A nationwide study found that the average yearly claim per automobile insurance

Question

A nationwide study found that the average yearly claim per automobile insurance policy (after deductible) is $1,000 with a standard deviation of $400. Consider these parameter values to be the true population parameters. Suppose that you are working for a small, start-up insurance company with only 625 automobile insurance policies, and the company has $605,000 to pay for claims next year. What is the probability that the company will not have enough money set aside to pay off the claims next year? If the company does not have enough money set aside, it will declare bankruptcy and you will lose your job. Should you start working on your resume? Show jour work and explain.

Explanation / Answer

According to Central Limit Theorem, the sampling distribution of sample means will become normal in shape as sample size increases for any variable, even when the variable is not normally distributed across the population. When N is large, the mean of sampling distribution approcahes normality with mean mu, and standard deviation, sigma/sqrt N. For a sample size of N=100 or more, the Central Limit theorem applies.

Therefore, using information given, mu=xbar=$1000, sigma=$400. In a sample of 625 automobiles, standard deviation, sigmaxbar=sigma/sqrt n=400/sqrt 625=16. The insurance per automobile in the current company is : $605000/625=$968.

Compute Z score, by using Z=(X-xbar)/sigmaxbar, where, X is the raw score. Then look into Z table to find area corresponding to Z score. That gives the required probability.

P(Z<968)=P[Z<(968-1000)/16]=P(Z<-2)=0.0228 (ans)

The probability is unusual, that is less than 5%. Therefore, one neednot be too worried about the anticipated situation.

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