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 6.5-in and a standard deviation of 1.1-in.

In what range would you expect to find the middle 95% of most head breadths?
Between ___ and ____ .

If you were to draw samples of size 35 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 ___.

Answer 1

solution

Given that,

mean = = 6.5

standard deviation = = 1.1

middle 95% of score is

P(-z < Z < z) = 0.95

P(Z < z) - P(Z < -z) = 0.95

2 P(Z < z) - 1 = 0.95

2 P(Z < z) = 1 + 0. 95= 1.95

P(Z < z) = 1.95/ 2 = 0.975

P(Z <1.96 ) = 0.975

z  ±1.96   

Using z-score formula  

x= z * +

x= ±1.96 *1.1 +6.5

x=4.344 and 8.656

part b

n = 35

= 6.5

= / n = 1.1 /35=0.19

middle 95% of score is

P(-z < Z < z) = 0.95

P(Z < z) - P(Z < -z) = 0.95

2 P(Z < z) - 1 = 0.95

2 P(Z < z) = 1 + 0. 95= 1.95

P(Z < z) = 1.95/ 2 = 0.975

P(Z <1.96 ) = 0.975

z  ±1.96   

Using z-score formula  

x= z * +

x= ±1.96 *0.19 +6.5

x=6.13 and 6.87