 and the supplied non-extreme population descriptions. | 1 |
| Justifies approximate normality for the difference of means. | 1 |
Part c
Evaluate these separate sampling models: (i) both populations normal with independent samples of 9 and 16; (ii) both populations moderately skewed with independent random samples of 40 and 50 and small sampling fractions; (iii) both populations non-normal with random samples of 25 and 50 and small sampling fractions. Explain the appropriate shape branch for each and why a large combined sample alone is insufficient in case (iii).
(i) Both normal population models yield a normal difference of independent sample means even at n=9 and 16. (ii) Both group sizes exceed 30, and the supplied moderate-skew descriptions and sampling assumptions support an approximately normal difference. (iii) The first sample has n=25<30 and its population is non-normal, so the stated two-group large-sample branch is not established; the second group’s 50 observations do not repair that missing justification. A different population-shape argument could matter, but none is supplied here.
| Evidence for credit | Points |
|---|---|
| Explains the both-normal branch at n=9/16. | 1 |
| Explains the approximate-normal branch at n=40/50. | 1 |
| Identifies the first non-normal sample’s missing size condition at 25/50 and rejects combined-n substitution. | 1 |
Part d
A separate experiment randomly assigns 80 independently produced test batteries, 40 to charging protocol C and 40 to protocol D, then measures lifetime in hours. The supplied protocol-specific response models are moderately skewed, finite-variance and without extreme outliers, with independent responses. Explain the experimental randomization requirement, whether two population SRSs or 10% population checks are needed, and whether the group sizes support a normal approximation for the mean difference. Distinguish treatment comparison from population representativeness.
Random assignment of batteries to protocols supplies the experimental randomization condition. Two population SRSs and their without-replacement 10% checks are not required merely for this assignment comparison. Under the explicitly supplied independent-response models, both group sizes are 40≥30 with moderate rather than extreme skew, supporting an approximate normal mean difference. Random assignment helps compare protocols; it does not make the 80 test batteries a representative random sample of all batteries sold.
| Evidence for credit | Points |
|---|---|
| Identifies batteries as experimental units and random protocol assignment as the randomization condition. | 1 |
| Separates experimental assignment from population SRS and 10% requirements. | 1 |
| Checks both group sizes 40 and the supplied response-model conditions. | 1 |
| Distinguishes treatment comparison from population generalization. | 1 |
Part e
For the original brand samples, estimate . Show the cutoff and area direction. Interpret D’s mean, SD and this probability in hours for repeated pairs of the specified brand samples.
, giving lower-tail probability about 0.10021. Across repeated independent SRSs of 64 A batteries and 100 B batteries, the average A-minus-B sample-mean lifetime difference is 1 hour; differences typically deviate from 1 by about 0.3905 hour. About 10.02% of such sample pairs would show A’s average lifetime at most 0.5 hour above B’s, including negative differences, under the approximate model. The event does not concern the difference between two individual batteries.
| Evidence for credit | Points |
|---|---|
| Uses the lower tail at z≈−1.28037 and obtains about 0.10021. | 1 |
| Interprets mean 1 hour and SD 0.3905 hour for the difference of sample means. | 1 |
| Interprets the probability for repeated 64/100-battery sample pairs, with correct signed event and units. | 1 |