AQA A-Level Mathematics: Hypothesis Testing for the Mean of a Normal Distribution with Known Variance — mark scheme explained

The question

Mark scheme, decoded

How the examiner actually awards the marks on this topic.

Gradora's own decode of the marking approach — not the exam board's published mark scheme.

Model answer

A full-mark response to the question above, worked through step by step.

⏱Timing: Allocate about 5-7 minutes to complete this type of question in an exam setting. This allows time for careful calculation and clear interpretation.

1. State the hypotheses: H 0 : μ = 100, H 1 : μ ≠ 100 (two-tailed test).1 mark
2. Choose the significance level: α = 0.05.0 marks
3. Calculate the test statistic: Z = (98 - 100) / (5 / √36) = -2.4.2 marks
4. Determine the critical values: For a two-tailed test at α = 0.05, the critical values are ±1.96.1 mark
5. Make a decision: Since -2.4 0 .1 mark
6. Interpret the results: There is sufficient evidence to conclude that the average weight of the chocolate bars is not 100 grams.0 marks

Final answer: Reject H 0

✓Work through every step correctly and you earn all 5 marks.

Common mistakes

• Confusing the null hypothesis with the alternative hypothesis.

  Why it happens: Students sometimes mix up which hypothesis represents no effect and which one represents the claim being tested. It's important to clearly state H 0 as the statement of no difference or no effect, and H 1 as the claim being tested.

  Fix: Always double-check that H 0 is correctly stated as μ = μ 0 and H 1 reflects the direction of the test (μ ≠ μ 0 , μ > μ 0 , or μ 0 ).

• Using the wrong formula for the test statistic.

  Why it happens: Students might use the t-test formula instead of the Z-test formula when the population standard deviation is known. The correct formula for a Z-test is Z = (X̄ - μ 0 ) / (σ / √n).

  Fix: Ensure that you use the Z-test formula when the population standard deviation (σ) is known, given, or assumed.

• Incorrectly identifying the critical value for a one-tailed test.

  Why it happens: Students might use the critical value for a two-tailed test when they should be using the critical value for a one-tailed test. The critical values are different for one-tailed and two-tailed tests.

  Fix: For a one-tailed test, use the appropriate critical value from the Standard Normal distribution table based on the direction of the alternative hypothesis (upper or lower tail).

• Failing to interpret the results in context.

  Why it happens: Students might correctly perform the hypothesis test but fail to explain what their decision means in terms of the population parameter and the practical implications of the problem.

  Fix: Always provide a clear interpretation of your decision, explaining what it means for the population mean and the practical context of the problem.

• Using the wrong significance level (α).

  Why it happens: Students might use a different significance level than the one specified in the problem. The significance level should be clearly stated and used consistently throughout the test.

  Fix: Always check the problem statement for the given significance level and use it to determine the critical value or p-value threshold.

• Incorrectly calculating the sample mean (X̄) or standard deviation (σ).

  Why it happens: Students might make arithmetic errors when calculating the sample mean or using the wrong value for the population standard deviation.

  Fix: Double-check your calculations for X̄ and ensure you are using the correct value for σ. If the problem provides a known or assumed σ, use that value in the test statistic formula.

• Confusing the p-value with the significance level (α).

  Why it happens: Students might think that the p-value is the same as the significance level. The p-value is the probability of observing a test statistic as extreme or more extreme than the one calculated, assuming H 0 is true.

  Fix: Understand that the p-value is compared to α to make a decision. If the p-value is less than α, reject H 0 ; otherwise, fail to reject H 0 .

• Failing to state the conclusion clearly.

  Why it happens: Students might make a decision but not clearly state whether they are rejecting or failing to reject the null hypothesis. A clear and concise conclusion is essential for full marks.

  Fix: Always state your decision clearly: 'Reject H 0 ' or 'Fail to reject H 0 '. Follow this with an interpretation in context.

Where the marks go

The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.

Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.

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