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
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.
How marks are awarded
Marks are typically awarded for correctly stating hypotheses, calculating the test statistic, finding critical values or p-values, making a decision based on α, and interpreting the results. Ensure each step is clearly shown.
What the command words demand
- State
- Clearly write down the null and alternative hypotheses.
- Determine
- Calculate the test statistic and find the critical values or p-value.
- Test
- Compare the test statistic to the critical values or the p-value to α.
- Conclude
- State whether you reject or fail to reject H 0 , and interpret the result in context.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate about 5-7 minutes per hypothesis testing question to ensure you have enough time to state hypotheses, calculate necessary values, and interpret results accurately.
- State the null hypothesis (H 0 ): p = 0.51 mark
- State the alternative hypothesis (H 1 ): p ≠ 0.51 mark
- Determine the test statistic: X = 14 (number of heads)0 marks
- 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 ≥ 13.2 marks
- Since X = 14 falls in the critical region, we reject H 0 .1 mark
Final answer: Reject H 0
Work through every step correctly and you earn all 5 marks.
Another worked example
A sample of data has a Pearson's correlation coefficient (r) of -0.85. The p-value for this correlation is 0.01, and the significance level α = 0.05. Interpret the results.
- State the null hypothesis (H 0 ): There is no linear relationship between the variables.1 mark
- State the alternative hypothesis (H 1 ): There is a linear relationship between the variables.0 marks
- The p-value (0.01) is less than α (0.05), so we reject H 0 .1 mark
- We conclude that there is a significant negative linear relationship between the variables.2 marks
Final answer: Reject H 0 ; significant negative linear relationship
Work through every step correctly and you earn all 4 marks.
Common mistakes
Confusing the null hypothesis with the alternative hypothesis.
Why it happens: Students sometimes mix up which hypothesis is being tested and which one they are trying to reject.
Fix: 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).
Why it happens: Students might use a different α value than what is given in the problem or forget to specify it.
Fix: Always check the problem for the specified α value and clearly state it at the beginning of your solution.
Misinterpreting the p-value.
Why it happens: Students might think that a high p-value means they should reject H 0 , or vice versa.
Fix: 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.
Why it happens: Students might use the critical values for a 2-tail test when they should be using those for a 1-tail test, or vice versa.
Fix: 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.
Why it happens: Students might forget to interpret their results in the context of the problem.
Fix: 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.
Why it happens: Students might interpret a significant correlation as evidence of causation.
Fix: 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.
Why it happens: Students might use an incorrect formula or method to calculate the test statistic.
Fix: 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.
Why it happens: Students might perform a hypothesis test without verifying that the necessary conditions are met (e.g., independence of observations, normality).
Fix: 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
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 type | What you’re asked to do | Marks |
|---|---|---|
| Hypothesis Test | Test whether a coin is fair using a binomial two-tail hypothesis test at 5%. | 5 |
| Interpret Correlation Test | Interpret a correlation coefficient and p-value to reach and explain a hypothesis test conclusion. | 4 |
| Hypothesis Test Proportion | Test a claim about a population proportion using a binomial hypothesis test at 1%. | 5 |
| Interpret Correlation Significance | Compare the p-value to alpha, decide on significance, and interpret the correlation in context. | 4 |
| Total across these question types | 18 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.