Ch 5 · 6.5 Hypothesis Tests · A Level Mathematics 9709
Topic 5 of 5 in Probability & Statistics 2 — A Level Mathematics 9709 — 71 lessons.
Introduction
A company claims its new drug works better; a shopkeeper suspects a coin is biased; a council asks whether a speed camera reduced accidents. A hypothesis test decides, using a sample, whether the evidence is strong enough to reject the "nothing has changed" position. Every conclusion must be stated in context and without overclaiming. (Syllabus section 6.5, Paper 6.)
Key ideas — The language of testing
Conclusions: "there is sufficient evidence at the 5% level that …" or "there is insufficient evidence …". Never say H_0 is proved true.
🧑🏫 Teacher's explanation
Imagine you are a judge in a courtroom. The defendant is assumed innocent until proven guilty. In statistics, we assume the 'status quo' is true—this is the Null Hypothesis (H_0). We only change our mind if the evidence is so overwhelming that it is extremely unlikely to have happened by pure chance.
📖 Definitions
🔍 Worked example — Standard
A manufacturer claims that the mean weight of cereal boxes is 500g with a known standard deviation of σ = 10g. A sample of 25 boxes is taken. Test the claim that the mean weight is different from 500g at a 5% significance level. Use the normal approximation.
🔍 Worked example — Exam-style
In a large population, the proportion of defective components is p = 0.1. A sample of 100 components is tested. Test the claim that the proportion is greater than 0.1 at a 5% significance level. Use a normal approximation.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
(a) State the null and alternative hypotheses. [2] (b) Calculate the z-score for this sample. [1] (c) Test the researcher's claim at the 5% significance level. State your conclusion clearly. [2] [5 marks]
Key ideas — Testing a binomial or Poisson observation
For large n (or λ), use the normal approximation with a continuity correction.
🧑🏫 Teacher's explanation
Imagine you are a judge in a courtroom. The defendant is assumed innocent (H_0) until proven guilty beyond a reasonable doubt. A hypothesis test is a mathematical way of deciding if a piece of evidence is so unusual that we should stop believing the defendant is innocent and instead accept the alternative theory (H_1).
📖 Definitions
🔍 Worked example — Easy
A factory claims that 5\% of its lightbulbs are defective. A sample of 20 bulbs is tested. Test whether there is evidence at the 5\% significance level that the proportion of defective bulbs is greater than 5\%.
🔍 Worked example — Standard
The number of emails received per hour follows a Poisson distribution with mean λ = 4. A researcher observes 8 emails in an hour. Test at the 5\% significance level whether the mean number of emails is greater than 4.
🔍 Worked example — Exam-style
A Poisson distribution has mean λ = 50. Test at the 5\% significance level whether the mean is different from 50, given that 62 events were observed in a period. Use a normal approximation.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Critical regions for discrete distributions
Find the largest tail with probability no more than the significance level. Because the distribution is discrete, the actual significance level (the true probability of the critical region) is usually less than the nominal level.
🧑🏫 Teacher's explanation
Imagine you are a referee in a football match. You have a rule: 'If a player commits more than 3 fouls, they get a red card.' The 'red card zone' is your rejection region. In statistics, we do the same. We decide on a 'significance level' α (like a threshold for bad behavior) and find a range of extreme values that are so unlikely to happen by chance that we reject our starting assumption (the null hypothesis).\n\nHowever, there is a catch with discrete distributions like Binomial or Poisson. Because you can't have 2.5 successes (you can only have 2 or 3), the 'boundary' of your rejection region is a jumpy, staircase-like function. If you want a significance level of 5\%, you might find that a rejection region of x ≥ 5 actually gives you a probability of 4.2\%, while x ≥ 4 gives you 7.1\%. You must always pick the region that keeps the probability at or below your target α. This means your 'actual' significance level is often smaller than the one you intended.
📖 Definitions
🔍 Worked example — Easy
A Poisson distribution has parameter λ = 4. Test H_0: λ = 4 against H_1: λ < 4 at a 5\% significance level. Find the rejection region.
🔍 Worked example — Standard
A Binomial distribution X sim B(20, 0.5) is used to test H_0: p = 0.5 against H_1: p > 0.5 at a 5\% significance level. Find the rejection region.
🔍 Worked example — Exam-style
In a two-tailed test for a discrete distribution, the significance level is 5\%. If the probability of the upper tail is 0.022 and the probability of the lower tail is 0.025, what is the actual significance level?
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Testing a population mean
If X is normal with known σ, or n is large (use s for σ by the CLT):
🧑🏫 Teacher's explanation
Imagine you are a judge in a courtroom. The defendant is assumed innocent until proven guilty. In statistics, we assume the 'status quo' (the null hypothesis) is true until the evidence (the data) is so extreme that it's highly unlikely to have happened by chance. We call this extreme evidence the 'rejection region'.
📖 Definitions
🔍 Worked example — Easy
A factory produces bolts with a mean diameter of 50 mm. The population standard deviation is 0.2 mm. A sample of 25 bolts is taken. Test whether the mean diameter has changed at a 5\% significance level.
🔍 Worked example — Standard
A population is normally distributed with μ = 100 and σ = 15. A sample of 36 observations is taken. Test the claim that the mean is greater than 100 at a 1\% significance level.
🔍 Worked example — Exam-style
A sample of 100 items has a mean barx = 55 and a sample standard deviation s = 8. The population is large enough to assume the sampling distribution of the mean is normal. Test if the population mean μ is different from 50 at a 5\% significance level.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Type I and Type II errors
🧑🏫 Teacher's explanation
In hypothesis testing, we are like a jury in a courtroom. The 'null hypothesis' (H_0) is the assumption that the defendant is innocent. We only reject this assumption if the evidence is overwhelming. However, we can make two specific mistakes.
📖 Definitions
🔍 Worked example — Easy
A manufacturer claims that a machine produces 95\% defect-free items. A quality control test has a significance level of 5\%. If the null hypothesis H_0 is that the machine is working correctly, what is the probability of a Type I error?
🔍 Worked example — Standard
A test for a new drug has a significance level of 0.01. The null hypothesis H_0 is that the drug has no effect. If the drug actually increases recovery rates (meaning H_0 is false), and the probability of the test statistic falling in the acceptance region is 0.94, calculate the probability of a Type II error.
🔍 Worked example — Exam-style
A test statistic Z follows a standard normal distribution N(0, 1) under H_0. The rejection region is Z < -1.645 or Z > 1.645. If the true mean under the alternative hypothesis H_1 is μ = 2, calculate the probability of a Type II error.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples — Example 1: binomial, one-tailed
A coin is tossed 20 times and shows 15 heads. Test at the 5% level whether it is biased towards heads.
🔍 Worked example — Easy
A fair six-sided die is rolled 10 times. Test at the 5% level whether the die is biased towards showing 6s. (Use H_0: p = 1/6)
🔍 Worked example — Standard
A coin is tossed 20 times and shows 15 heads. Test at the 5% level whether it is biased towards heads. (Use H_0: p = 0.5)
🔍 Worked example — Exam-style
In the previous example (20 tosses, 15 heads, H_0: p=0.5, H_1: p>0.5), calculate the probability of a Type II error if the true probability is p=0.7.
⚠️ Common errors
Worked examples — Example 2: Poisson
A road had an average of 4 accidents a week. After new signs, there is 1 accident in a week. Test at 5% whether the rate has fallen.
🔍 Worked example — Easy
A bakery claims that an average of 3 customers enter the shop every hour. A manager observes the shop for one hour and sees only 1 customer. Test at the 5% significance level whether the number of customers has decreased.
🔍 Worked example — Standard
A call center receives an average of 6 calls per hour. After a system update, they observe 2 calls in an hour. Test at the 5% significance level whether the rate of calls has fallen.
🔍 Worked example — Exam-style
A factory produces lightbulbs with an average of 10 defects per batch. A quality control inspector finds 4 defects in a batch. Test at the 1% significance level whether the defect rate has decreased.
⚠️ Common errors
Worked examples — Example 3: a mean with known σ
Bags should have mean mass 500 g; σ = 12 g. A sample of 36 has bar x = 496 g. Test at 5% whether the mean has changed.
🔍 Worked example — Easy
A factory produces bags of sugar with a known population standard deviation σ = 12g and a target mean μ = 500g. A sample of n = 36 bags is taken and found to have a mean barx = 496g. Test at the 5% significance level whether the mean mass has changed.
🔍 Worked example — Exam-style
A sample of n=100 items has barx=10.5 where μ=10 and σ=2. Test at 1% significance level if the mean is greater than 10.
⚠️ Common errors
Worked examples — Example 4: normal approximation
40% of customers usually buy a dessert. After a menu change, 50 of 100 customers do. Test at 5% whether the proportion has increased.
🔍 Worked example — Easy
A factory produces lightbulbs with a 10\% failure rate. In a sample of 100 bulbs, 15 are faulty. Test at the 5\% significance level whether the failure rate has increased.
🔍 Worked example — Standard
A coin is claimed to be fair (p=0.5). It is flipped 200 times and results in 110 heads. Test at the 1\% significance level whether the coin is biased towards heads.
🔍 Worked example — Exam-style
In a large population, 30\% of people prefer tea over coffee. In a sample of 400 people, 130 prefer tea. Test at the 5\% significance level whether the proportion of tea drinkers has decreased.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) H_0: p = tfrac16, H_1: p > tfrac16. P(X ≥ 9) = 0.0506 > 0.05: do not reject (insufficient evidence of bias). P(X ≥ 10) = 0.0197, so the critical region is X ≥ 10. (2) Po(7.5): P(X ≤ 2) = 0.0203 < 0.05, reject H_0: evidence the rate has decreased. Critical region X ≤ 2 (P(X ≤ 3) = 0.0591), so P(Type I) = 0.0203. (3) z = 0.9/3/5 = 1.5 < 1.96: do not reject; insufficient evidence that μ ≠ 10. (4) z = 1.2/5/10 = 2.4 > 2.326: reject H_0; evidence at 1% that the mean exceeds 50.