Answer to Question #342808 in Statistics and Probability for Neil

Question

A population consists of the five numbers 2, 3, 6, 8, and 11.

Consider the samples of size 2 that can be drawn from this

population.

A. List all the possible samples and the corresponding mean.

B. Construct the sampling distribution of the sample mean.

Answer 1

1 -- All possible samples and their means:

1. 2, 3 with mean (2+3)/2=2,5
2. 2, 6 with mean 4
3. 2, 8 with mean 5
4. 2,11 with mean 6.5
5. 3, 6 with mean 4.5
6. 3, 8 with mean 5.5
7. 3, 11 with mean 7
8. 6, 8 with mean 7
9. 6, 11 with mean 8.5
10. 8, 11 with mean 9.5

2 -- Sampling distribution of the sample mean:

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c:c} & \bar{X} & f & f(\bar{X}) & Xf(\bar{X})& X^2f(\bar{X}) \\ \hline & 2.5 & 1 & 0.1 & 0.25 & 0.625\\ \hdashline & 4 & 1 & 0.1 & 0.4 & 1.6 \\ \hdashline & 5 & 1 & 0.1 & 0.5 & 2.5\\ \hdashline & 6.5 & 1 & 0.1 & 0.65 & 4.225 \\ \hdashline & 4.5& 1 & 0.1 & 0.45 & 2.025 \\ \hdashline & 5.5 & 1 & 0.1 & 0.55 & 3.025 \\ \hdashline & 7 & 2 & 0.2& 1.4 & 9.8 \\ \hdashline & 8.5 & 1 & 0.1& 0.85 & 7.225 \\ \hdashline & 9.5 & 1 & 0.1 & 0.95 & 9.025 \\ \hdashline Total= & & 10 & 1 & 6 & 40.05 \\ \hdashline \end{array}$$