Business Week surveyed MBA alumni 10 years after graduation...

Business Week surveyed MBA alumni 10 years after graduation (Business Week, September $22,$ 2003 . One finding was that alumni spend an average of $\$ 115.50$ per week eating out socially. You have been asked to conduct a follow-up study by taking a sample of 40 of these MBA alumni. Assume the population standard deviation is $\$ 35 .$ a. Show the sampling distribution of $\overline{x},$ the sample mean weekly expenditure for the 40 MBA alumni. b. What is the probability that the sample mean will be within $\$ 10$ of the population mean? c. Suppose you find a sample mean of $\$ 100 .$ What is the probability of finding a sample mean of $\$ 100$ or less? Would you consider this sample to be an unusually low spending group of alumni? Why or why not?

Business Week surveyed MBA alumni 10 years after graduation (Business Week, September $22,$ 2003 . One finding was that alumni spend an average of $\$ 115.50$ per week eating out socially. You have been asked to conduct a follow-up study by taking a sample of 40 of these MBA alumni. Assume the population standard deviation is $\$ 35 .$
a. Show the sampling distribution of $\overline{x},$ the sample mean weekly expenditure for the 40 MBA alumni.
b. What is the probability that the sample mean will be within $\$ 10$ of the population mean?
c. Suppose you find a sample mean of $\$ 100 .$ What is the probability of finding a sample mean of $\$ 100$ or less? Would you consider this sample to be an unusually low spending group of alumni? Why or why not?

Key Concepts

Central Limit Theorem

The central limit theorem explains that as the sample size increases, the distribution of the sample mean will approximate a normal distribution regardless of the shape of the population distribution. This theorem is fundamental to making inferences about the population using the sample mean.

Sampling Distribution of the Sample Mean

This concept pertains to the probability distribution of all possible sample means from samples of a fixed size taken from a population. The mean of this distribution is the same as the population mean, and its spread is measured by the standard error, which is the population standard deviation divided by the square root of the sample size.

Standard Error

Standard error is the measure of the dispersion of the sample means around the population mean. It is calculated by dividing the population standard deviation by the square root of the sample size, and it quantifies the expected variability among different sample means.

Z-Score Calculation

A z-score quantifies how many standard deviations an observed value is from the mean of a distribution. By converting a sample mean into a z-score, one can use the standard normal distribution to calculate the probability of observing such a value under the assumption that the sample comes from the population.

Hypothesis Testing and Interpretation of Unusual Outcomes

This concept involves determining whether an observed sample statistic (like the sample mean) is likely to have occurred by random chance under the assumption of a known population parameter. By comparing the probability (p-value) of the observed outcome, one can assess if the result is unusually extreme, which might lead to rejecting the assumption that the sample came from the intended population.

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