Question

Engineers must consider the breadths of male heads when designing helmets. The company researchers have determined that the population of potential clientele have head breadths that are normally distributed with a mean of 6.9-in and a standard deviation of 0.8-in.In what range would you expect to find the middle 98% of most head breadths?Between and .If you were to draw samples of size 47 from this population, in what range would you expect to find the middle 98% of most averages for the breadths of male heads in the sample?Between and

Answer 1

The population of potential clientele have head breadths normally distributed with mean 6.9 in and standard deviation 0.8 in

98% of the breadths must differ from the mean by no more than 2.33 times the standard deviation.

This mean the the range we would expect to find the middle 98% of most head breadths is represented by the interval [6.9 - 2.33*0.8, 6.9 + 2.33*0.8] = [5.04, 8.76] in

If we are considering samples of size 47, the standard deviation for the mean of each sample is given by:

`\frac{\sigma}{\sqrt[]{n}}=\frac{0.8}{\sqrt[]{47}}=0.117\text{ in}`

In this case, we would expect to find 98% of most averages for the breadths of male heads in the interval [6.9 - 2.33*0.117, 6.9 + 2.33*0.117] = [6.63, 7.17] in