An insurance company wants to see if costs on a certain medical procedure have changed in the last three months. They have much historical data showing that the distribution shape stayed the same over the years even when costs changed. They are not aware of anything that would have influenced a change in the distribution. The historical average price for this procedure was $387.61, with a sigma of $12.76. They want to be able to identify a cost change as low as $4.00 and be 95% confident of the answer (so they know whether to change rates).

a. What is the minimum number of random procedure costs they will have to sample?

Using the sample number n obtained above, the recent sample cost data on this procedure indicates an average price of $391.04 and a sigma of $14.23.

b. Does the sigma change indicate that the cost variation is no longer the same as historical?

c. If the above test does not show a significant sigma change, is the average change significantly different?

d. After analyzing the above sample, the insurance company decides to take a similar sample from a separate district to see if the two district sample costs are different. The two districts have much historical data showing that the distribution shapes are similar and have stayed the same over the years even when costs changed. They are not aware of anything that would have influenced a change in the distributions. As given above, the first district sample had an average price of $391.04 and a sigma of $14.23. The second district sample had an average price of $384.97 with a sigma of $16.06. Are the two districts different at a 95% confidence?