Statistics: Hypothesis Testing
Hypothesis testing is a formal statistical method for using sample data to decide whether there is enough evidence to reject a claim (the null hypothesis) about a population parameter in favour of an alternative hypothesis. A Level Statistics covers binomial hypothesis tests for a proportion, finding critical regions, one-tailed and two-tailed tests, and testing the product moment correlation coefficient.
Before you start
Make sure you're comfortable with these topics first:
Method
- State the null hypothesis H0 (the claimed value of the parameter, e.g. p = 0.3) and the alternative hypothesis H1 (p > 0.3 for a one-tailed test, or p not equal to 0.3 for a two-tailed test).
- Identify the test statistic and its distribution under H0, e.g. X ~ B(n, p0) using the value of p claimed in H0.
- For a given significance level, find the critical region: the most extreme values of X whose total probability is as close as possible to (but not exceeding) the significance level, working from the relevant tail(s).
- Compare the observed value of X with the critical region: if it lies inside the critical region, reject H0; if not, there is insufficient evidence to reject H0.
- Always write the conclusion in context, referring back to the original claim, and state whether the result is significant at the given level.
- For a two-tailed test, split the significance level equally between both tails (e.g. use 2.5% at each end for a 5% two-tailed test) and find both critical regions.
Worked example
A machine is claimed to produce faulty components with probability 0.15. After a repair, the manager suspects the true probability has decreased. A random sample of 20 components is tested, and none are found to be faulty. Test, at the 5% significance level, whether there is evidence that the probability of a faulty component has decreased.
- State the hypotheses: H0: p = 0.15, H1: p < 0.15 (a one-tailed test, since the manager suspects a decrease).
- Under H0, X ~ B(20, 0.15). Find P(X = 0) = 0.85^20 = 0.0388 (4dp).
- Since this is a one-tailed lower-tail test, compare P(X <= 0) = P(X = 0) = 0.0388 with the significance level 0.05.
- 0.0388 < 0.05, so the observed value X = 0 lies in the critical region, and H0 is rejected.
Practice questions
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Q1State what is meant by the significance level of a hypothesis test.Show answer
Answer: The probability of incorrectly rejecting H0 when H0 is actually true.
Q2For a two-tailed test at the 10% significance level, state the significance level used in each tail.Show answer
Answer: 5% (10% divided equally between the two tails).
Q3A test uses H0: p = 0.2 against H1: p > 0.2. State what type of error is made if H0 is rejected when p is actually equal to 0.2.Show answer
Answer: A Type I error.
Q4Under H0, X ~ B(15, 0.3). Given that P(X >= 8) = 0.0500 (awrt) and P(X >= 7) = 0.1181 (awrt), state the critical region for a one-tailed test at the 5% significance level.Show answer
Answer: X >= 8 (the smallest upper-tail region with probability not exceeding 0.05).
Q5A gardener claims a new fertiliser has changed the proportion of seeds, historically 0.25, that fail to germinate. A random sample of 30 seeds is treated, and 2 fail to germinate. Given X ~ B(30, 0.25) under H0 and P(X <= 2) = 0.0106, test at the 5% significance level whether there is evidence of a change (two-tailed).Show answer
Answer: Compare 0.0106 with 0.025 (half of 5%): since 0.0106 < 0.025, reject H0; there is evidence the proportion has changed.
Q6A test statistic gives a p-value of 0.038. State, with a reason, the conclusion of the test at the 1% significance level and at the 5% significance level.Show answer
Answer: At 1%: do not reject H0, since 0.038 > 0.01. At 5%: reject H0, since 0.038 < 0.05.
Exam-style questions
Written in the style of a A Level Maths exam paper, with a full mark scheme.
A machine is set to produce ball bearings with a historic defect rate of 0.12. Following maintenance, the plant manager, Owen, suspects the defect rate has decreased. He tests a random sample of 25 ball bearings and finds that 1 is defective. Let X be the number of defective ball bearings, out of 25. (a) State suitable null and alternative hypotheses for Owen's test. (2) (b) Assuming X ~ B(25, 0.12), find P(X <= 1). (3)
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A call centre historically finds that 18% of outbound calls result in a sale. Following a new script, the manager wants to test whether the proportion has changed (either increased or decreased). A random sample of 30 calls is monitored. Let X be the number of calls, out of 30, that result in a sale. (a) State suitable hypotheses for a two-tailed test. (2) (b) Given that, under H0, P(X <= 1) = 0.0197 and P(X <= 2) = 0.0741 (both awrt), find the lower critical region for a two-tailed test at the 10% significance level. (2) (c) State the actual significance level contributed by the lower tail. (1) (d) In the sample, 8 of the 30 calls resulted in a sale. Given that the upper critical region is X >= 10, state, with a reason, the conclusion of the test. (1)
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For 10 randomly selected towns, the annual rainfall (x, mm) and average crop yield (y, tonnes per hectare) are recorded. The product moment correlation coefficient is calculated to be r = 0.712. (a) State suitable hypotheses to test, at the 5% significance level, whether there is positive correlation between rainfall and crop yield. (2) (b) Given that the critical value for a one-tailed test at the 5% level, with n = 10, is 0.5494, determine whether there is significant evidence of positive correlation. Justify your answer. (2)
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Free printable worksheet
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