Question

Asked Apr 11, 2019

Step 1

**Sample proportions:**

The sample proportion of even numbers = (number of even numbers in the sample of two households)/2.

For example, for observations (2, 2), there are 2 even numbers in the sample of two households. So, corresponding proportion of even numbers is: 2/2 = 1.

Again, for the observations (2, 7), there is 1 odd number and 1 even number. So, proportion of even numbers is: 1/2.

The following table lists the possible samples with replacement, from the three given observations, along with the sample proportion of even numbers for each sample:

Step 2

**Probability distribution:**

Observe that, there are a total of 9 possible samples. In the 9 samples, the proportion 0 occurs 1 time. So, the probability of proportion 0 is 1/9.

The proportion 1/2 occurs 4 times in 9 samples, so that the probability of the proportion 1/2 is 4/9.

Now, each of the proportion 1 occurs 4 times in 9 samples, so that the probability of the proportion 1 is 4/9.

The total probability is: 1/9 + 4/9 + 4/9 = 9/9 = 1.

Using these probabilities, the probability distribution of the sample proportions is shown as follows:

Step 3

**Explanation:**

The mean of the sample proportions is: *p* = (0 × 1/9) + (1/2 × 4/9) + (1 × 4/9) = 6/9 = 2/3.

In the population, there are 2 even numbers and 1 odd number, out of the three numbers. So, the population proportion of even numbers is 2/3.

Evidently, the **mean of the sample proportions is equal to the population proportion of even numbers**.

Thus, the sample proportion targets to find the population proportion.

Now, the mean of the samp...

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