4.1 Z-test: known population standard deviation

The z-test is appropriate when you have access to the population standard deviation. This often occurs when you have comprehensive industry reports or extensive historical data. Key characteristics of the z-test include:

• It uses the known population standard deviation in calculations.
• The test statistic follows a standard normal distribution (bell-shaped curve).
• It generally provides more precise results when population parameters are available.

Using the z-statistic formula, this can be explained:

 = sample mean

 = population mean under the null hypothesis

 = population standard deviation

n = sample size

This formula reflects the z-test’s characteristics by directly incorporating the known population standard deviation (σ) and transforming the difference between the sample mean and population mean into a standardised score that follows the standard normal distribution.

It may seem odd to you that in Unit 1, you learned how to calculate the z-score, which is shown in this formula.

Where:

 = raw score

 = population mean

 = population standard deviation

You may question how this relates to the z-statistic we have been discussing. While these concepts are related, they serve different purposes in statistical analysis.

The key difference lies in their focus:

• The z-score focuses on a particular data point within a population. It tells us how many standard deviations an individual value is from the population mean. Imagine you are analysing the performance of a new marketing campaign. A z-score to determine how a specific customer’s spending compares to the average customer spending across all campaigns.

• The z-statistic, on the other hand, focuses on the sample mean in relation to the population parameters. It tells us how many standard errors the sample mean is from the hypothesised population mean. This is crucial for hypothesis testing and making inferences about populations based on sample data. Using the same marketing campaign example, a z-statistic to test whether the average spending in your new campaign (based on a sample) is significantly different from the average spending (population mean).