Question

The scores of individual students on the American College Testing (ACT), college readiness assessment, have a Normal distribution with a mean of 18.6 and a standard deviation of 6.0. At Northside High, 36 seniors take the test. Assume the scores at this school have the same distribution as national scores. What is the standard deviation of the sampling distribution of the sample mean score for a random sample of 36 students

Answer 1

The standard deviation of the sampling distribution of the sample mean score for a random sample of 36 students is 1.

Explanation:

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.

In this problem, we have that:

`\sigma = 6, n = 36`

Then, by the Central Limit Theorem:

`s = (\sigma)/(√(n))`

`s = (6)/(√(36))`

`s = 1`

The standard deviation of the sampling distribution of the sample mean score for a random sample of 36 students is 1.