Question

Suppose that in one region of the country, the mean amount of credit card debt per household in households having credit card debt is $8,000, with standard deviation $1,000. Find the probability that the mean amount of credit card debt in a sample of 400 such households will be within $7,925 and $$8,100.

Answer 1

Solution:

Given:  the mean amount of credit card debt per household in households having credit card debt is $8,000, with standard deviation $1,000.

Mean =

Standard deviation =

Sample size = n= 400

We have to find:

Population distribution of amount of credit card debt per household in households having credit card debt is unknown, but sample size = n = 400 is large.

Since sample size n = 400 is large , we can use Central limit theorem which states that for large sample size n ,
sampling distribution of sample mean is approximately normal with mean of sample means:

and standard deviation of sample means is:

thus find z score for and for

thus we get:

Look in z table for z = 2.0 and 0.00 as well as for  z = -1.5 and 0.00  and find corresponding area.

P( Z< 2.00 ) = 0.9772

and

P( Z< -1.50 ) = 0.0668

thus

Thus the probability that the mean amount of credit card debt in a sample of 400 such households will be within $7,925 and $$8,100 is 0.9104