A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Statistical Hypothesis Testing and Correlation Coefficients — mark scheme explained
The short answer
Statistical hypothesis testing is a fundamental tool in statistics used to make decisions about population parameters based on sample data. This section covers the language and application of statistical hypothesis testing using the binomial model, as well as the interpretation of correlation coefficients.
The question
A coin is flipped 20 times and lands heads up 14 times. Test the hypothesis that the coin is fair (p = 0.5) at the 5% significance level. [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 null hypothesis (H 0 ): p = 0.5
- S2
State the alternative hypothesis (H 1 ): p ≠ 0.5
- S3
Determine the test statistic: X = 14 (number of heads)
- S4
Find the critical values for a 2-tail test with n = 20 and p = 0.5 using binomial tables or software. The critical region is X ≤ 7 or X ≥ 13.
- S5
Since X = 14 falls in the critical region, we reject H 0 .
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
State the null hypothesis (H 0 ): p = 0.5
- S2
State the alternative hypothesis (H 1 ): p ≠ 0.5
- S3
Determine the test statistic: X = 14 (number of heads)
- S4
Find the critical values for a 2-tail test with n = 20 and p = 0.5 using binomial tables or software. The critical region is X ≤ 7 or X ≥ 13.
- S5
Since X = 14 falls in the critical region, we reject H 0 .
Final answer: Reject H 0
Common mistakes
- Confusing the null hypothesis with the alternative hypothesis. — Always clearly state H 0 and H 1 at the beginning of the test, and remember that H 0 is the statement you assume to be true unless evidence suggests otherwise.
- Using the wrong significance level (α). — Always check the problem for the specified α value and clearly state it at the beginning of your solution.
- Misinterpreting the p-value. — Remember that if the p-value is less than or equal to α, you reject H 0 . If it is greater than α, you fail to reject H 0 .
- Using the wrong critical values for a 1-tail test or 2-tail test. — Always check whether the alternative hypothesis specifies a direction (1-tail) or not (2-tail), and use the appropriate critical values.
- Failing to state the conclusion in context. — Always conclude by stating whether you reject or fail to reject H 0 , and explain what this means in the context of the problem (e.g., 'There is a significant linear relationship between the variables').
- Confusing correlation with causation. — Remember that correlation does not imply causation. A significant correlation only indicates a relationship, not a cause-and-effect link.
- Using the wrong formula for the test statistic. — Always double-check the formula and method you are using to ensure they are appropriate for the given problem (e.g., binomial test, t-test).
- Failing to check assumptions before performing a hypothesis test. — Always verify the assumptions required for the specific hypothesis test you are using. If an assumption is not met, consider alternative methods or transformations.
Where the marks go
- Full worked solution (all marking points)5 marks