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

Answer:

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

Step-by-step 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:

$Z = \frac{X - \mu}{\sigma}$

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 $s = \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 168000, \sigma = 40000, n = 40, s = \frac{40000}{\sqrt{40}} = 6324.55$

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

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{178000 - 168000}{6324.55}$

$Z = 1.58$

$Z = 1.58$ has a pvalue of 0.9429

Z = 158000

$Z = \frac{X - \mu}{s}$

$Z = \frac{158000 - 168000}{6324.55}$

$Z = -1.58$

$Z = -1.58$ 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