Statistics questions

Question

Analyzing many years of data, a car insurance company finds that its customers' claims have a mean of $250 and a standard deviation of $1000, and the distribution is skewed to the right. In a year with 10,000 insurance claims, what is the probability that total of all the claims will be less than $2,700,000?

Give your answer either as a percentage between 0 and 100, rounded to 1 decimal place, omitting the percent sign, or as a proportion between 0 and 1, rounded to 3 decimal places.

Answer 1

This distribution is clearly non-normal. Skewed to the high values and has a standard deviation (250) larger than the mean (1000), which would predict negative values for the distribution.

Use the fact that means with many samples are nearly normal.

The mean (250) that is being compared against a limit is m=2,700,000/10,000 = 270.

The standard error of the nearly normal distribution of means would be

SE = sigma/sqrt(N) = 1000/sqrt(10000) =1000/100 = 10,

and the question becomes the percentile represented by

z = (270-250)/10 = 20/10 =2.0

P(z <= 2.0) = 0.977. Thus, only 0.023 = 2.3% probability of going over the limit of 270.