Oven+Repair

=** The Oven Repair Problem **= ** Back to Central Limit Theorem **

**PART I**
The average repair costs of a microwave oven is $55 with a standard deviation of $8. The costs are normally distributed.

a) If one oven is repaired, find the probability that the repair bill is greater than $57. Step 1: Find the z-score z = (57 - 55) / 8 = 0.25 Step 2: Locate the probability from the z-table or other source. P(x > 0.25) = 40.13%

b) If 42 ovens are repaired, find the probability that the mean of the repair bills will be greater than $57. Step 1: Find the z-score z = (57 - 55) / [8/SQRT(42)] = 1.62 Step 2: Locate the probability from the z-table or other source. P(x-bar > 1.62) = 5.26%  Probability Calculator in GeoGebra is shown below. Make sure that you have N(0,1) in the first row.  Type in the z-score and click the greater than button on the far right.

Why does the z-score formula change when the problem changes from a single oven to a random sample of 42 ovens? Instead of using the standard deviation, we used the standard error of the mean.
 * Question **

z = (observed - mean) / standard deviation z = (observed - mean) / [standard deviation / SQRT(n)]

According to the Central Limit Theorem, the standard error of the mean is 8 / SQRT (42) = 1.23 or 1.23 dollars. What follows is a simulation that illustrates what I like to call the "new improved" standard deviation.

PART II
First, we need to generate samples of 42 microwave ovens. Use the randomNormal(55,8) function in a case table and add 42 cases. A dot plot indicates that the mean of ONE RANDOM SAMPLE of repair cost of 42 ovens is 54.6966 dollars. The cases in Microwave Oven Repair Cost plot are actual repair bills (simulated of course).



In order to use the simulation to generate the standard error of the mean, I collected 1,000 samples of size n=42. Use the history tool and select the sample mean. The History of Microwave Oven Repair Cost plot shows the distribution of the 1,000 sample means. This graph is called the distribution of the sample means. Note that the mean of the sample means is $55.0367, which is very close to the population mean of $55. Why is it not exactly $55?

The standard deviation hat plot and ruler tool show that the standard deviation in the distribution of the sample means is 1.20 dollars. According to the Central Limit Theorem, the standard error of the mean is 8 / SQRT (42) = or 1.23 dollars (see Part I above). This simulation is off by only 0.03.

The next plot shows a new simulation with 1,000 sample means when n=42. In this simulation the standard deviation is $1.24 which is really close to $1.23.

Completed TinkerPlots File