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 5.8-in and a standard deviation of 0.9-in.

If you were to draw samples of size 40 from this population, in what range would you expect to find the middle 95% 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.
In what range would you expect to find the middle 95% of most head breadths?
Between ___and___ .

Answer 1

The range in which we would expect to find the middle 95% of most averages for the breadths of male heads in a sample size of 40 is between 5.55 and 6.05 inches.

Step-by-step explanation:

To find the range in which we would expect to find the middle 95% of most averages for the breadths of male heads in a sample size of 40, we need to calculate the margin of error with a 95% confidence level. First, we find the critical value for a 95% confidence level. Since the data is normally distributed, we use the Z-distribution. The critical value for a 95% confidence level is 1.96.The margin of error is calculated by multiplying the critical value by the standard deviation of the population divided by the square root of the sample size. In this case, the standard deviation is 0.9 inches and the sample size is 40. So the margin of error is 1.96 * (0.9 / sqrt(40)) = 0.252 inches.

To find the range, we subtract and add the margin of error from the population mean. The population mean is 5.8 inches. So the range in which we would expect to find the middle 95% of most averages for the breadths of male heads in a sample size of 40 is 5.8 - 0.252 to 5.8 + 0.252, which is approximately between 5.55 and 6.05 inches.