Question

Suppose the heights of 18-year-old men are approximately normally distributed, with mean 72 inches and standard deviation 4 inches.(a) what is the probability that an 18-year-old man selected at random is between 71 and 73 inches tall? (round your answer to four decimal places.) .2013 incorrect: your answer is incorrect. (b) if a random sample of twenty-three 18-year-old men is selected, what is the probability that the mean height x is between 71 and 73 inches? (round your answer to four decimal places.)

Answer 1

a) There is a 19.74% probability that an 18-year-old man selected at random is between 71 and 73 inches tall.

b) There is a 76.98% probability that the mean height x is between 71 and 73 inches.

Explanation:

The Central Limit Theorem estabilishes that, for a random variable X, with mean
`\mu` and standard deviation
`\sigma`, a large sample size can be approximated to a normal distribution with mean
`\mu` and standard deviation
`(\sigma)/(√(n))`.

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

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:

Mean of 72 inches and standard deviation 4 inches. This means that
`\mu = 72, \sigma = 4`.

(a) what is the probability that an 18-year-old man selected at random is between 71 and 73 inches tall?

This probability is the pvalue of Z when
`X = 73` and the pvalue of Z when
`X = 71`.

X = 73

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

`Z = (73 - 72)/(4)`

`Z = 0.25`

`Z = 0.25` has a pvalue of 0.5987

X = 71

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

`Z = (71 - 72)/(4)`

`Z = -0.25`

`Z = -0.25` has a pvalue of 0.4013

This mean that there is a 0.5987-0.4013 = 0.1974 = 19.74% probability that an 18-year-old man selected at random is between 71 and 73 inches tall.

b) if a random sample of twenty-three 18-year-old men is selected, what is the probability that the mean height x is between 71 and 73 inches?

Now we use the Central Limit Theorem, so:

`s = (\sigma)/(√(n)) = (4)/(√(23)) = 0.8341`.

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

`Z = (73 - 72)/(0.8341)`

`Z = 1.20`

`Z = 1.20` has a pvalue of 0.8849

X = 71

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

`Z = (71 - 72)/(0.8341)`

`Z = -1.20`

`Z = -1.20` has a pvalue of 0.1151

This means that there is a 0.8849 - 0.1151 = 0.7698 = 76.98% probability that the mean height x is between 71 and 73 inches.

Answer 2

(a) 0.1974
(b) 0.7695