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

Answer:

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.

Step-by-step 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:

$Z = \frac{X - \mu}{\sigma}$

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 $s = \frac{\sigma}{\sqrt{n}}$

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.

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{64 - 63.5}{2.5}$

$Z = 0.2$

$Z = 0.2$ 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 $n = 34, s = \frac{2.5}{\sqrt{34}} = 0.4287$

This is the pvalue of Z when X = 64.

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{64 - 63.5}{0.4287}$

$Z = 1.17$

$Z = 1.17$ has a pvalue of 0.8790

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