Z-tests

Definition

A z-test is a statistical hypothesis test that uses the standard normal distribution to determine if the mean of a population is significantly different from a hypothesized value. It is commonly used when the sample size is large and the population standard deviation is known or can be estimated.

5 Must Know Facts For Your Next Test

1. The z-test is used when the population standard deviation is known or can be estimated, and the sample size is large (typically greater than 30).
2. The z-statistic is calculated as the difference between the sample mean and the hypothesized population mean, divided by the standard error of the sample mean.
3. The z-test is used to test hypotheses about the population mean, such as whether the mean is equal to a specified value, greater than a specified value, or less than a specified value.
4. The z-test can be used to compare the means of two independent samples, as long as the population standard deviations are known or can be estimated.
5. The p-value obtained from a z-test is used to determine the statistical significance of the results and make a decision about the null hypothesis.

Review Questions

• Explain the purpose of a z-test and when it is appropriate to use it.
  • The purpose of a z-test is to determine if the mean of a population is significantly different from a hypothesized value. It is appropriate to use a z-test when the sample size is large (typically greater than 30) and the population standard deviation is known or can be estimated. The z-test uses the standard normal distribution to standardize the sample mean and compare it to the hypothesized population mean, allowing researchers to make inferences about the population parameter.
• Describe the steps involved in conducting a z-test and interpreting the results.
  • To conduct a z-test, the researcher first states the null and alternative hypotheses, which typically involve the population mean. They then calculate the z-statistic, which is the difference between the sample mean and the hypothesized population mean, divided by the standard error of the sample mean. The z-statistic is then compared to the critical value from the standard normal distribution, or the p-value is calculated, to determine if the null hypothesis should be rejected or retained. The interpretation of the results depends on the specific hypotheses being tested and the level of statistical significance chosen by the researcher.
• Analyze the relationship between z-tests and the distribution needed for hypothesis testing, as described in Section 9.3.
  • $$\text{The z-test is a type of hypothesis test that requires the use of the standard normal distribution, as described in Section 9.3 of the course material. Specifically, the standard normal distribution is needed to standardize the sample mean and calculate the z-statistic, which is then used to determine the statistical significance of the results. The standard normal distribution is appropriate for z-tests because it has a mean of 0 and a standard deviation of 1, which allows the researcher to compare the sample mean to the hypothesized population mean. The choice of using the z-test versus other hypothesis tests, such as the t-test, depends on the known or estimated value of the population standard deviation and the sample size, as outlined in Section 9.3.}$$

Related terms

Hypothesis Testing:

The process of using statistical analysis to determine whether a hypothesis about a parameter of a population is likely to be true or false.

Standard Normal Distribution:

A probability distribution that is normal with a mean of 0 and a standard deviation of 1, which is used to standardize data and perform z-tests.

Sample Mean:

The average of a set of observations drawn from a population, used to estimate the population mean in a z-test.