2.12 Sampling and Randomization Distributions — Answers and Rubrics

The population has four individual values with probabilities 1/4. The particular sample has two observations 2 and 8, with mean 5. The sampling distribution above describes the mean over all 16 possible size-two samples from that population. It concerns means, not pooled original observations.

The means are 6,6,2,7,2,7,5,7,6,4,6,7. The observed average is 65/12≈5.417, and the observed range is 2–7. This small run has more recorded means at 6 and 7 than elsewhere and does not mirror the exact symmetric table. Dividing counts by 12 gives its probability estimates; finite simulation variation explains the discrepancy from the true center 5.

1,000 repetitions estimate the same n=2 sampling distribution more reliably. Changing each sample to 20 observations changes the mean’s distribution, generally reducing its spread and making its shape more normal for independent draws from this bounded finite-variance population. CLT concerns means and increasing sample size; it does not guarantee a perfect normal shape or make a highly skewed population safe at a universal threshold.

Valid results vary. Each of the 80 records must show the specified number of independent values from 2,4,6,8 with equal probabilities and the correct mean. There must be 40 means at each sample size. Both distributions estimate sampling distributions centered at 5; the larger-n distribution is expected to be more concentrated and more bell-shaped, but an honest description must match the learner’s actual recorded run. Credit requires actual samples, calculated means and a usable table/plot, not settings alone.

Each run selects one of the six equally likely two-person A sets, with the other two forming B; times 10,12,16,18 remain fixed. Calculate D for all 24 assignments and tabulate its frequencies. Possible values are−6,−2,0,2,6 seconds. The expected reference center is 0, but actual counts/observed center may vary. Treatment labels change; responses and group sizes do not. This recorded distribution approximates the statistic’s behavior under repeated random reassignment.

Each assignment has probability 1/6, so D has probabilities 1/6,1/6,2/6,1/6,1/6 at−6,−2,0,2,6. Center 0 and possible range−6 to 6 follow from the full enumeration. A 24-run table estimates it with random frequency variation. It reuses four fixed responses; it does not sample new values from a population.

For independent random draws with finite variance, increasing sample size improves the mean’s normal approximation; a much larger n may be needed for extreme skew. More repetitions improve the estimate of a fixed-n distribution without changing that theoretical distribution. D here has only five attainable values from six fixed-response assignments at two-per-group; the mean CLT does not force this small randomization distribution to be normal.