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Z-test

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

The z-test is a statistical hypothesis test that uses the standard normal distribution to determine if the mean of a population is equal to a specified value. It is commonly used when the population standard deviation is known or when the sample size is large enough to assume a normal distribution.

5 Must Know Facts For Your Next Test

  1. The z-test is used when the population standard deviation is known or when the sample size is large enough (typically n > 30) to assume a normal distribution.
  2. The z-test statistic is calculated as the difference between the sample mean and the hypothesized population mean, divided by the standard error of the mean.
  3. The z-test is used to construct confidence intervals for a population mean, where the confidence level determines the probability that the true population mean falls within the interval.
  4. The null and alternative hypotheses in a z-test are typically formulated to evaluate whether the population mean is equal to, less than, or greater than a specified value.
  5. The z-test is a powerful tool for making inferences about a population mean when the assumptions of normality and known (or large sample) standard deviation are met.

Review Questions

  • Explain how the z-test is used in the context of the Standard Normal Distribution.
    • The z-test utilizes the Standard Normal Distribution to make inferences about a population mean. When the population standard deviation is known or the sample size is large enough, the z-test statistic follows a Standard Normal Distribution under the null hypothesis. This allows the researcher to calculate the probability of observing the sample mean (or a more extreme value) if the null hypothesis is true, and then use this probability to determine whether to reject or fail to reject the null hypothesis.
  • Describe how the z-test is used to construct a Confidence Interval for a population mean.
    • The z-test is used to construct a confidence interval for a population mean by first calculating the z-test statistic, which is the difference between the sample mean and the hypothesized population mean, divided by the standard error of the mean. The z-test statistic is then used to determine the critical value from the Standard Normal Distribution corresponding to the desired confidence level. The confidence interval is then calculated as the sample mean plus or minus the product of the critical value and the standard error of the mean. This provides a range of values that is likely to contain the true population mean.
  • Evaluate how the z-test is used to test Null and Alternative Hypotheses about a population mean.
    • The z-test is a fundamental tool in hypothesis testing for making inferences about a population mean. The null hypothesis typically states that the population mean is equal to a specified value, while the alternative hypothesis can be that the population mean is less than, greater than, or not equal to the specified value. The z-test statistic is calculated and compared to the critical value from the Standard Normal Distribution to determine the p-value, which represents the probability of observing the sample mean (or a more extreme value) if the null hypothesis is true. Based on the p-value and the predetermined significance level, the researcher can then decide whether to reject or fail to reject the null hypothesis, providing evidence to support or refute the claim about the population mean.