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 7.1-in and a standard deviation of 1.2-in.

In what range would you expect to find the middle 68% of most head breadths?
Between and .

If you were to draw samples of size 60 from this population, in what range would you expect to find the middle 68% of most averages for the breadths of male heads in the sample?
Between and .

Enter your answers as numbers. Your answers should be accurate to 2 decimal places.

Answer 1

Here X: Head breadths are normally distributed with

According to the Empirical rule (which is applicable for normal distribution only) 68% of data values fall within 1 standard deviation from the mean. That is between

and

So we would expect middle 68% of most head breadths between 5.9 in and 8.3 in.

The same rule will apply for average that is mean.

According to the Empirical rule, 68% data fall within  

n = sample size = 60

We would expect middle 68% of most averages for the breadths of male heads in the sample between 6.94 in and 7.25 inches.