Chapter 7 Q22

Q. The annual cost of automobile insurance is $939. Assume that the population standard deviation is $245. Find the probability that a SRS of insurance policies will have a sample mean within $25 of the population mean for sample sizes 30, 50, 100, and 400.

A. The interval is between 939 – 25, and 939 + 25 or 914 to 964. We can do this in Excel. For the first sample, where n =30, find the standard error. This is the population standard deviation divided by the square root of the sample size. Where n =30, the standard error is 44.73.

First find the probability from the extreme left of the distribution to 964. In Excel that is =norm.dist(964,939,44.73,true) = 0.71. 

Now find the probability from the extreme left of the distribution to 914. In Excel this is =norm.dist(914,939,44.73,true) = 0.29. 

The final step is to subtract the smaller probability from the larger one. This gives the probability of the area between 914 and 964. This is 0.71 – 0.29 = 0.42.

Follow the same steps with the larger samples. You will see that the probability increases with the sample size. As the sample size increases, the standard error decreases and we are more confident of the location of the unknown population parameter µ. For example, with a sample size of n=400, the probability increases to 0.96.

In this question we know µ. But make the intellectual leap to see that we can use the same method to estimate the location of µ if we did not know it.