Question

From past data, a credit card company knows that the monthly balance, for those customers who carry a balance and whose credit limit is at least $2000, is approximately normally distributed with a mean of $825 and a standard deviation of $390.

The amount owed by those whose debt is at the 90th percentile and above (i.e., the top ten percent) is at least?

Answer 1

Solution:

Given that,

mean = = 825

standard deviation = = 390

Using standard normal table,

P(Z < z) = 90%

P(Z < z) = 0.90

P(Z < 1.282 ) = 0.90

z = 1.28

Using z-score formula,

x = z * +

x = 1.28 * 390 + 825

x = 1324.2

The 90^th percentile = 1324.2

Answer 2

To find the amount owed by those whose debt is at the 90th percentile and above (i.e., the top ten percent), we need to find the value of the monthly balance at the 90th percentile of the normal distribution.

The 90th percentile of a normal distribution corresponds to the value below which 90% of the data lies. In other words, it is the value that separates the top 10% of the data from the rest.

To find the value at the 90th percentile, we can use the cumulative distribution function (CDF) of the normal distribution. The CDF gives us the probability that a random variable is less than or equal to a specific value.

In this case, we want to find the value (let's call it x) such that P(X ≤ x) = 0.9 (since the 90th percentile is the point where 90% of the data is below x). We can use the standard normal distribution table or a calculator to find the z-score corresponding to the cumulative probability of 0.9.

Once we have the z-score, we can use the formula for transforming a z-score back to the original data scale:

x = μ + z * σ

where: μ = mean of the distribution ($825 in this case) σ = standard deviation of the distribution ($390 in this case) z = z-score corresponding to the cumulative probability of 0.9

Find the z-score and then calculate the value x using the formula above to get the amount owed by those whose debt is at the 90th percentile and above.