Question

1. The distribution of heights of adult females: We assume that height is normally distributed with a population mean of 65 inches and a population standard deviation of 4 inches.

2. The distribution of heights of adult males: We assume that height is normally distributed with a population mean of 70 inches and a population standard deviation of 5 inches.

a. Above what Z-score value does 2.5% of the normal distribution fall? Using the formula for Z-scores and the Z-score value you just found, figure out what raw-score value (i.e., in inches) has 2.5% of the distribution of female heights above it.

b. Between what two Z-scores does the middle 80% of the normal distribution fall? Using the formula for Z-scores and the Z-score values you just found, figure out what raw-score values have 80% of the distribution of male heights between them.

c. Below what raw-score value does 5% of the distribution of male heights fall? Below what raw-score value does 5% of the distribution of female heights fall?

Answer 1

Solution:-

Given that,

mean = = 65 in. ( female)

standard deviation = = 4 in.

mean = = 70 in. ( male)

standard deviation = = 5 in.

a) Using standard normal table,

P(Z > z) = 2.5%

= 1 - P(Z < z) = 0.025  

= P(Z < z) = 1 - 0.025

= P(Z < z ) = 0.975

= P(Z < 1.96 ) = 0.975  

z = 1.96

Using z-score formula,

x = z * +

x = 1.96 * 4 + 65

x = 72.84 in.

b) Using standard normal table,

P( -z < Z < z) = 80%

= P(Z < z) - P(Z <-z ) = 0.80

= 2P(Z < z) - 1 = 0.80

= 2P(Z < z) = 1 + 0.80

= P(Z < z) = 1.80 / 2

= P(Z < z) = 0.90

= P(Z < 1.282) = 0.90

= z  ± 1.282

Using z-score formula,

x = z * +

x = -1.282 * 5 + 70

x = 63.59 in.

Using z-score formula,

x = z * +

x = 1.282 * 5 + 70

x = 76.41 in.

The middle 80% are from 63.59 in. to 76.41 in.

c) Using standard normal table,

P(Z < z) = 5%

= P(Z < z) = 0.05  

= P(Z < -1.645) = 0.05

z = -1.645

Using z-score formula,

x = z * +

x = -1.645 * 5 + 70

x = 61.78 in.( male)

Using z-score formula,

x = z * +

x = -1.645 * 4 + 65

x = 58.42 in.(female)