One-Sample Z-Test

It is important to differentiate when a one-sample Z-Test should be used when comparing a sample and a population.

Formula for Z-Score

The z-score formula shows that the z-score is equal to the mean of the sample, minus the mean of the population divided by the standard error of the mean, which is the population standard deviation divided by the square root of the sample size.

The result of this formula gives us a Z-score. A positive value will indicate the area under a curve to the right of the mean, whereas a negative value will indicate the area under a curve to the left of the mean. The percent under the curve is gotten from taking the z-score and looking it up on a Z-table. It is important to make sure that you are looking at the appropriate z-table based on if the z-score is positive or negative. Z-table for Positive Values. Z-table for Negative Values.

What does it mean?

x̅ = mean of the sample

μ = population mean

σ = population standard deviation

n= sample size

Z= z-score

Hypothesis Testing

Z-tests can be used for hypothesis testing. For example, they can be used to help determine if a null hypothesis (H₀) is accepted or rejected and if a research hypothesis(H₁) is true. For example, the null hypothesis could indicate that the sample mean and the population mean are the same, or H_0: x̅ = μ or that the sample mean and population mean are not the same in a research hypothesis H₁: x̅ ≠ μ (Salkind, 2017, 201)

Lets Try It

Below we will go through the step to calculate a Z-Score and report the results in APA format. Excel and manual calculations and directions for using SPSS to calculate the one-sample Z-score will be shown.

The data for these examples was collected among 75 3rd and 4th grade students. They were administered a one-minute typing test to gauge their typing speed. The numbers in the data set are typing speeds in words per minute.

Example: Excel Calculation

Example: SPSS

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How2Stat.net. (n.d.) Statisticians do it better p < .05. Retrieved April 10, 2018. http://www.how2stats.net/2014/03/one-sample-z-test.html

Salkind, N. J. (2017). Statistics for People Who (Think They) Hate Statistics (6th ed.). Thousand Oaks, CA: Sage Publications.