Z-test

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 in the context of evaluating null and alternative hypotheses, determining the appropriate probability distribution for hypothesis testing, and conducting hypothesis testing for a single mean or proportion.

5 Must Know Facts For Your Next Test

1. The z-test is used when the population standard deviation is known, and the sample size is large (typically greater than 30).
2. The z-test statistic is calculated by subtracting the hypothesized population mean from the sample mean and dividing the result by the standard error of the mean.
3. The z-test is used to determine the probability of obtaining the observed sample mean or a more extreme value under the assumption that the null hypothesis is true.
4. The z-test is a two-tailed test, meaning that the test statistic can fall in either the positive or negative direction, depending on the alternative hypothesis.
5. The z-test is commonly used in hypothesis testing for a single mean or a single proportion, where the population standard deviation is known.

Review Questions

• Explain how the z-test is used in the context of null and alternative hypotheses.
  • The z-test is used to determine whether to reject or fail to reject the null hypothesis based on the sample data. The null hypothesis typically states that the population mean is equal to a specified value, while the alternative hypothesis states that the population mean is not equal to the specified value (two-tailed test), greater than the specified value (one-tailed test), or less than the specified value (one-tailed test). The z-test statistic is calculated and compared to the critical value from the standard normal distribution to determine the p-value and make a decision about the null hypothesis.
• Describe the probability distribution needed for the z-test in the context of hypothesis testing.
  • The z-test uses the standard normal distribution, which has a mean of 0 and a standard deviation of 1, as the probability distribution for hypothesis testing. This is because the z-test statistic is calculated by standardizing the sample mean using the known population standard deviation. The standardized z-score follows the standard normal distribution, allowing for the calculation of the p-value and the comparison to the critical value to determine the significance of the test.
• Explain how the z-test is used in hypothesis testing for a single mean and a single proportion.
  • The z-test is commonly used in hypothesis testing for a single mean and a single proportion when the population standard deviation is known. For a single mean, the z-test is used to determine if the sample mean is significantly different from a hypothesized population mean. For a single proportion, the z-test is used to determine if the sample proportion is significantly different from a hypothesized population proportion. In both cases, the z-test statistic is calculated and compared to the critical value from the standard normal distribution to make a decision about the null hypothesis.

Related terms

Null Hypothesis: The null hypothesis is a statistical hypothesis that states that there is no significant difference between a population parameter and a hypothesized value or that there is no significant difference between two population parameters.

Alternative Hypothesis: The alternative hypothesis is a statistical hypothesis that states that there is a significant difference between a population parameter and a hypothesized value or that there is a significant difference between two population parameters.

Standard Normal Distribution: The standard normal distribution is a probability distribution that has a mean of 0 and a standard deviation of 1. It is used to standardize data and determine the probability of a value occurring within a certain range.