A simple random sample of size nequals10 is obtained from a population with muequals68 and sigmaequals15. (a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities involving the sample mean? Assuming that this condition is true, describe the sampling distribution of x overbar. (b) Assuming the normal model can be used, determine P(x overbarless than71.4). (c) Assuming the normal model can be used, determine P(x overbargreater than or equals69.1).

(a) The distribution of the sample mean () is N (68, 4.74²).

(b) The value of is 0.7642.

(c) The value of is 0.3670.

Step-by-step explanation:

A random sample of size n = 10 is selected from a population.

Let the population be made up of the random variable X.

The mean and standard deviation of X are:

(a)

According to the Central Limit Theorem if we have a population with mean μ and standard deviation σ and we take appropriately huge random samples (n ≥ 30) from the population with replacement, then the distribution of the sample mean will be approximately normally distributed.

Since the sample selected is not large, i.e. n = 10 < 30, for the distribution of the sample mean will be approximately normally distributed, the population from which the sample is selected must be normally distributed.

Then, the mean of the distribution of the sample mean is given by,

And the standard deviation of the distribution of the sample mean is given by,

Thus, the distribution of the sample mean () is N (68, 4.74²).

(b)

Compute the value of as follows:

*Use a z-table for the probability.

Thus, the value of is 0.7642.

(c)

Compute the value of as follows:

Apply continuity correction as follows:

Thus, the value of is 0.3670.