Question

simple random sample of size nequals38 is obtained from a population with muequals62 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 than65.1​). ​(c) Assuming the normal model can be​ used, determine ​P(x overbargreater than or equals64.1​).

Answer 1

a) Either the underlying distribution must be normal, or the sample size has to be at least 30.

Sampling distribution approximately normal, with mean 62 and standard deviation of 2.4333

b) 0.8990 = 89.90%

c) 0.1949 = 19.49%

Explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem to solve this question.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

`\mu = 62, \sigma = 15`

​(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.

Either the underlying distribution must be normal, or the sample size has to be at least 30.

In this case, n = 38.

By the Central Limit Theorem, the sampling distribution is approximately normal, with mean 62 and standard deviation
`s = (15)/(√(38)) = 2.4333`

(b) Assuming the normal model can be​ used, determine ​P(x overbar less than 65.1​). ​

This is the pvalue of Z when X = 65.1. So

`Z = (X - \mu)/(\sigma)`

By the Central Limit Theorem

`Z = (X - \mu)/(s)`

`Z = (65.1 - 62)/(2.4333)`

`Z = 1.27`

`Z = 1.27` has a pvalue of 0.8980

So the answer is 0.8990 = 89.90%

(c) Assuming the normal model can be​ used, determine ​P(x overbargreater than or equals64.1​).

This is 1 subtracted by the pvalue of Z when X = 64.1. So

`Z = (X - \mu)/(s)`

`Z = (64.1 - 62)/(2.4333)`

`Z = 0.86`

`Z = 0.86` has a pvalue of 0.8051

1 - 0.8051 = 0.1949

So the answer is 0.1949 = 19.49%