Define a null hypothesis for a population mean mu with known sigma.

Define a null hypothesis for a population mean mu with known sigma.

• A. H0: mu > mu0
• B. H0: mu < mu0
• C. H0: mu ≠ mu0
• D. H0: mu = mu0

Answer: (D)

In this setting, the quantity under test is a precise claim about the population mean. The null hypothesis should state that the true mean equals the value you’re testing against, mu0. This exact statement lets us know exactly how the sample mean behaves under the null: the standard z statistic Z = (X̄ − mu0) / (sigma / sqrt(n)) follows a standard normal distribution when the null is true. That known distribution is what we use to determine p-values and rejection regions. If you used a directional or “not equal” statement for the null, you wouldn’t have a single fixed distribution to reference under the null, which is essential for calibrated testing. Those forms describe possible alternatives rather than the baseline claim you’re testing against. So the correct null is mu equals mu0.

In this setting, the quantity under test is a precise claim about the population mean. The null hypothesis should state that the true mean equals the value you’re testing against, mu0. This exact statement lets us know exactly how the sample mean behaves under the null: the standard z statistic Z = (X̄ − mu0) / (sigma / sqrt(n)) follows a standard normal distribution when the null is true. That known distribution is what we use to determine p-values and rejection regions.

If you used a directional or “not equal” statement for the null, you wouldn’t have a single fixed distribution to reference under the null, which is essential for calibrated testing. Those forms describe possible alternatives rather than the baseline claim you’re testing against. So the correct null is mu equals mu0.