Normal Probability & Random Number Generation

Engineers must consider the breadths of male heads when designing motorcycle helmets. Men have had breadths that are normally distributed with the mean of 6.0 inches and a standard deviation of 1.0 inches (based on anthropometric survey data from Gordon, Churchill, et al.).

A. If one male is randomly selected, find the probability that his head breadth is less than 6.2 inches.

B. The Safeguard Helmet Company plans an initial production run of 100 helmets. Find the probability that 100 randomly selected men have a mean head breadth less than 6.2 inches.

C. The production manager sees the result from part (B) and reasons that all helmets should be made for men with head breadths less than 6.2 inches, because they would fit all but a few men. What is wrong with that reasoning?

D. The production manager has just received an order from management to produce 250 helmets for inventory. Assuming a symmetric, normal distribution, he's been asked to inventory stock that will cover head sizes for the middle 80% of the average male population. Please provide the range of head sizes we are talking about with a lower boundary value and an upper boundary value.

E. What type of statistical estimation does this problem exemplify?

What effect does sample size have on our estimate of the mean? Use the random number generator in Excel's Data Analysis Pak to demonstrate this effect. You can access the Data Analysis Pak by selecting the 'Data' menu choice in Excel. When you do it should appear in the 'Analysis' box on the far right of ribbon. If it's not there, you can install it using the Excel Options and 'Add-in' choices just as you did with MegaStat. Simply select the first choice for Data Analysis Pak.

Let's consider simulating throwing a single dice. Use the random generator in the Data Analysis Pak to simulate dice throwing. In the Data Analysis Pak select 'Random Number Generator'. When this opens you have to input 7 pieces of information initially. Under 'Variables' select 5. This will generate 5 columns of data. Under 'Number of Random Numbers' select 15. This selects a sample size of 15. For distribution select "Uniform". Under parameters input '1' to '6'. For random seed enter '2453'. Then select 'Output Range' and click in cell A1. When you hit 'OK' the you will generate 5 columns by 15 rows. Each column represents a sample of size 15. You have now generated 5 samples of size 15. Compute the mean and standard deviation for each column using MegaStat's 'Descriptive Statistics' procedure. Do not use the population standard deviation value. Copy the row of means into another worksheet and transpose them using the 'Paste Special' function using your right mouse button. Make sure you select 'Values' and 'Transpose' before hitting Ok. Use MegaStat's Histogram function in the 'Frequency Distribution' procedure with a starting value of 1 and an interval size of 0.1. Examine your histogram. If the shape is not bell-shaped describe it as 'Somewhat Uniform'.

Now reopen the 'Random Number' generator. It will have remembered all your input data values. All you need to change is the "Number of Variables' selection. Change it to 10. Keep everything else the same except for your 'Output range'. Put your cursor in Cell A18 or enter A18 into the 'Output Range Choice. Select Ok and it will generate 10 samples of size 15. Compute the mean for each of your newly generated samples of size 15. Copy and transpose these means into a column adjacent to the first sample of means of size 15. Label this new column Size 10. Use MegaStat's Histogram function in the 'Frequency Distribution' procedure again with a starting value of 1 and an interval size of 0.1. Examine your histogram. Compare it to the first histogram. You should do this at least two more times for samples of size 30, 50, and 500. Generate your histogram and compare the histogram for each of the five sample sizes. (Remember that the parameter you change for each sample size is 'Number of Variables'.

You will need to assign the following descriptors to each histogram's shape.
i. Somewhat Uniform
ii. Somewhat Bell-shaped
iii. Bell-shaped

General conclusions:

A) What happens to the mean as the sample size increases from 5 to 10 to 30 to 50 to 500?
B) What happens to the standard deviation as the sample size increases?
C) What happens to the distribution shape as the sample size increases?
D) How do these results illustrate the Central Limit Theorem?