Question

Heights of 10 year olds, regardless of gender, closely follow a normal distribution with mean 55 inches and standard deviation 6 inches. Round all answers to two decimal places.

1. What is the probability that a randomly chosen 10 year old is shorter than 56 inches?

2. What is the probability that a randomly chosen 10 year old is between 61 and 66 inches?

3. If the tallest 15% of the class is considered very tall, what is the height cutoff for very tall? inches

4. What is the height of a 10 year old who is at the 28 th percentile? inches

Answer 1

Solution:

Given that,

mean = = 55

standard deviation = = 6

1 ) p ( x < 56 )

= p ( x -  / ) < ( 56 - 55 / 6 )

= p ( z < 1 / 6)

= p ( z < 0.17 )

Using z table

= 0.5675

Probability = 0.5675

2 ) p ( 61 < x < 66 )

= p ( 61 - 55 / 6) < ( x -  / ) < ( 66 - 55 / 6 )

= p ( 6 / 6 < z < 11 / 6 )

= p (1 < z < 1.83 )

= p (z < 1.83 ) - p ( z < 1 )

Using z table

= 0.9664 - 0.8413

= 0.1251

Probability = 0.1251

Using standard normal table,

3 ) P(Z > z) = 15%

1 - P(Z < z) = 0.15

P(Z < z) = 1 - 0.15 = 0.85

P(Z < 1.04) = 0.85

z = 1.04

Using z-score formula,

x = z * +

x = 1.04 * 6 + 55

x = 61.24

The height cutoff for very tall is 61.24 inches.

4 ) P(Z < z) = 28 %

P(Z < z) = 0.28

P(Z < -0.58 ) = 0.28

z = -0.58

Using z-score formula,

x = z * +

x = -0.58 * 6 + 55

x = 51.52

The height is 51.52 inches.