Homework Soursea How many ordered with replacement samples of size 3 are th
a. How many ordered with replacement samples of size 3 are there from {1,2,3,4. Let the answer be N. Write these down. Then write the sampling distribution of X b. Collect all the similar values of X and combine the probabilities. So for example if there are n samples for which X equals 6, then the total probability that equals 6 is n/N c. Draw the histogram of the population. Is it normal looking? d. Draw the histogram of the sampling distribution of X. Is it more normal looking? e. Finally write a non-silly, non-humorous paragraph on how magical the central limit theorem is.
Solution
b. the probability distribution is shown below.
the histogram for population is uniform.
the histogram for sampling distribution is approximately normal.
the sampling distribution is approximately normal, no matter what is the distribution of population.
| Sample of size 3 | Sample Mean | ||
| 1 | 1 | 1 | 1 |
| 1 | 1 | 2 | 1.333333 |
| 1 | 1 | 3 | 1.666667 |
| 1 | 1 | 4 | 2 |
| 1 | 2 | 1 | 1.333333 |
| 1 | 2 | 2 | 1.666667 |
| 1 | 2 | 3 | 2 |
| 1 | 2 | 4 | 2.333333 |
| 1 | 3 | 1 | 1.666667 |
| 1 | 3 | 2 | 2 |
| 1 | 3 | 3 | 2.333333 |
| 1 | 3 | 4 | 2.666667 |
| 1 | 4 | 1 | 2 |
| 1 | 4 | 2 | 2.333333 |
| 1 | 4 | 3 | 2.666667 |
| 1 | 4 | 4 | 3 |
| 2 | 1 | 1 | 1.333333 |
| 2 | 1 | 2 | 1.666667 |
| 2 | 1 | 3 | 2 |
| 2 | 1 | 4 | 2.333333 |
| 2 | 2 | 1 | 1.666667 |
| 2 | 2 | 2 | 2 |
| 2 | 2 | 3 | 2.333333 |
| 2 | 2 | 4 | 2.666667 |
| 2 | 3 | 1 | 2 |
| 2 | 3 | 2 | 2.333333 |
| 2 | 3 | 3 | 2.666667 |
| 2 | 3 | 4 | 3 |
| 2 | 4 | 1 | 2.333333 |
| 2 | 4 | 2 | 2.666667 |
| 2 | 4 | 3 | 3 |
| 2 | 4 | 4 | 3.333333 |
| 3 | 1 | 1 | 1.666667 |
| 3 | 1 | 2 | 2 |
| 3 | 1 | 3 | 2.333333 |
| 3 | 1 | 4 | 2.666667 |
| 3 | 2 | 1 | 2 |
| 3 | 2 | 2 | 2.333333 |
| 3 | 2 | 3 | 2.666667 |
| 3 | 2 | 4 | 3 |
| 3 | 3 | 1 | 2.333333 |
| 3 | 3 | 2 | 2.666667 |
| 3 | 3 | 3 | 3 |
| 3 | 3 | 4 | 3.333333 |
| 3 | 4 | 1 | 2.666667 |
| 3 | 4 | 2 | 3 |
| 3 | 4 | 3 | 3.333333 |
| 3 | 4 | 4 | 3.666667 |
| 4 | 1 | 1 | 2 |
| 4 | 1 | 2 | 2.333333 |
| 4 | 1 | 3 | 2.666667 |
| 4 | 1 | 4 | 3 |
| 4 | 2 | 1 | 2.333333 |
| 4 | 2 | 2 | 2.666667 |
| 4 | 2 | 3 | 3 |
| 4 | 2 | 4 | 3.333333 |
| 4 | 3 | 1 | 2.666667 |
| 4 | 3 | 2 | 3 |
| 4 | 3 | 3 | 3.333333 |
| 4 | 3 | 4 | 3.666667 |
| 4 | 4 | 1 | 3 |
| 4 | 4 | 2 | 3.333333 |
| 4 | 4 | 3 | 3.666667 |
| 4 | 4 | 4 | 4 |
