Question

If a sample of n = 4 scores is obtained from a population with μ = 70 and σ = 12, then what is the z-score corresponding to a sample mean of M = 76?​ Group of answer choices

Answer 1

`Z = 1`

Explanation:

To solve this question, we have to know what a z-score is and the central limit theorem.

Z-score:

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 = 70, \sigma = 12, n = 4, s = (12)/(√(4)) = 6`

What is the z-score corresponding to a sample mean of M = 76?

This is Z when X = 76. So

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

By the Central Limit Theorem

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

`Z = (76 - 70)/(6)`

`Z = 1`