Question

6. A bank auditor claims that credit card balances are normally distributed, with a mean of $3570

and a standard deviation of $980.

You randomly select 5 credit card holders. What is the probability that their mean credit card

balance is less than $2500?

A. 0.1305

B. 0.0021

C. 0.0073

D. 0.1385

Answer 1

Probability that their mean credit card balance is less than $2500 is 0.0073.

Explanation:

We are given that a bank auditor claims that credit card balances are normally distributed, with a mean of $3570 and a standard deviation of $980.

You randomly select 5 credit card holders.

Let
`\bar X` = sample mean credit card balance

The z score probability distribution for sample mean is given by;

Z =
`(\bar X-\mu)/((\sigma)/(√(n) ) )` ~ N(0,1)

where,
`\mu` = population mean credit card balance = $3570

`\sigma` = standard deviation = $980

n = sample of credit card holders = 5

Now, the probability that their mean credit card balance is less than $2500 is given by = P(
`\bar X` < $2500)

P(
`\bar X` < $2500) = P(
`(\bar X-\mu)/((\sigma)/(√(n) ) )` <
`(2500-3570)/((980)/(√(5) ) )` ) = P(Z < -2.44) = 1 - P(Z
`\leq` 2.44)

= 1 - 0.9927 = 0.0073

The above probability is calculated by looking at the value of x = 2.44 in the z table which has an area of 0.9927.

Therefore, probability that their mean credit card balance is less than $2500 is 0.0073.