Question

The Normal model​ N(65, 2.5) describes the distribution of heights of college women​ (inches). Which of the following questions asks for a probability and which asks for a measurement​ (and is thus an inverse Normal​ question)? a. nbsp What is the probability that a random college woman has a height of 68 inches or​ more? b. nbsp To be in the Tall​ Club, a woman must have a height such that only​ 2% of women are taller. What is this​ height?

Answer 1

a) 11.51% probability that a random college woman has a height of 68 inches or​ more

b) This height is 70.135 inches.

Explanation:

Problems of normally distributed samples can be solved using the z-score formula.

The normal distribution has two parameters, which are the mean and the standard deviation.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 65, \sigma = 2.5`

a. What is the probability that a random college woman has a height of 68 inches or​ more?

This is 1 subtracted by the pvalue of Z when X = 68. So

`Z = (X - \mu)/(\sigma)`

`Z = (68 - 65)/(2.5)`

`Z = 1.2`

`Z = 1.2` has a pvalue of 0.8849

1 - 0.8849 = 0.1151

11.51% probability that a random college woman has a height of 68 inches or​ more

b. To be in the Tall​ Club, a woman must have a height such that only​ 2% of women are taller. What is this​ height?

This weight is the 100-2 = 98th percentile, which is the value of X when Z has a pvalue of 0.98. So X when Z = 2.054.

`Z = (X - \mu)/(\sigma)`

`2.054 = (X - 65)/(2.5)`

`X - 65 = 2.054*2.5`

`X = 70.135`

This height is 70.135 inches.