Question

BusinessWeek conducted a survey of graduates from 30 top MBA programs. On the basis of the survey, assume that the mean annual salary for male and female graduates 10 years after graduation is $168,000 and $117,000, respectively. Assume the standard deviation for the male graduates is $40,000, and for the female graduates it is $25,000. What is the probability that a simple random sample of 40 male graduates will provide a sample mean within $10,000 of the population mean, $168,000

Answer 1

88.58% probability that a simple random sample of 40 male graduates will provide a sample mean within $10,000 of the population mean, $168,000

Explanation:

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

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 random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sample means with size n of at least 30 can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`

In this problem, we have that:

`\mu = 168000, \sigma = 40000, n = 40, s = (40000)/(√(40)) = 6324.55`

What is the probability that a simple random sample of 40 male graduates will provide a sample mean within $10,000 of the population mean, $168,000

This is the pvalue of Z when X = 168,000 + 10,000 = 178,000 subtracted by the pvalue of Z when X = 168,000 - 10,000 = 158,000. So

Z = 178000

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

By the Central Limit Theorem

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

`Z = (178000 - 168000)/(6324.55)`

`Z = 1.58`

`Z = 1.58` has a pvalue of 0.9429

Z = 158000

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

`Z = (158000 - 168000)/(6324.55)`

`Z = -1.58`

`Z = -1.58` has a pvalue of 0.0571

0.9429 - 0.0571 = 0.8858

88.58% probability that a simple random sample of 40 male graduates will provide a sample mean within $10,000 of the population mean, $168,000