Final answer:
To draw all possible samples of 3 letters without replacement from the word "PAKISTAN", use combinations. Find the proportion of letter A in each sample by counting the number of samples that contain the letter A. Construct the sampling distribution of proportions by repeating this process for all possible samples. Verify that µp^ = P and Var(P^) = Pq/n * (N-n)/(N-1).
Step-by-step explanation:
To draw all possible samples of 3 letters without replacement from the word "PAKISTAN", we can use combinations. There are 8 letters in the word, so we have 8 options for the first letter, 7 options for the second letter, and 6 options for the third letter. The total number of possible samples is 8 × 7 × 6 = 336.
To find the proportion of letter A in each sample, we count the number of samples that contain the letter A. The letter A appears twice in the word "PAKISTAN", so there are 2 options for choosing the position of the first A and 1 option for choosing the position of the second A. The total number of samples with the letter A is 2 × 1 × 6 = 12.
Therefore, the proportion of letter A in each sample is 12/336 = 0.0357 or 3.57%.
The sampling distribution of proportions can be constructed by repeating this process for all possible samples and calculating the proportion of letter A in each sample. The mean of the sampling distribution, µp^, is the average proportion of letter A across all samples. In this case, µp^ would be the same as the proportion of letter A in the population.
The variance of the sampling distribution, Var(P^), can be calculated using the formula Pq/n * (N-n)/(N-1), where P is the proportion of letter A in the population, q is the proportion of non-A letters in the population, n is the sample size (3 in this case), and N is the population size (8 in this case).