Question

Describe the sampling distribution of p. Assume the size of the population is 15,000. n=700​, p=0.6 Question content area bottom Part 1 Choose the phrase that best describes the shape of the sampling distribution of p below. A. Approximately normal because n≤0.05N and np(1−p)<10. B. Approximately normal because n≤0.05N and np(1−p)≥10. C. Not normal because n≤0.05N and np(1−p)<10. D. Not normal because n≤0.05N and np(1−p)≥10. Part 2 Determine the mean of the sampling distribution of p. μp=enter your response here ​(Round to one decimal place as​ needed.) Part 3 Determine the standard deviation of the sampling distribution of p. σp=enter your response here ​(Round to three decimal places as​ needed.)

Answer 1

Part 1:

The shape of the sampling distribution of p can be determined using the following criteria:
- np(1-p) ≥ 10
- n ≤ 0.05N

Here, n=700, N=15000, and p=0.6.

np(1-p) = 700 * 0.6 * (1-0.6) = 168.

Since np(1-p) ≥ 10, the first criterion is met.

n/N = 700/15000 = 0.0467, which is less than 0.05. So the second criterion is also met.

Therefore, the sampling distribution of p is approximately normal.

The correct answer is (B) Approximately normal because n≤0.05N and np(1−p)≥10.


Part 2:

The mean of the sampling distribution of p is given by the formula:

μp = p = 0.6

Therefore, the mean of the sampling distribution of p is 0.6.

The answer is μp = 0.6.


Part 3:

The standard deviation of the sampling distribution of p is given by the formula:

σp = sqrt [ p(1-p) / n ]

Substituting the values, we get:

σp = sqrt [ 0.6 * (1-0.6) / 700 ]

σp = 0.026

Therefore, the standard deviation of the sampling distribution of p is 0.026.

The answer is σp = 0.026.

Answer 2

Part 1:

The correct answer is B. The sampling distribution of p is approximately normal because n is less than 5% of the population size (n ≤ 0.05N) and np(1-p) is greater than or equal to 10 (np(1-p) ≥ 10).

Part 2:

The mean of the sampling distribution of p is equal to the population proportion, which is 0.6. Therefore, μp = 0.6.

Part 3:

The standard deviation of the sampling distribution of p is given by the formula:

σp = sqrt((p(1-p))/n)

Substituting p = 0.6 and n = 700, we get:

σp = sqrt((0.6(1-0.6))/700)

σp ≈ 0.024

Therefore, the standard deviation of the sampling distribution of p is approximately 0.024.