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The language and logic of hypothesis testing: null and alternative hypotheses, significance level, test statistic, one-tailed tests, critical value, critical region, acceptance region, p-value and actual significance level. Every hypothesis test on Paper 2 uses these ideas, and some parts ask you to define or explain them.
Test statistic: the value from the sample used to make the decision
Critical region: the values of the test statistic that lead to rejecting \(H_0\)
Critical value: the boundary (first value) of the critical region
p-value: probability of the observed value or a more extreme one, if \(H_0\) is true
Reject \(H_0\) if the p-value is less than the significance level, or if the test statistic is in the critical region
Actual significance level \(= P(\text{in the critical region} \mid H_0 \text{ true})\)
Notes
Hypotheses
A hypothesis test uses a sample to test a claim about a population parameter: a proportion \(p\), a correlation coefficient \(\rho\) or a mean \(\mu\)
The null hypothesis \(H_0\) gives the parameter a single value, e.g. \(H_0: p = 0.3\). It is assumed true unless there is enough evidence against it.
The alternative hypothesis \(H_1\) is what you suspect instead: \(p \gt 0.3\) or \(p \lt 0.3\) (one-tailed), or \(p \ne 0.3\) (two-tailed).
Hypotheses are always about the population parameter, never about the sample result.
The key terms
diagram
The test statistic is the value from the sample used in the test, e.g. the number of successes \(X\)
The significance level (e.g. 5%) sets how unlikely a result must be before \(H_0\) is rejected. It is also the probability of rejecting \(H_0\) when \(H_0\) is actually true (for a binomial test the true probability is at most this; see below).
The critical region is the set of values of the test statistic that lead to rejecting \(H_0\). The critical value is the first value in it (its boundary). The acceptance region is the rest.
H0: p = 0.25, X ~ B(30, 0.25). Reject H0 only if X lands in the red tail
The p-value is the probability, if \(H_0\) is true, of getting the observed value or a more extreme one. If the p-value is less than the significance level, reject \(H_0\)
Actual significance level
With a discrete test statistic, the critical region usually cannot have a probability exactly equal to the significance level. The actual significance level is the probability of being in the critical region when \(H_0\) is true.
Conclusions
State the decision (reject or do not reject \(H_0\)), then a conclusion in context that does not claim certainty, e.g. ‘there is sufficient evidence to suggest that the proportion has increased’.
Not rejecting \(H_0\) does not prove that it is true; there is just not enough evidence against it.
How to answer each type of question
M = method mark, A = answer mark, B = a mark on its own.
Write suitable hypotheses
1 to 2 marksGrade E
Identify the parameter and define it in words.
\(H_0\): the parameter equals the claimed value.
\(H_1\): choose \(\gt\), \(\lt\) or \(\ne\) from the wording of the suspicion.
Example. A coin is thought to be biased towards heads. It is to be tossed 20 times to test this. Write down suitable hypotheses, defining your parameter.
Show the model answerHide the model answer
\(H_0: p = 0.5\) (B1),\(H_1: p \gt 0.5\) (B1), where \(p\) is the probability that the coin lands heads
Critical region, actual significance level and what they mean
3 to 4 marksGrade B
Define terms in words, linked to the test statistic.
Actual significance level: find the probability of the critical region using the distribution under \(H_0\)
If asked why it is not exactly the significance level, refer to the test statistic being discrete.
Example. A test of \(H_0: p = 0.25\) against \(H_1: p \gt 0.25\) uses \(X \sim B(30, p)\). At the 5% level the critical region is \(X \ge 13\). (a) Explain what is meant by the critical region. (b) Find the actual significance level of the test. (c) Explain why the actual significance level is not 5%.
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(a) The set of values of \(X\) for which \(H_0\) would be rejected
(c) \(X\) is discrete, so the tail probabilities go up in steps: \(P(X \ge 12) = 0.0507\) is more than 5%, so the largest allowed critical region has probability 0.0216
(B1)
Don’t lose marks
Writing hypotheses about the sample (e.g. \(x\) or \(r\)) instead of the parameter.
Writing conclusions that claim certainty, e.g. ‘this proves the coin is biased’.
Confusing the critical value (a value of the test statistic) with a probability.
More tips
Memory tricks
\(H_0\) always has an equals sign, and both hypotheses use the parameter (\(p\), \(\mu\) or \(\rho\)).
Conclusion template: ‘There is (in)sufficient evidence at the 5% level to suggest that …’, finished in the words of the question.
Exam technique
Define the parameter in words, e.g. ‘\(p\) is the proportion of customers who pay with cash’.
Always end with a conclusion in context that refers back to the claim in the question.
If asked for the actual significance level, give the probability of the critical region, not the stated 5%.
What each grade needs
What you need to be able to do, from the first marks up to the top grade.
Grade E
Write null and alternative hypotheses\(H_0\) gives the parameter a single value; \(H_1\) says it is greater, less or different.
Grade D
Explain what the significance level meansIt sets how unlikely a result must be to reject \(H_0\), and is the probability of rejecting \(H_0\) when it is true.
Grade C
Define critical region and critical valueThe critical region is the set of values that lead to rejecting \(H_0\); the critical value is its boundary.
Grade C
Use a p-value to reach a decisionReject \(H_0\) if the p-value is less than the significance level.
Grade B
Find the actual significance level of a testCalculate the probability that the test statistic falls in the critical region when \(H_0\) is true.
Grade A
Interpret test results and their limitationsExplain that not rejecting \(H_0\) does not prove it, and why the actual and stated significance levels differ.
Quick recall
Cover the answers and test yourself. The app has these as flashcards that come back just before you'd forget them.
State, with a reason, the conclusion the student should draw about the null hypothesis.
Reject \(H_0\), because \(0.03 \lt 0.05\).
Sample questions
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy4 marks
A student carries out a hypothesis test at the 5% level of significance and obtains a \(p\)-value of 0.03.
(a) State, with a reason, the conclusion the student should draw about the null hypothesis.[1]
(b) The student writes: “This means there is a 3% chance that the null hypothesis is true.” Explain why this interpretation is not correct.[1]
(c) Explain what is meant by the critical region of a hypothesis test.[1]
(d) Explain why the significance level should be chosen before the data are collected.[1]
Show the answer and mark scheme
(a)Answer: Reject \(H_0\), because \(0.03 \lt 0.05\).
B1 for reject \(H_0\) since \(0.03 \lt 0.05\) (the result is significant)
Worked solution: The \(p\)-value 0.03 is less than the significance level 0.05, so the result is significant and \(H_0\) is rejected.
(b)Answer: The \(p\)-value is the probability, assuming \(H_0\) is true, of a result at least as extreme as the one observed; it is not the probability that \(H_0\) is true.
B1 for the \(p\)-value is calculated assuming \(H_0\) is true: it is the probability of a result as extreme as (or more extreme than) the one observed, not the probability that \(H_0\) is true
Worked solution: The \(p\)-value is worked out by assuming \(H_0\) is true. It is the probability of getting a result at least as extreme as the observed one if \(H_0\) is true, not the probability that \(H_0\) itself is true.
(c)Answer: The set of values of the test statistic for which the null hypothesis would be rejected.
B1 for the range (set) of values of the test statistic that would lead to rejecting \(H_0\)
Worked solution: The critical region is the set of values of the test statistic that are so unlikely under \(H_0\) (at the chosen significance level) that, if the observed value falls in it, \(H_0\) is rejected.
(d)Answer: So that the level cannot be chosen, after seeing the data, to give the result the tester wants.
B1 for to avoid bias: choosing the level after seeing the result could make any desired conclusion possible
Worked solution: If the significance level were chosen after seeing the data, it could be picked to give whichever conclusion the tester preferred. Fixing it in advance keeps the test fair.
Question 2Medium3 marks
The time taken by a commuter to travel to work is modelled by a normal distribution with mean \(\mu\) minutes. It is claimed that \(\mu = 35\). After roadworks begin, the commuter believes that the journey takes longer on average. A random sample is to be used to test this.
(a) Write down suitable hypotheses.[2]
(b) Explain why the hypotheses are written in terms of \(\mu\) rather than the sample mean.[1]
Show the answer and mark scheme
(a)Answer: \(H_0: \mu = 35, \ H_1: \mu \gt 35\)
B1 for \(H_0: \mu = 35\)
B1 for \(H_1: \mu \gt 35\)
(b)Answer: The test uses the sample to make an inference about the population: \(\mu\) is the population mean, whereas the sample mean is known exactly once the sample is taken.
B1 for the hypotheses being about the population parameter, with the sample used to make an inference about it
Question 3Hard4 marks
Last year, 20% of the emails received by a company were spam. The IT manager wants to find out whether the proportion of emails that are spam has changed. A random sample of 24 emails is taken, and \(X\) is the number that are spam. A test of \(H_0: p = 0.2, \ H_1: p \ne 0.2\) at the 5% level has critical region \(X = 0 \text{ or } X \ge 10\).
(a) Find the probability of incorrectly rejecting \(H_0\) using this test.[3]
(b) Explain why this probability is less than 0.05.[1]
Show the answer and mark scheme
(a)Answer: \(0.0047 + 0.0126 = 0.0173\)
M1 for \(P(X = 0)\) or \(P(X \ge 10)\)
M1 for adding the two tail probabilities
A1 for awrt 0.0173
(b)Answer: \(X\) is discrete, so each tail can only take certain probabilities; each tail is chosen to be as large as possible without exceeding 0.025, so the total is less than 0.05.
B1 for the discreteness of the binomial distribution, with each tail not exceeding 0.025