For each situation, write down suitable null and alternative hypotheses in terms of p, and say whether the test is one-tailed or two-tailed.
Harry thinks a coin is biased towards landing on heads. p is the probability the coin lands on heads.
A firm says 4% of the phone cases it makes are faulty. After a machine is serviced the manager believes the proportion has gone down. p is the proportion of faulty cases.
Records show 8% of households in a town have a pet rabbit. A vet thinks this has changed. p is the proportion of households with a pet rabbit.
3 marks
Show answer
a) H0: p = 0.5, H1: p > 0.5, one-tailedb) H0: p = 0.04, H1: p < 0.04, one-tailedc) H0: p = 0.08, H1: p ≠ 0.08, two-tailed
2
Callum claims he can tell the difference between two brands of cola by taste. In a test he is given 10 cups, each filled at random with one of the two brands, and he identifies 9 of them correctly.
Test, at the 5% level of significance, whether there is evidence that Callum does better than guessing. State your hypotheses clearly.
4 marks
Show answer
H0: p = 0.5, H1: p > 0.5; P(X ≥ 9) = 0.0107 < 0.05, so reject H0: there is evidence that Callum can tell the brands apart.
3
Last year 30% of the students at a school walked to school. A new bus route has started and the head teacher believes that fewer students now walk. She takes a random sample of 20 students and finds that 2 of them walk to school.
Test, at the 5% level of significance, whether there is evidence to support the head teacher’s belief. State your hypotheses clearly.
5 marks
Show answer
H0: p = 0.3, H1: p < 0.3; P(X ≤ 2) = 0.0355 < 0.05, so reject H0: there is evidence that fewer students walk to school.
4
A garden centre says that 20% of its tomato plants produce yellow tomatoes. Joy thinks the proportion is higher. She plans to test this using a random sample of 25 plants.
Write down suitable hypotheses for her test.
Find the critical region for the test at the 5% level of significance.
Write down the actual significance level of the test.
5 marks
Show answer
a) H0: p = 0.2, H1: p > 0.2b) X ≥ 9c) 0.0468
5
A survey company has found that 40% of adults in a city cycle at least once a week. A researcher believes this proportion is different for adults who live in the city centre. She will take a random sample of 30 adults who live in the city centre and test at the 10% level of significance.
Write down suitable hypotheses.
Find the critical region for the test. The probability in each tail should be less than 0.05
Find the actual significance level of the test.
6 marks
Show answer
a) H0: p = 0.4, H1: p ≠ 0.4b) X ≤ 7 or X ≥ 17c) 0.0916
6
A restaurant owner says that 20% of diners order a dessert. The new chef believes that since he changed the menu, the proportion of diners who order a dessert has increased. On one evening, 10 out of 30 randomly chosen diners order a dessert.
Test, at the 5% level of significance, whether there is evidence to support the chef’s belief. State your hypotheses clearly.
5 marks
Show answer
H0: p = 0.2, H1: p > 0.2; P(X ≥ 10) = 0.0611 > 0.05, so do not reject H0: insufficient evidence that the proportion ordering dessert has increased.
7
A national survey found that 15% of Year 10 students play a musical instrument. Mr Okafor believes the proportion at his school is different. In a random sample of 40 Year 10 students from his school, 11 play a musical instrument.
Test, at the 5% level of significance, whether there is evidence to support Mr Okafor’s belief. State your hypotheses clearly.
Explain how the conclusion would change if Mr Okafor had believed that the proportion at his school was higher than 15%.
6 marks
Show answer
a) H0: p = 0.15, H1: p ≠ 0.15; P(X ≥ 11) = 0.0299 > 0.025, so do not reject H0: insufficient evidence that the proportion is differentb) a one-tailed test compares 0.0299 with 0.05; 0.0299 < 0.05 so there would be evidence that the proportion is higher
8
A delivery firm finds that 8% of its parcels arrive late. It changes its routes and wants to know if the proportion of late parcels has gone down. A random sample of 60 parcels is taken after the change.
Using a 5% level of significance, find the critical region for a suitable test.
State the actual significance level of the test.
In the sample, 1 parcel arrives late. State, with a reason, the conclusion of the test.
6 marks
Show answer
a) X ≤ 1 (H0: p = 0.08, H1: p < 0.08)b) 0.0418c) 1 is in the critical region, so reject H0: evidence that the proportion of late parcels has gone down
9
A quiz app says that a player who is guessing gets each question right with probability 0.25. Zainab thinks that she does better than guessing. She answers 15 questions and X is the number she gets right.
Find the critical region for a test of H0: p = 0.25 against H1: p > 0.25 at the 5% level of significance.
Find the probability of rejecting H0 when it is in fact true.
Zainab gets 7 questions right. Find P(X ≥ 7) and hence explain whether this result is significant at the 5% level and at the 10% level.
7 marks
Show answer
a) X ≥ 8b) 0.0173c) P(X ≥ 7) = 0.0566: not significant at 5% (0.0566 > 0.05), significant at 10% (0.0566 < 0.1)
10
A cereal company says that 30% of its boxes contain a free toy. Ella believes the proportion is lower. She will buy n boxes and test H0: p = 0.3 against H1: p < 0.3 at the 5% level of significance.
Find the smallest value of n for which finding no toys at all would lead Ella to reject H0.
Ella buys 9 boxes. Show that the critical region is X = 0 and find the actual significance level.
Ella finds 1 toy in her 9 boxes. State the conclusion of her test in context.
7 marks
Show answer
a) n = 9b) P(X = 0) = 0.79 = 0.0404 0.05; actual significance level 0.0404c) 1 is not in the critical region, so do not reject H0: insufficient evidence that fewer than 30% of boxes contain a toy
11
Mia wants to find out whether a coin is biased. She will toss it 20 times and record X, the number of heads.
Write down suitable hypotheses.
Using a 5% level of significance, find the critical region. The probability in each tail should be as close as possible to 2.5% without going over.
Find the actual significance level of the test.
Mia gets 14 heads. She says “this shows the coin is biased”. Comment on her statement.
7 marks
Show answer
a) H0: p = 0.5, H1: p ≠ 0.5b) X ≤ 5 or X ≥ 15c) 0.0414d) 14 is not in the critical region, so there is insufficient evidence that the coin is biased; her statement is not supported
12
A garden centre claims that 85% of its tulip bulbs flower in the first spring after planting.
Assuming the claim is true, find the probability that at least 10 bulbs in a pack of 12 flower.
Olu thinks that fewer than 85% of the bulbs flower. He plants a random sample of 50 bulbs to test the claim at the 5% level of significance.
Write down suitable hypotheses and find the critical region for his test.
State the actual significance level of the test.
In the spring, 37 of Olu’s 50 bulbs flower. State the conclusion of his test in context.
Give one reason why the bulbs in a single garden might not flower independently of each other.
9 marks
Show answer
a) 0.7358b) H0: p = 0.85, H1: p < 0.85; X ≤ 37c) 0.0301d) 37 is in the critical region, so reject H0: evidence that fewer than 85% of the bulbs flowere) bulbs planted together share the same soil, weather and pests, so whether one flowers may be linked to whether another does
Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.