Question

The height of women ages​ 20-29 is normally​ distributed, with a mean of 64.9 inches. Assume sigmaequals2.3 inches. Are you more likely to randomly select 1 woman with a height less than 66 inches or are you more likely to select a sample of 27 women with a mean height less than 66 ​inches? Explain.

Answer 1

To select a sample of 27 women with a mean height less than 66 ​inches is more likely than to randomly select 1 woman with a height less than 66 inches

Explanation:

1) The probability of randomly selecting 1 woman with a height less than 66 inches is

P(z<z(66)) where z(66) is the z-score of the woman whose height is 66 inches.

z score can be calculated using the formula

z(66)=
`(X-M)/(s)` where

• X =66 inches

• M is the mean height of women aged​ 20-29 (64.9 inches)

• s is the standard deviation (2.3 inches)

Then z(66)=
`(66-64.9)/(2.3)` ≈ 0.48

and P(z<0.48) = 0.6844

2) The probability of selecting a sample of 27 women with a mean height less than 66 ​inches can be calculated using the equation

t=
`(X-M)/((s)/(√(N) ) )` where

• X = 66 inches

• M is the average height of women aged 20-29 (64.9 inches)

• s is the standard deviation (2.3 inches)

• N is the sample size (27)

t=
`(66-64.9)/((2.3)/(√(27) ) )` ≈ 2.49

looking t-table P(t<2.49)≈0.9903

Since 0.9903>0.6844 we can conclude that to select a sample of 27 women with a mean height less than 66 ​inches is more likely than to randomly select 1 woman with a height less than 66 inches