3.1 Point Estimates and Unbiased Estimators — 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.
Justify Bias and Calculate Point Estimates: worked answer
A toy population consists of values 2,4,6, so its population mean is 4. Select an unordered simple random sample of two without replacement; its three possible pairs are equally likely. Estimator M is the sample mean; estimator T is the sample maximum. Separately, a random sample of 50 residents contains 18 who commute by bicycle; their population bicycle proportion is unknown. Another population’s random sample of travel times is 12,14,16,18,20 minutes; its population mean is unknown.
Part a
Construct the values of M and T for every possible two-value sample. Calculate each estimator’s average over all equally likely samples and justify which is unbiased for the population mean and which overestimates it on average. Explain why M=3 in one sample would not prove bias.
| Sample | M | T |
|---|---|---|
| 2,4 | 3 | 4 |
| 2,6 | 4 | 6 |
| 4,6 | 5 | 6 |
M averages(3+4+5)/3=4, equal to the population mean, so it is unbiased. T averages(4+6+6)/3=16/3≈5.333, above 4, so it is biased upward. A particular M=3 underestimates, but unbiasedness concerns the average over all samples, not each observed estimate.
| Evidence for credit | Points |
|---|---|
| Computes every M/T value correctly. | 1 |
| Calculates centers 4 and 16/3. | 1 |
| Justifies unbiased versus upward bias and rejects a one-sample bias inference. | 1 |
Part b
Calculate the bicycle-proportion point estimate and travel-time mean point estimate. Name each corresponding population parameter, distinguish statistic/estimate from parameter, and explain whether the observed estimates become known true parameters.
estimates the population proportion p of residents who bicycle. minutes estimates the travel-time population mean μ. These observed sample statistics provide point estimates; the population parameters remain fixed but unknown and need not equal .36 or 16.
| Evidence for credit | Points |
|---|---|
| Calculates .36 and 16 minutes. | 1 |
| Names p and μ for the correct populations. | 1 |
| Distinguishes observed statistics/point estimates from unknown true parameters. | 1 |