Question

In a survey of women in a certain country​ (ages 20minus​29), the mean height was 65.6 inches with a standard deviation of 2.85 inches. Answer the following questions about the specified normal distribution. ​(a) What height represents the 95th ​percentile? ​(b) What height represents the first​ quartile?

Answer 1

SOLUTION:

From given data,

In a survey of women in a certain country​ (ages 20minus​29), the mean height was 65.6 inches with a standard deviation of 2.85 inches. Answer the following questions about the specified normal distribution.

mean = = 65.6

standard deviation = = 2.85

​(a) What height represents the 95th ​percentile

Where,

95% confidence interval

Confidence interval is 95%

95% = 95/100 = 0.95

= 1 - Confidence interval = 1-0.95 = 0.05

Z = Z_0.05 = 1.645

Use this formula,

z = (x - µ)/σ

1.645 = (x - 65.6)/2.85

=> x = 65.6 + ( 1.645 * 2.85 )

x = 65.6 + 4.68825

x = 70.28825

70.28825  inches height represent the 95th percentile.

​(b) What height represents the first​ quartile?

The z scores for the first quartile (25%) are -0.6745

Use this formula,

z = (x - µ)/σ
Therefore,
-0.6745 = (x - 65.6)/2.85

=> x = 65.6 + ( -0.6745 * 2.85 )

x = 65.6 -1.922325

x = 63.677675

63.677675 inches height represent the first quartile.