Question

4. Assume that women's heights are normally distributed with a mean given by μ=63.5 in.and a standard deviation given by σ=2.5 in. a. If 1 woman is randomly selected, find the probability that her height is less than 64 in. b. If 34 women are randomly selected, find the probability that they have a mean height less than 64 in.

Answer 1

a) 57.93% probability that her height is less than 64 in

b) 87.90% probability that they have a mean height less than 64 in.

Explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution:

Problems of normally distributed samples are 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.

Central limit theorem:

The Central Limit Theorem estabilishes that, for a random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sample means with size n of at least 30 can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`

In this problem, we have that:

`\mu = 63.5, \sigma = 2.5`

a. If 1 woman is randomly selected, find the probability that her height is less than 64 in.

This is the pvalue of Z when X = 64.

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

`Z = (64 - 63.5)/(2.5)`

`Z = 0.2`

`Z = 0.2` has a pvalue of 0.5793

57.93% probability that her height is less than 64 in

b. If 34 women are randomly selected, find the probability that they have a mean height less than 64 in.

Now we have
`n = 34, s = (2.5)/(√(34)) = 0.4287`

This is the pvalue of Z when X = 64.

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

By the Central Limit Theorem

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

`Z = (64 - 63.5)/(0.4287)`

`Z = 1.17`

`Z = 1.17` has a pvalue of 0.8790

87.90% probability that they have a mean height less than 64 in.