Question

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities. Do you try to pad an insurance claim to cover your deductible? About 45% of all U.S. adults will try to pad their insurance claims! Suppose that you are the director of an insurance adjustment office. Your office has just received 122 insurance claims to be processed in the next few days. Find the following probabilities. (Round your answers to four decimal places.) (a) half or more of the claims have been padded (b) fewer than 45 of the claims have been padded (c) from 40 to 64 of the claims have been padded (d) more than 80 of the claims have not been padded

Answer 1

checking normality condition -

sample size (n) = 122

p = 0.45

1-p = 0.55

n*p = 122 * 0.45 = 54.9 >= 10

n * (1 - p ) = 67.1‬ > = 10

Hence we can assume population approximately normal.

a)

The population proportion of success is p = 0.45, and the sample size is n= 122. We need to compute Pr(X≥61):

The population mean is computed as:

and the population standard deviation is computed as:

Therefore, we get that

b)

The population proportion of success is p = 0.45, and the sample size is n= 122. We need to compute Pr(X≤45):

Therefore, we get that

c)

The population proportion of success is p = 0.45, and the sample size is n= 122. We need to compute Pr(40≤X≤64):

Therefore, we get that

d)

Pr[more than 80 claims not padded] = 1 - Pr[more than 80 claims padded]

The population proportion of success is p = 0.45, and the sample size is n= 122. We need to compute Pr(X≥80):

Therefore, we get that

Pr[more than 80 claims not padded] = 1 - Pr[more than 80 claims padded] = 1 - 0 = 1