Question

A sample from a population with μ = 40 and σ = 8 has a mean of M = 36. If the sample mean corresponds to a z = –1.00, then how many scores are in the sample?

Answer 1

Answer:

There are 4 scores in the sample.

Step-by-step 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 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 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 $s = \frac{\sigma}{\sqrt{n}}$.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this question, we have that:

$\mu = 40, \sigma = 8, X = 36, Z = -1, s = \frac{8}{\sqrt{n}}$

We want to find n. So

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

By the Central Limit Theorem

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

$-1 = \frac{36 - 40}{\frac{8}{\sqrt{n}}}$

$-4\sqrt{n} = -8$

$4\sqrt{n} = 8$

Simplifying by 4

$\sqrt{n} = 2$

$(\sqrt{n})^2 = 2^2$

$n = 4$

There are 4 scores in the sample.