Hypothesis testing – Year 12 A level maths worksheet with answers to download as PDF

Ravi suspects a coin is biased towards heads. He tosses it 20 times and gets 15 heads. Let \(p\) be the probability of heads.

1. State suitable null and alternative hypotheses. [2 marks]
2. State the distribution of the number of heads \(X\) if the null hypothesis is true. [1 mark]
3. Find \(P(X\ge15)\). [2 marks]
4. Test at the 5% significance level and state your conclusion in context. [3 marks]

A drug is known to cure 40% of patients. A new drug is tested on 20 patients and 12 are cured. Test at the 5% level whether the new drug is more effective.

1. State the hypotheses. [2 marks]
2. Calculate the probability of 12 or more cures if \(p=0.4\). [3 marks]
3. State the conclusion in context. [3 marks]

A spinner is supposed to land on red with probability 0.25. In 40 spins it lands on red 17 times. Test at the 5% level whether the probability of red has changed.

1. State the hypotheses and explain why the test is two-tailed. [3 marks]
2. Find \(P(X\ge17)\) under \(H_0\). [3 marks]
3. Compare with the correct significance level for each tail and state the conclusion. [4 marks]

\(X\sim B(20,p)\). A test of \(H_0:p=0.3\) against \(H_1:p<0.3\) is carried out at the 5% level.

1. Find the critical region. [3 marks]
2. State the actual significance level of the test. [1 mark]
3. If the observed value of \(X\) is 3, what is the conclusion? [2 marks]

\(X\sim B(20,0.5)\). Find the critical region for a two-tailed test of \(H_0:p=0.5\) at the 10% significance level, with at most 5% in each tail.

1. Find the lower tail. [3 marks]
2. Find the upper tail. [3 marks]
3. State the actual significance level. [2 marks]

Rewrite each conclusion correctly in context.

1. "Reject \(H_0\)" in a test about a new fertiliser, where \(p\) is the proportion of plants that flower. [2 marks]
2. "Accept \(H_0\)" in a test about a bus company claiming 90% of buses are on time. [2 marks]
3. Why is it wrong to say that a test has "proved" the null hypothesis? [2 marks]

A cereal company says 25% of its boxes contain a prize. A shopper buys 15 boxes and finds prizes in 7 of them. Test at 5% whether the proportion of prize boxes is greater than the company says.

1. State the hypotheses. [2 marks]
2. Find \(P(X\ge7)\) under \(H_0\). [3 marks]
3. State the conclusion. [2 marks]
4. Find the critical region for the test and its actual significance level. [3 marks]

A seed company claims that 80% of its seeds germinate. A gardener plants 25 seeds and 16 germinate. She suspects the percentage is lower. Test at the 5% level.

1. State the hypotheses. [2 marks]
2. Find the probability of 16 or fewer germinating if the claim is true. [3 marks]
3. State your conclusion. [2 marks]
4. The gardener repeats the test and 17 germinate. What is the conclusion now? [3 marks]

A fair die has probability \(\tfrac16\) of showing a six. A student rolls a die 30 times and gets only one six. She suspects the die rarely shows a six.

1. State the hypotheses. [2 marks]
2. Find \(P(X\le1)\) under \(H_0\) to 4 decimal places. [3 marks]
3. Test at the 5% level. [2 marks]
4. Explain why getting zero sixes would be stronger evidence. [1 mark]

The recovery rate for a condition is 65%. A new treatment is tried on 12 patients and all 12 recover.

1. State the hypotheses. [2 marks]
2. Calculate the probability that all 12 recover if \(p=0.65\). [2 marks]
3. Conclude at the 1% level. [3 marks]
4. Another trial gives 11 recoveries out of 12. Is that significant at the 5% level? [3 marks]

Before an update, 10% of users reported a crash. After the update, 50 users are asked and 3 report a crash. The developers hope the crash rate has fallen.

1. State the hypotheses. [2 marks]
2. Find \(P(X\le3)\) under \(H_0\). [2 marks]
3. Test at the 5% level. [2 marks]
4. Find the critical region and actual significance level. [4 marks]

In a previous poll, half the voters in a town supported a plan. A new poll of 30 people finds that only 8 support it. The council asks whether support has changed.

1. State the hypotheses. [2 marks]
2. Find \(P(X\le8)\) under \(H_0\). [2 marks]
3. Carry out the test at the 5% level. [3 marks]
4. Give the conclusion in context. [2 marks]

A manufacturer says that only 2% of its components are defective. A buyer tests 40 components and finds 4 defective.

1. State the hypotheses. [2 marks]
2. Find \(P(X\ge4)\) under \(H_0\). [3 marks]
3. Test at the 1% level and at the 5% level. [3 marks]
4. What would you advise the buyer? [1 mark]

A netball player scores 60% of her shots. After a training programme she scores 19 out of 25 shots. Test at the 5% level whether she has improved.

1. State the hypotheses. [2 marks]
2. Find \(P(X\ge19)\) under \(H_0\). [3 marks]
3. State the conclusion. [2 marks]
4. How many successes out of 25 would have been needed to show improvement at the 5% level? [3 marks]

A teacher claims that 30% of a school\'s students walk to school. A sample of 20 students is taken to test whether the proportion is lower. Here are cumulative probabilities for \(X\sim B(20,0.3)\).

1. Find the critical region at the 5% level and the actual significance level. [3 marks]
2. Find the critical region at the 10% level. [2 marks]
3. The sample has 3 who walk. State the conclusions at 5% and 10%. [3 marks]
4. Explain why the actual significance level is less than the nominal one. [2 marks]

A drug is known to be effective for 40% of patients. A two-tailed test at the 5% level (at most 2.5% in each tail) is carried out on a sample of 50 patients. Some probabilities for \(X\sim B(50,0.4)\) are shown.

1. Find the lower critical value. [2 marks]
2. Find the upper critical value. [2 marks]
3. Write down the critical region and the actual significance level. [3 marks]
4. The sample gives 28 effective cases. State the conclusion. [2 marks]

\(X\sim B(20,p)\). A test of \(H_0:p=0.2\) against \(H_1:p>0.2\) uses the critical region \(X\ge8\).

1. Find the actual significance level. [2 marks]
2. The true value is actually \(p=0.4\). Find the probability that the test correctly rejects \(H_0\). [3 marks]
3. Find the probability that the test fails to reject \(H_0\) when \(p=0.4\). [1 mark]
4. Suggest two ways of making the test better at detecting \(p=0.4\). [2 marks]

To test whether a coin is biased towards heads, a student plans to toss it \(n\) times and reject \(H_0:p=0.5\) in favour of \(H_1:p>0.5\) if the results are extreme enough, at the 5% level.

1. Show that if all \(n\) tosses give heads, the probability under \(H_0\) is \(0.5^n\) and find the smallest \(n\) for which this is below 5%. [3 marks]
2. Show that \(P(X\ge n-1)=\dfrac{n+1}{2^n}\). [3 marks]
3. Find the smallest \(n\) for which getting at least \(n-1\) heads would be significant at 5%. [2 marks]
4. Comment on the usefulness of tosses with small \(n\). [2 marks]

A school claims that 70% of students prefer the new canteen menu. Five classes of 10 students each are surveyed independently, with the following numbers who prefer the new menu.

1. Test the claim for class D at the 5% level (one-tailed, \(p<0.7\)). [4 marks]
2. Combine the classes into a single sample of 50 and test again. [5 marks]
3. Explain why the combined test is more reliable. [2 marks]

A coin is tossed 30 times and lands heads 20 times.

1. Test at the 5% level whether the coin is biased towards heads (one-tailed). [4 marks]
2. Test at the 5% level whether the coin is biased (two-tailed). [4 marks]
3. Explain why the two conclusions differ and why the choice of \(H_1\) must be made before looking at the data. [3 marks]