Question

A population has a mean of 180 and a standard deviation of 24. A sample of 100 observations will be taken. The probability that the mean from that sample will be between 183 and 186 is _____.

Answer 1

The probability that the mean from that sample will be between 183 and 186 is 0.0994 = 9.94%.

Explanation:

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

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score 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 p-value, 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.

A population has a mean of 180 and a standard deviation of 24.

This means that
`\mu = 180, \sigma = 24`

A sample of 100 observations will be taken.

This means that
`n = 100, s = (24)/(√(100)) = 2.4`

The probability that the mean from that sample will be between 183 and 186 is:

This is the pvalue of Z when X = 186 subtracted by the pvalue of Z when X = 183. So

X = 186

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

By the Central Limit Theorem

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

`Z = (186 - 180)/(2.4)`

`Z = 2.5`

`Z = 2.5` has a pvalue of 0.9938

X = 183

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

`Z = (183 - 180)/(2.4)`

`Z = 1.25`

`Z = 1.25` has a pvalue of 0.8944

0.9938 - 0.8944 = 0.0994

The probability that the mean from that sample will be between 183 and 186 is 0.0994 = 9.94%.