Conduct a statistical hypothesis test for the mean of a Normal…

The idea

When a variable X is modelled as Normal with unknown mean μ but variance known, given or assumed, a sample mean x̄ from n independent observations is also Normal with mean μ and variance σ²/n, so its standard error is σ/√n. A hypothesis test sets H₀: μ = μ₀ against a one- or two-tailed H₁, then standardises: z = (x̄ − μ₀)/(σ/√n). For a two-tailed test at significance level α, reject H₀ if |z| exceeds the critical value z_(α/2); for a one-tailed test compare z with z_α in the correct direction. Equivalently compare x̄ with a critical region. The conclusion must refer to the original context: state whether there is sufficient evidence at the α level to reject H₀, and phrase the finding in terms of the real quantity measured, not merely 'accept H₀'.

What earns the marks

• State H₀ and H₁ clearly, choosing one- or two-tailed form from the wording of the problem and defining μ in context.
• Identify n, the sample mean x̄, the hypothesised mean μ₀ and the known, given or assumed standard deviation σ, then compute the standard error σ/√n.
• Calculate the test statistic z = (x̄ − μ₀)/(σ/√n), or find the critical region for x̄, showing the substitution accurately.
• Compare the test statistic with the appropriate critical value, or compare the sample mean with the critical region, and state the decision about H₀.
• Interpret the decision in the context of the question, referring to evidence about the population mean at the stated significance level.

Common mistakes and how to put them right

• Using the sample standard deviation in place of the known, given or assumed population standard deviation when computing the standard error.

  Put it right: The correct divisor is σ/√n with the stated σ.

• Dividing by σ rather than σ/√n, which ignores sample size.

  Put it right: The corrected standard error is σ/√n.

• Choosing a two-tailed critical value for a one-tailed test, or comparing against the wrong tail.

  Put it right: The corrected approach selects the critical value matching the direction of H₁.

Exam tips

• Write down H₀, H₁ and the significance level before any calculation so the tail direction is fixed.
• Show the standard error and the substitution into z explicitly; method marks are available for correct structure even if arithmetic slips.
• Finish with a sentence that names the context variable and states the strength of evidence at the given significance level.