Illustrative example on the use of the Z score table

Example:

A random sample of size 100 is taken from a normal population with σ = 25. What is the probability that the mean of the sample will differ from the mean of the population by 3 or more either way?

Solution:

Step 1: Introduction

Suppose a random variable X follows the normal distribution with mean µ and standard deviation σ. Then the sample mean (M) also has a normal distribution with the same mean, µM= µ, and the standard deviation is as follows,

Symbolically, we can write this as follows.

Moreover when we scale a normal random variable by subtracting mean and dividing by its standard deviation. Then it becomes a standard normal random variable with the mean 0 and standard deviation of 1. Symbolically, we can write this as follows.

We have the sample size (n) = 100, and the standard deviation (σ) = 25. The probability that the mean of the sample (M) will differ from the mean of the population (µ) by 3 or more, either way, is equals, P( | M-µ | ≥ 3).

Step 2: Calculations

Here Z follows the standard normal distribution. To find these Z probabilities we will use the Z score table as follows:

Z table gives left-tailed probabilities for a given Z score. For example, let Z0 be the Z-score then the Z table gives the probability of kind, P(Z < Z0).

This gives, P( | M-µ | ≥ 3) = 2*P( Z ≤ -1.2) = 2*0.1151 = 0.2302.

The probability that the sample mean (M) will differ from the mean of the population (µ) by 3 or more, either way, is 0.2351.

To know how to use the Z score table in detail refer to the full article on “How to use Z score tables?”