When is a z-test appropriate for testing a population mean?

When is a z-test appropriate for testing a population mean?

• A. The population standard deviation is unknown and the sample is small.
• B. The population standard deviation is known and the population is normal or the sample size is large.
• C. The sample standard deviation is known and the population is non-normal.
• D. The data are categorical.

Answer: (B)

Z-testing a population mean relies on knowing the population standard deviation and on the sampling distribution of the sample mean being normal. When sigma is known, the test statistic uses sigma in the denominator: Z = (X̄ − μ0) / (σ/√n). This follows a standard normal distribution under the null hypothesis if the population is normal or if the sample size is large enough for the Central Limit Theorem to make the sampling distribution of the mean approximately normal. That combination—known sigma plus a normal/population-wide normality or a large n—is what makes a z-test appropriate. If sigma is unknown, you’d use a t-test with the sample standard deviation and the t distribution, and with categorical data a z-test for a mean isn’t applicable.

Z-testing a population mean relies on knowing the population standard deviation and on the sampling distribution of the sample mean being normal. When sigma is known, the test statistic uses sigma in the denominator: Z = (X̄ − μ0) / (σ/√n). This follows a standard normal distribution under the null hypothesis if the population is normal or if the sample size is large enough for the Central Limit Theorem to make the sampling distribution of the mean approximately normal. That combination—known sigma plus a normal/population-wide normality or a large n—is what makes a z-test appropriate. If sigma is unknown, you’d use a t-test with the sample standard deviation and the t distribution, and with categorical data a z-test for a mean isn’t applicable.