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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?

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