A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Hypothesis Testing for the Mean of a Normal Distribution with Known Variance — mark scheme explained
The short answer
Hypothesis testing is a fundamental statistical method used to make decisions about population parameters based on sample data. In this context, we will focus on conducting a hypothesis test for the mean of a Normal distribution when the variance is known, given, or assumed.
The question
A factory claims that the average weight of its chocolate bars is 100 grams. A sample of 36 chocolate bars has a mean weight of 98 grams and a known population standard deviation of 5 grams. Test the claim at a significance level of 0.05. [Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
What each mark is really for — in plain English — and the wording trap that loses it.
- S1
State the hypotheses: H 0 : μ = 100, H 1 : μ ≠ 100 (two-tailed test).
- S2
Choose the significance level: α = 0.05.
- S3
Calculate the test statistic: Z = (98 - 100) / (5 / √36) = -2.4.
- S4
Determine the critical values: For a two-tailed test at α = 0.05, the critical values are ±1.96.
- S5
Make a decision: Since -2.4 0 .
- S6
Interpret the results: There is sufficient evidence to conclude that the average weight of the chocolate bars is not 100 grams.
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
State the hypotheses: H 0 : μ = 100, H 1 : μ ≠ 100 (two-tailed test).
- S2
Choose the significance level: α = 0.05.
- S3
Calculate the test statistic: Z = (98 - 100) / (5 / √36) = -2.4.
- S4
Determine the critical values: For a two-tailed test at α = 0.05, the critical values are ±1.96.
- S5
Make a decision: Since -2.4 0 .
- S6
Interpret the results: There is sufficient evidence to conclude that the average weight of the chocolate bars is not 100 grams.
Final answer: Reject H 0
Common mistakes
- Confusing the null hypothesis with the alternative hypothesis. — 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. — 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. — 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. — 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). — 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). — 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). — 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. — 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
- Full worked solution (all marking points)5 marks