Question

Suppose a simple random sample of size n = 37 is obtained from a population with μ = 64 and σ = 17.

(a) What must be true regarding the distribution of the population to use the normal model to compute probabilities regarding the sample mean?

(α) To minimize the deviation of the sample mean from the population mean μ, what should be the minimum sample size (n)?

A. The population must be normally distributed.
B. The population must be normally distributed, and the sample size must be large.
C. There are no requirements on the shape of the distribution of the population.
D. Since the sample size is large enough, the population distribution does not need to be normal.

Assuming the normal model can be used, describe the sampling distribution X.

17 A. Approximately normal, with μ = 64 and σ = 17
B. Approximately normal, with μ = 64 and σ = √17
C. Approximately normal, with μ = 64 and σ = 17/√37

(b) Assuming the normal model can be used, determine P(X < 68.3).

(c) Assuming the normal model can be used, determine P(X > 65.3).

Answer 1

The distribution of the population must be normal for the normal model to be used. A large sample size is needed to minimize the deviation of the sample mean from the population mean. The sampling distribution of X is approximately normal, with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

Step-by-step explanation:

To use the normal model to compute probabilities regarding the sample mean, the population must be normally distributed. Answer option A is correct.

In order to minimize the deviation of the sample mean from the population mean, the minimum sample size (n) should be large. Answer option B is correct.

Assuming the normal model can be used, the sampling distribution of X is approximately normal, with a mean (μ) equal to the population mean (64) and a standard deviation (σ) equal to the population standard deviation divided by the square root of the sample size (17/√37). Answer option C is correct.