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.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 42 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

we have that

For 95% confidence interval

Using a Z-scores table

the value of z=1.96

Remember that

z =(x - μ)/σ

we have

μ=5.8 in

σ=0.9 in

substitute

For z=1.96

1.96=(x-5.8)/0.9

x=1.96*0.9+5.8

x=7.564 in

For z=-1.96

-1.96=(x-5.8)/0.9

x=-1.96*0.9+5.8

x=4.036 in

therefore

between 4.036 in and 7.564 in

Part 2

If you were to draw samples of size 42 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?

In this part, divide the standard deviation by the square root of the sample size

`\frac{0.9}{\sqrt[\square]{42}}=0.1389`

the middle is equal to the average of

5.8(+/-)1.96*0.1389

-----> 6.07

-----> 5.53

Find out the average

(6.07+5.5278)/2=5.8

the middle is equal to the mean in this problem

between 5.57 and 6.07