Statistics: What Is a Z-Test?

Z-Test

A z-test is a statistical test used to determine whether two population means are different when the sample size is large (typically larger than 30) and is normally distributed (bell-curve shaped). This is unlike the T-Test that can be used with non-normal distributions.

Nonetheless, t-test and z-tests are extremely similar in their methodology. You basically do the exact same hypothesis test steps but with z-score formulas instead of t-score formulas.

The five z-tests steps are as follows:

1. State the null hypothesis and alternate hypotheses ($H_0, H_A$)

2. Choose an "alpha" $\alpha$-level

3. Find the critical value of z in a z-table

4. Calculate the z-test statistic

5. Compare the z-test statistic to the critical z-value and decide if you should support or reject the null hypothesis

(Video) One Sample Z-Test

by statslectures

In this video by statslectures, you will learn about the one sample z-test.

Z-Test

A z-test is a statistical test used to determine if there is a significant difference between sample and population means or between the means of two samples. It is particularly useful when dealing with large sample sizes. The purpose of a one sample z-test is to compare the sample mean to the known population mean when the population standard deviation is known.

First, we want to state both the null hypotheses and the alternative hypotheses. Remember that the null hypothesis is the hypothesis that there is no effect or no difference. For example, if you're testing a new drug, the null hypothesis might be that the drug has no effect on recovery rates. The alternative hypothesis is the hypothesis that there is an effect or a difference. For the drug example, the alternative hypothesis would be that the drug does have an effect on recovery rates.

To assess whether the average height of the sample differs significantly from the known population mean, we conducted a z-test assuming the population variance is accurately known. If it not, we can use the sample variance to perform a z-test and determine if the sample mean significantly differed from the hypothesized population mean.

Once we've set the hypotheses, we need to choose a level of significance. This is your cutoff for deciding if the result is statistically significant.

Next, we can use the following formula to calculate the z-score.

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

Remember that $x=$ the standard value, $\mu=$ the mean of the sample, and $\sigma=$ the standard deviation of the sample.

After we've found the z-score, we need to determine the critical value. This depends on the level of significance and whether the test is one-tailed or two-tailed. Compare the calculated z-score to this critical value.

Lastly, you can decide to either reject the null hypothesis ($H_0$) or fail to reject the null hypothesis.

Looking Further

While a z-test is used to compare sample means to population means, the chi-square test is employed to assess the relationship between categorical variables, addressing different aspects of statistical analysis.