Question

The height of women (ages 20 to 29) are approximaltely normally distributed with a mean of 68 inches and standard deviation of 3.8 inches. The heights of men (ages 20 to 29) are approximately normally distributed with a mean height of 71.5 inches and a standard deviation of 3.4 inches.

A) Use the z- score to compare a woman that is 5 feet 7 inches and a man that is 5 feet 7 inches tall.

B) If a z-score of 3 considered very tall, what is thet height of a man that is very tall ? What is the height of a woman that is very tall?

C) If a z-score of -3 is considered very short, what is the height of a woman that is very short? What is the height of a man that is very short?

D) The interval between a z-score of -1 and a z-score of 1 is considered "normal". Identify the interval for the "middle 68% (the normal height) for women (20 to 29 in age). Identify the interval for the "middle 68%" (the normal height) for men (20 to 29 in age).

Answer 1

Solution:

     We are given that:

The height of women (ages 20 to 29) are approximaltely normally distributed with a mean of 68 inches and standard deviation of 3.8 inches.

Thus     and

The heights of men (ages 20 to 29) are approximately normally distributed with a mean height of 71.5 inches and a standard deviation of 3.4 inches.

The and

Part A) Use the z- score to compare a woman that is 5 feet 7 inches and a man that is 5 feet 7 inches tall.

1 foot = 12 inches then 5 feet = 12*5=60 inches

thus 5 feet 7 inches = 60+7=67 inches

now find z scores for x = 67

i) z score using womens mean and SD

ii) z score using men mean and SD

Since z score for men = -1.32 which is less than that of women, hence men at 5 feet 7 inch height is relatively small than that of women.

Part B) If a z-score of 3 considered very tall, what is thet height of a man that is very tall ? What is the height of a woman that is very tall?

z = 3

To find height by using z , we use following formula:

inches.

Thus man is very tall when height is 81.7 inches.

Now find woman height:

inches.

Thus woman is very tall when height = 79.4 inch.

Part C) If a z-score of -3 is considered very short, what is the height of a woman that is very short? What is the height of a man that is very short?

For Man:

Thus man is very short when height = 61.3 inches.

Now find woman height:

Thus woman is very short when height = 56.6 inches.

Part D) The interval between a z-score of -1 and a z-score of 1 is considered "normal".

Identify the interval for the "middle 68% (the normal height) for women (20 to 29 in age).

That is find :

Thus then interval is: ( 64.2 , 71.8 )

The interval for the "middle 68% (the normal height) for women (20 to 29 in age) is:

( 64.2 inches , 71.8 inches)

Identify the interval for the "middle 68%" (the normal height) for men (20 to 29 in age).

Thus then interval is: ( 68.1 , 74.9 )

The interval for the "middle 68% (the normal height) for men (20 to 29 in age) is:

( 68.1 inches , 74.9 inches )