Consider the population (1,3). Suppose you take all possible samples of size n=3 with replacement. What is the sampling distribution of the sample total?
To find the sampling distribution of the sample total, we need to consider all possible samples of size 3 that can be selected with replacement from the population (1, 3).
In this case, each sample of size 3 can be thought of as a sequence of 3 elements, where each element can either be 1 or 3 (as these are the only values in the population). Since replacement is allowed, the same element can appear multiple times in a single sample.
To find the sampling distribution, we need to calculate the sample total for each possible sample. The sample total is the sum of all elements in the sample.
Let's go through the process step by step to find the sampling distribution:
Step 1: Generate all possible samples
Since the population (1, 3) has only 2 elements, there are a total of 2^3 = 8 possible samples of size 3 that can be selected with replacement. We can generate all possible samples as follows:
(1, 1, 1)
(1, 1, 3)
(1, 3, 1)
(1, 3, 3)
(3, 1, 1)
(3, 1, 3)
(3, 3, 1)
(3, 3, 3)
Step 2: Calculate the sample total for each sample
For each sample, we need to calculate the sum of its elements. Let's calculate the sample total for each possible sample:
(1, 1, 1) => 1 + 1 + 1 = 3
(1, 1, 3) => 1 + 1 + 3 = 5
(1, 3, 1) => 1 + 3 + 1 = 5
(1, 3, 3) => 1 + 3 + 3 = 7
(3, 1, 1) => 3 + 1 + 1 = 5
(3, 1, 3) => 3 + 1 + 3 = 7
(3, 3, 1) => 3 + 3 + 1 = 7
(3, 3, 3) => 3 + 3 + 3 = 9
Step 3: Organize the sample totals in a frequency distribution
To obtain the sampling distribution, we need to organize the sample totals in a frequency distribution. We list each unique sample total and calculate its frequency, which represents the number of samples that had that particular total.
Sample totals: 3, 5, 7, 9
Frequency: 1, 2, 3, 1
Thus, the sampling distribution of the sample total from all possible samples of size n = 3 with replacement from the given population is as follows:
Sample Total | Frequency
-----------------------
3 | 1
5 | 2
7 | 3
9 | 1