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

Answer:

$Z = 1$

Step-by-step 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:

$Z = \frac{X - \mu}{\sigma}$

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 $s = \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 70, \sigma = 12, n = 4, s = \frac{12}{\sqrt{4}} = 6$

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

This is Z when X = 76. So

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{76 - 70}{6}$

$Z = 1$