Business Week conducted a survey of graduates from 30 top...

Business Week conducted a survey of graduates from 30 top MBA programs (Business Week, September $22,2003$ . 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 $\$ 25,000$ . a. 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, $\$ 168,000 ?$ b. What is the probability that a simple random sample of 40 female graduates will provide a sample mean within $\$ 10,000$ of the population mean, $\$ 117,000$ ? c. In which of the preceding two cases, part (a) or part (b), do we have a higher probability of obtaining a sample estimate within $\$ 10,000$ of the population mean? Why? d. What is the probability that a simple random sample of 100 male graduates will provide a sample mean more than $\$ 4000$ below the population mean?

Business Week conducted a survey of graduates from 30 top MBA programs (Business Week, September $22,2003$ . 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 $\$ 25,000$ .
a. 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, $\$ 168,000 ?$
b. What is the probability that a simple random sample of 40 female graduates will provide a sample mean within $\$ 10,000$ of the population mean, $\$ 117,000$ ?
c. In which of the preceding two cases, part (a) or part (b), do we have a higher probability of obtaining a sample estimate within $\$ 10,000$ of the population mean? Why?
d. What is the probability that a simple random sample of 100 male graduates will provide a sample mean more than $\$ 4000$ below the population mean?

Key Concepts

Central Limit Theorem

The central limit theorem states that, regardless of the underlying population distribution, the distribution of the sample mean tends to be approximately normal when the sample size is sufficiently large. This is a crucial concept for making probability statements about the sample mean, as it allows us to use normal distribution techniques even when the original data might not be normally distributed.

Sampling Distribution of the Mean

The sampling distribution of the mean is the probability distribution of all possible sample means from a population. It is centered at the population mean, and its spread is determined by the standard error. This concept is fundamental in estimating probabilities about the sample mean and assessing the precision of the sample as an estimator of the population mean.

Standard Error

The standard error is the standard deviation of the sampling distribution of the mean. It is calculated by dividing the population standard deviation by the square root of the sample size. This measure quantifies the extent of variability in sample means from sample to sample, which is critical in evaluating how close a sample mean is likely to be to the population mean.

Z-Score

The z-score represents how many standard errors a data point (or sample mean) is away from the population mean. By converting a difference into a z-score, one can use the standard normal distribution to calculate probabilities, which is essential for tasks such as determining the likelihood of a sample mean falling within a specified range of the population mean.

Transcript

0:00 All right.

00:03 We're given some data about graduates and their salaries 10 years after graduation, and we've divided them into data from men and data from women.

00:16 So for part a, given these statistics, we are given sample of 40 men, and we want to find the probability that our sample mean will be within 10 ,000 of our population mean.

00:42 Hold on, going to correct that notation.

00:47 There we go.

00:48 Not my need is penmanship, but you get the general idea.

00:51 Right, so let's find our standard deviation of the sampling distribution.

00:56 That's going to be the standard deviation over sample size.

01:04 So that's going to be 40 ,000 divided by the square root of 40.

01:13 And that equals 6 ,324 .56.

01:21 We're going to find a z lower and z upper like so.

01:27 So for a z lower, it's going to be negative 10 ,000, divided by our standard deviation for the sample, or, yeah, our sampling distribution.

01:43 And then for our upper, it's going to be 10 ,000 positive.

01:49 When you compute these out, you get a negative.

01:51 1 .58 from 1 .58.

01:57 Comparing this to our normal probabilities table, this gives you probability lower, 0 .0571, probability upper, 0 .9429.

02:15 This means our probability is going to be probability upper minus probability lower, which is 0 .8858.

02:26 All right, part b.

02:33 This time we're looking at a sample of 40 women, and now we need to find the probability that our sample mean that we find is within 10 ,000 of the mean for the women.

02:52 So once again, we're going to find our standard deviation, sampling distribution.

02:58 That's going to be this time we're looking at the statistics for the women.

03:05 So this is going to be...

03:10 25 ,000 over square root of 40.

03:17 Calculate that out, that's 3 ,952 .47, or sorry, 847.

03:32 Anyway, let's find our z lower and z upper, so zl, zu.

03:38 Again, this is going to be negative 10 ,000 over our standard deviation of the sampling distribution for women...