# Normal Distribution and Central Limit Theorem

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= ...