Question

Credit card holders carry an average debt of $3,027 per month, with a standard deviation of $1,060. A credit card company is looking to drop its accounts with their customers who are not using their cards as much, that is customers whose monthly debts are in the bottom 2.5% of all accounts. Use a calculator to find how much debt must a customer have to be dropped by the credit card company if the company only looks at 30 accounts.

Answer 1

z-score =-1.96

standard deviation= 193.53

x= 2647.68

Explanation:

σx¯=1,06030−−√=193.53

By plugging all the numbers into the formula z=x¯−μσx¯ we find that

−1.96=x¯−3,027193.53

−379.32=x¯−3,027

2,647.68=x¯

Answer 2

To be dropped, the client must have debts of $949.40 or lower.

Explanation:

When the distribution is normal, we use the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question:

`\mu = 3027, \sigma = 1060`

Bottom 2.5%

The 2.5th percentile and lower.

The 2.5th percentile is X when Z has a pvalue of 0.025. So X when Z = -1.96. Then

`Z = (X - \mu)/(\sigma)`

`-1.96 = (X - 3027)/(1060)`

`X - 3027 = -1.96*1060`

`X = 949.4`

To be dropped, the client must have debts of $949.40 or lower.