# Probability of sample mean and range

In the Department of Education at UR University, student records suggest that the population of students spends an average of 5.5 hrs per week playing organized sports. The population's standard deviation is 2.2 hrs per week. Based on a sample of 121 students, healthy lifestyles Inc. (HLI) would like to apply the central limit theorem to make various estimates.

A.) Calculate the probability that the sample mean will be between 5.3 and 5.7 hrs

B.) How strange would it be to obtain a sample mean greater than 6.5 hrs?

https://brainmass.com/statistics/central-limit-theorem/probability-sample-mean-range-417165

#### Solution Summary

This solution is comprised of detailed explanation and step-by-step calculation of the given problem and provides students with a clear perspective of the underlying concepts.

Sampling Distribution

Problem 1:

The mean television viewing time for Americans is 15 hours per week (Money, November 2003). Suppose a sample of 40 Americans is taken to further investigate viewing habits. Assume the population standard deviation for weekly viewing time is o = 3 hours.

a. What is probability the sample mean will be within 1 hour of the population mean (to 4 decimals)?

b. What is the probability the sample mean will be within 45 minutes of the population mean (to 4 decimals)?

Problem 2:

The Democrat and Chronicle reported that 25% of the flights arriving at the San Diego airport during the first five months of 2001 were late (Democrat and Chronicle, July 23, 2001). Assume the population proportion is p = .25.

a. Calculate o (p) with a sample size of 1000 flights (to 4 decimals).

b. What is the probability that the sample proportion will be within +/-.03 of the population proportion if a sample of size 1000 is selected (to 4 decimals)?

c. What is the probability that the sample proportion will be within +/-0.3 of the population proportion if a sample of size 500 is selected (to 4 decimals)?

Problem 3:

The president of Doerman Distributors, Inc., believes that 30% of the firm's orders come from first-time customers. A simple random sample of 100 orders will be used to estimate the proportion of first-time customers.

a. What is the probability that the sample proportion will be between .20 and .40 (to 4 decimals)?

b. What is the probability that the sample proportion will be between .25 and .35 (to 4 decimals)?

Problem 4:

Americans have become increasingly concerned about the rising cost of Medicare. In 1990, the average annual Medicare spending per enrollee was $3267; in 2003, the average annual Medicare spending per enrollee was $6883 (Money, Fall 2003). Suppose you hired a consulting firm to take a sample of fifty 2003 Medicare enrollees to further investigate the nature of expenditures. Assume the population standard deviation for 2003 was $2200.

a. Calculate the standard error of the mean amount of Medicare spending for a sample of fifty 2003 enrollees (to 2 decimals).

b. What is the probability the sample mean will be within +/-$300 of the population mean (to 2 decimals).

c. What is the probability the sample mean will be greater than $7500 (to 4 decimals)?