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 1.1-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 39 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 ______.

Answer 1

Solution:-

Given that,

a) mean = = 5.8

standard deviation = = 1.1

Using standard normal table,

P( -z < Z < z) = 68%

= P(Z < z) - P(Z <-z ) = 0. 68

= 2P(Z < z) - 1 = 0. 68

= 2P(Z < z) = 1 + 0. 68

= P(Z < z) = 1. 68 / 2

= P(Z < z) = 0.84

= P(Z < 0.99 ) = 0.84

= z  ± 0.99

Using z-score formula,

x = z * +

x = - 0.99 * 1.1 + 5.8

x = 4.7

x = z * +

x = 0.99 * 1.1 + 5.8

x = 6.9

Between 4.7 and 6.9

b) n = 39

=   = 5.8

= / n = 1.1 / 39 = 0.18

Using z-score formula  

= z * +

= -0.99 * 0.18 + 5.8

= 5.6

Using z-score formula  

= z * +

= 0.99 * 0.18 +5.8

= 6.0

Between 5.6  and 6.0