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# Normal Distribution and Central Limit Theorem

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The mean amount purchased by a typical customer at Churchill's Grocery Store is \$23.50 with a standard deviation of \$5.00. Assume the distribution of amounts purchased follows the normal distribution. For a sample of 50 customers, answer the following questions.

a. What is the likelihood the sample mean is at least \$25.00?
b. What is the likelihood the sample mean is greater than \$22.50 but less than \$25.00?
c. Within what limits will 90 percent of the sample means occur?

https://brainmass.com/statistics/central-limit-theorem/normal-distribution-and-central-limit-theorem-231004

#### Solution Preview

The mean amount purchased by a typical customer at Churchill's Grocery Store is \$23.50 with a standard deviation of \$5.00. Assume the distribution of amounts purchased follows the normal distribution. For a sample of 50 customers, answer the following questions.

a. What is the likelihood the sample mean is at least \$25.00?

Mean=M = \$23.50
Standard deviation =s= \$5.00
sample size=n= 50
sx=standard error of mean=s/square root of n= 0.7071 = ( 5 /square root of 50)
xbar= \$25.00
z=(xbar-M )/sx= 2.1213 =(25-23.5)/0.7071
Cumulative Probability corresponding to z= ...

\$2.19