Question

he heights of adults in a certain town have a mean of 65.42 inches with a standard deviation of 2.32 inches. A random sample of 144 adults living in the center of the town was selected and their mean height was found to be 64.82 inches. Find the probability that a sample of this size would have a mean of 64.82 inches or less.

Answer 1

0.001 = 0.1% probability that a sample of this size would have a mean of 64.82 inches or less.

Explanation:

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

Normal probability distribution

When the distribution is normal, we use 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 normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

The heights of adults in a certain town have a mean of 65.42 inches with a standard deviation of 2.32 inches.

This means that
`\mu = 65.42, \sigma = 2.32`

Sample of 144:

This means that
`n = 144, s = (2.32)/(√(144)) = 0.1933`

Find the probability that a sample of this size would have a mean of 64.82 inches or less.

This is the pvalue of Z when X = 64.82. So

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

By the Central Limit Theorem

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

`Z = (64.82 - 65.42)/(0.1933)`

`Z = -3.1`

`Z = -3.1` has a pvalue of 0.001

0.001 = 0.1% probability that a sample of this size would have a mean of 64.82 inches or less.