Question

A typical adult has an average IQ score of 105 with a standard deviation of 20. If 20 randomly selected adults are given an IQ test, what is the probability that the sample mean scores will be between 87 and 124 points

Answer 1

100% probability that the sample mean scores will be between 87 and 124 points

Explanation:

To solve this question, we have 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`, a large sample size can be approximated to a normal distribution with mean \mu and standard deviation, which is also called standard error
`s = (\sigma)/(√(n))`

In this problem, we have that:

`\mu = 105, \sigma = 20, n = 20, s = (20)/(√(20)) = 4.47`

What is the probability that the sample mean scores will be between 87 and 124 points

This is the pvalue of Z when X = 124 subtracted by the pvalue of Z when X = 87. So

X = 124

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

By the Central Limit Theorem

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

`Z = (124 - 105)/(4.47)`

`Z = 4.25`

`Z = 4.25` has a pvalue of 1

X = 87

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

`Z = (87 - 105)/(4.47)`

`Z = -4.25`

`Z = -4.25` has a pvalue of 0

1 - 0 = 1

100% probability that the sample mean scores will be between 87 and 124 points