2.05b Hypothesis test for binomial proportion

5 A travel company finds from its records that \(40 \%\) of its customers book with travel agents. The company redesigns its website, and then carries out a survey of 10 randomly chosen customers. The result of the survey is that 1 of these customers booked with a travel agent.

1. Test at the \(5 \%\) significance level whether the percentage of customers who book with travel agents has decreased.
2. The managing director says that "Our redesigned website has resulted in a decrease in the percentage of our customers who book with travel agents." Comment on this statement.

3 It is known that on average one person in three prefers the colour of a certain object to be blue. In a psychological test, 12 randomly chosen people were seated in a room with blue walls, and asked to state independently which colour they preferred for the object. Seven of the 12 people said that they preferred blue. Carry out a significance test, at the \(5 \%\) level, of whether the statement "on average one person in three prefers the colour of the object to be blue" is true for people who are seated in a room with blue walls.

8 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , 8 ^ { 2 } \right)\). A test is carried out, at the \(5 \%\) significance level, of \(\mathrm { H } _ { 0 } : \mu = 30\) against \(\mathrm { H } _ { 1 } : \mu > 30\), based on a random sample of size 18 .

1. Find the critical region for the test.
2. If \(\mu = 30\) and the outcome of the test is that \(\mathrm { H } _ { 0 }\) is rejected, state the type of error that is made. On a particular day this test is carried out independently a total of 20 times, and for 4 of these tests the outcome is that \(\mathrm { H } _ { 0 }\) is rejected. It is known that the value of \(\mu\) remains the same throughout these 20 tests.
3. Find the probability that \(\mathrm { H } _ { 0 }\) is rejected at least 4 times if \(\mu = 30\). Hence state whether you think that \(\mu = 30\), giving a reason.
4. Given that the probability of making an error of the type different from that stated in part (ii) is 0.4 , calculate the actual value of \(\mu\), giving your answer correct to 4 significant figures. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}

7 An online shopping company takes orders through its website. On average \(80 \%\) of orders from the website are delivered within 24 hours. The quality controller selects 10 orders at random to check when they are delivered.

1. Find the probability that
   (A) exactly 8 of these orders are delivered within 24 hours,
   (B) at least 8 of these orders are delivered within 24 hours. The company changes its delivery method. The quality controller suspects that the changes will mean that fewer than \(80 \%\) of orders will be delivered within 24 hours. A random sample of 18 orders is checked and it is found that 12 of them arrive within 24 hours.
2. Write down suitable hypotheses and carry out a test at the \(5 \%\) significance level to determine whether there is any evidence to support the quality controller's suspicion.
3. A statistician argues that it is possible that the new method could result in either better or worse delivery times. Therefore it would be better to carry out a 2 -tail test at the \(5 \%\) significance level. State the alternative hypothesis for this test. Assuming that the sample size is still 18, find the critical region for this test, showing all of your calculations.

7 To withdraw money from a cash machine, the user has to enter a 4-digit PIN (personal identification number). There are several thousand possible 4-digit PINs, but a survey found that \(10 \%\) of cash machine users use the PIN '1234'.

1. 16 cash machine users are selected at random.
   (A) Find the probability that exactly 3 of them use 1234 as their PIN.
   (B) Find the probability that at least 3 of them use 1234 as their PIN.
   (C) Find the expected number of them who use 1234 as their PIN. An advertising campaign aims to reduce the number of people who use 1234 as their PIN. A hypothesis test is to be carried out to investigate whether the advertising campaign has been successful.
2. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis.
3. A random sample of 20 cash machine users is selected.
   (A) Explain why the test could not be carried out at the \(10 \%\) significance level.
   (B) The test is to be carried out at the \(k \%\) significance level. State the lowest integer value of \(k\) for which the test could result in the rejection of the null hypothesis.
4. A new random sample of 60 cash machine users is selected. It is found that 2 of them use 1234 as their PIN. You are given that, if \(X \sim \mathrm {~B} ( 60,0.1 )\), then (to 4 decimal places) $$\mathrm { P } ( X = 2 ) = 0.0393 , \quad \mathrm { P } ( X < 2 ) = 0.0138 , \quad \mathrm { P } ( X \leqslant 2 ) = 0.0530 .$$ Using the same hypotheses as in part (ii), carry out the test at the \(5 \%\) significance level. \section*{END OF QUESTION PAPER}

9 Briony suspects that a particular 6-sided dice is biased in favour of 2. She plans to throw the dice 35 times and note the number of times that it shows a 2 . She will then carry out a test at the \(4 \%\) significance level. Find the rejection region for the test.

12 The table shows information for England and Wales, taken from the UK 2011 census. A random sample of 10000 people in another country was chosen in 2011 , and the number, \(m\), of children aged 5-17 was noted.
It was found that there was evidence at the \(2.5 \%\) level that the proportion of children aged 5-17 in the same year was higher than in the UK.
Unfortunately, when the results were recorded the value of \(m\) was omitted. Use an appropriate normal distribution to find an estimate of the smallest possible value of \(m\). TURN OVER FOR THE NEXT QUESTION

1. Past records show that \(15 \%\) of customers at a shop buy chocolate. The shopkeeper believes that moving the chocolate closer to the till will increase the proportion of customers buying chocolate.

After moving the chocolate closer to the till, a random sample of 30 customers is taken and 8 of them are found to have bought chocolate. Julie carries out a hypothesis test, at the 5\% level of significance, to test the shopkeeper's belief.
Julie's hypothesis test is shown below. \(\mathrm { H } _ { 0 } : p = 0.15\) \(\mathrm { H } _ { 1 } : p \geqslant 0.15\) Let \(X =\) the number of customers who buy chocolate. \(X \sim \mathrm {~B} ( 30,0.15 )\) \(\mathrm { P } ( X = 8 ) = 0.0420\) \(0.0420 < 0.05\) so reject \(\mathrm { H } _ { 0 }\) There is sufficient evidence to suggest that the proportion of customers buying chocolate has increased.

1. Identify the first two errors that Julie has made in her hypothesis test.
2. Explain whether or not these errors will affect the conclusion of her hypothesis test. Give a reason for your answer.
3. Find, using a 5\% level of significance, the critical region for a one-tailed test of the shopkeeper's belief. The probability in the tail should be less than 0.05
4. Find the actual level of significance of this test.

1. Afrika works in a call centre.

She assumes that calls are independent and knows, from past experience, that on each sales call that she makes there is a probability of \(\frac { 1 } { 6 }\) that it is successful. Afrika makes 9 sales calls.

1. Calculate the probability that at least 3 of these sales calls will be successful. The probability of Afrika making a successful sales call is the same each day.
   Afrika makes 9 sales calls on each of 5 different days.
2. Calculate the probability that at least 3 of the sales calls will be successful on exactly 1 of these days. Rowan works in the same call centre as Afrika and believes he is a more successful salesperson. To check Rowan's belief, Afrika monitors the next 35 sales calls Rowan makes and finds that 11 of the sales calls are successful.
3. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence to support Rowan's belief.

1. Past information shows that \(25 \%\) of adults in a large population have a particular allergy.

Rylan believes that the proportion that has the allergy differs from 25\%
He takes a random sample of 50 adults from the population.
Rylan carries out a test of the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.25\) using a \(5 \%\) level of significance.

1. Write down the alternative hypothesis for Rylan's test.
2. Find the critical region for this test. You should state the probability associated with each tail, which should be as close to \(2.5 \%\) as possible.
3. State the actual probability of incorrectly rejecting \(\mathrm { H } _ { 0 }\) for this test. Rylan finds that 10 of the adults in his sample have the allergy.
4. State the conclusion of Rylan's hypothesis test.

4. The random variable \(X \sim \mathrm {~B} ( 27,0.35 )\)

1. Find
   1. \(\mathrm { P } ( X = 10 )\)
   2. \(\mathrm { P } ( 12 \leqslant X < 15 )\) Historical records show that the proportion of defective items produced by a machine is 0.12 Following a maintenance service of the machine, a random sample of 60 items is taken and 3 defective items are found.
2. Carry out a suitable test to determine whether the proportion of defective items produced by the machine has decreased following the maintenance service. You should state your hypotheses clearly and use a \(5 \%\) level of significance.
3. Write down the \(p\)-value for your test in part (b)

1. A nursery has a sack containing a large number of coloured beads of which \(14 \%\) are coloured red.

Aliya takes a random sample of 18 beads from the sack to make a bracelet.

1. State a suitable binomial distribution to model the number of red beads in Aliya's bracelet.
2. Use this binomial distribution to find the probability that
   1. Aliya has just 1 red bead in her bracelet,
   2. there are at least 4 red beads in Aliya's bracelet.
3. Comment on the suitability of a binomial distribution to model this situation. After several children have used beads from the sack, the nursery teacher decides to test whether or not the proportion of red beads in the sack has changed. She takes a random sample of 75 beads and finds 4 red beads.
4. Stating your hypotheses clearly, use a 5\% significance level to carry out a suitable test for the teacher.
5. Find the \(p\)-value in this case.

1. A dentist knows from past records that \(10 \%\) of customers arrive late for their appointment.

A new manager believes that there has been a change in the proportion of customers who arrive late for their appointment. A random sample of 50 of the dentist's customers is taken.

1. Write down
   • a null hypothesis corresponding to no change in the proportion of customers who arrive late
   • an alternative hypothesis corresponding to the manager's belief
   • Using a \(5 \%\) level of significance, find the critical region for a two-tailed test of the null hypothesis in (a) You should state the probability of rejection in each tail, which should be less than 0.025
   • Find the actual level of significance of the test based on your critical region from part (b)
   The manager observes that 15 of the 50 customers arrived late for their appointment.
2. With reference to part (b), comment on the manager's belief.

11 It is known that, under the standard treatment for a certain disease, \(9.7 \%\) of patients with the disease experience side effects within one year. In a trial of a new treatment, a random sample of 450 patients with this disease was selected and the number \(X\) who experienced side effects within one year was noted.

1. State one assumption needed in order to use a binomial model for \(X\). It was found that 51 of the 450 patients experienced side effects within one year.
2. Test, at the \(10 \%\) significance level, whether the proportion of patients experiencing side effects within one year is greater under the new treatment than under the standard treatment.

10 Some packets of a certain kind of biscuit contain a free gift. The manufacturer claims that the proportion of packets containing a free gift is 1 in 4 . Marisa suspects that this claim is not true, and that the true proportion is less than 1 in 4 . She chooses 20 packets at random and finds that exactly 1 contains the free gift.

1. Use a binomial model to test the manufacturer's claim, at the \(2.5 \%\) significance level. The packets are packed in boxes, with each box containing 40 packets. Marisa chooses three boxes at random and finds that one box contains 19 packets with the free gift and the other two boxes contain no packets with the free gift.
2. Give a reason why this suggests that the binomial model used in part (a) may not be appropriate.

12 The variable \(X\) has the distribution \(\mathrm { B } \left( 50 , \frac { 1 } { 6 } \right)\). The probabilities \(\mathrm { P } ( X = r )\) for \(r = 0\) to 50 are given by the terms of the expansion of \(( a + b ) ^ { n }\) for specific values of \(a , b\) and \(n\).

1. State the values of \(a\), \(b\) and \(n\). A student has an ordinary 6 -sided dice. They suspect that it is biased so that it shows a 2 on fewer throws than it would if it were fair. In order to test the suspicion the dice is thrown 50 times and the number of 2 s is noted. The student then carries out a hypothesis test at the \(5 \%\) significance level.
2. Write down suitable hypotheses for the test.
3. Determine the rejection region for the test, showing the values of any relevant probabilities.

9 Last year, market research showed that \(8 \%\) of adults living in a certain town used a particular local coffee shop. Following an advertising campaign, it was expected that this proportion would increase. In order to test whether this had happened, a random sample of 150 adults in the town was chosen. The random variable \(X\) denotes the number of these 150 adults who said that they used the local coffee shop.

1. 1. Assuming that the proportion of adults using the local coffee shop is unchanged from the previous year, state a suitable binomial distribution with which to model the variable \(X\).
   2. The probabilities given by this model are the terms of the binomial expansion of an expression of the form \(( a + b ) ^ { n }\). Write down this expression, using appropriate values of \(a , b\) and \(n\). It was found that 18 of these 150 adults said that they use the local coffee shop.
2. Test, at the 5\% significance level, whether the proportion of adults in the town who use the local coffee shop has increased. It was later discovered by a statistician that the random sample of 150 adults had been chosen from shoppers in the town on a Friday and a Saturday.
3. Explain why this suggests that the assumptions made when using a binomial model for \(X\) may not be valid in this context.

8 According to the latest research there are 19.8 million male drivers and 16.2 million female drivers on the roads in the UK.

1. A driver in the UK is selected at random. Find the probability that the driver is male.
2. Calculate the probability that there are 7 female drivers in a random sample of 25 UK drivers. When driving in a built-up area, Rebecca exceeded the speed limit and was obliged to attend a speed awareness course. Her husband said "It's nearly always male drivers who are speeding." When Rebecca attends the course, she finds that there are 25 drivers, 7 of whom are female. You should assume that the drivers on the speed awareness course constitute a random sample of drivers caught speeding.
3. In this question you must show detailed reasoning. Conduct a hypothesis test to determine whether there is any evidence at the \(5 \%\) level to suggest that male drivers are more likely to exceed the speed limit than female drivers.
4. State a modelling assumption that is necessary in order to conduct the hypothesis test in part (c).

8 In 2018 research showed that 81\% of young adults in England had never donated blood.
Following an advertising campaign in 2021, it is believed that the percentage of young adults in England who had never donated blood in 2021 is less than \(81 \%\). Ling decides to carry out a hypothesis test at the 5\% level.
Ling collects data from a random sample of 400 young adults in England.

1. State the null and alternative hypotheses for the test, defining the parameter used.
2. Write down the probability that the null hypothesis is rejected when it should in fact be accepted.
3. Assuming the null hypothesis is correct, calculate the expected number of young adults in the sample who had never donated blood.
4. Calculate the probability that there were no more than 308 young adults who had never donated blood in the sample.
5. Determine the critical region for the test. In fact, the sample contained 314 young adults who had never donated blood.
6. Carry out the test, giving the conclusion in the context of the question.

12 Data collected in the twentieth century showed that the probability of a randomly selected person having blue eyes was 0.08 . A medical researcher believes that the probability in 2024 is less than this so they decide to carry out a hypothesis test at the \(5 \%\) significance level.

1. Write down suitable hypotheses for the test, defining the parameter used.
2. Assuming that the probability that a person selected at random has blue eyes is still 0.08 , calculate the probability that 3 or fewer people in a random sample of 92 people have blue eyes. The researcher collects a random sample of 92 people and finds that 3 of them have blue eyes.
3. Use your answer to part (b) to carry out the test, giving your conclusion in context.

4 In a certain country it is known that 11\% of people are left-handed.

1. Calculate the probability that, in a random sample of 98 people from this country, 5 or fewer are found to be left-handed, giving your answer correct to 3 significant figures. An anthropologist believes that the proportion of left-handed people is lower in a particular ethnic group. The anthropologist collects a random sample of 98 people from this particular ethnic group in order to test the hypothesis that the proportion of left-handed people is less than \(11 \%\). The anthropologist carries out the test at the \(1 \%\) level.
2. Determine the critical region for this test.

10 A company operates trains. The company claims that \(92 \%\) of its trains arrive on time. You should assume that in a random sample of trains, they arrive on time independently of each other.

1. Assuming that \(92 \%\) of the company's trains arrive on time, find the probability that in a random sample of 30 trains operated by this company
   1. exactly 28 trains arrive on time,
   2. more than 27 trains arrive on time. A journalist believes that the percentage of trains operated by this company which arrive on time is lower than \(92 \%\).
2. To investigate the journalist's belief a hypothesis test will be carried out at the \(1 \%\) significance level. A random sample of 18 trains is selected. For this hypothesis test,

12 The jaguar is a species of big cat native to South America. Records show that 6\% of jaguars are born with black coats. Jaguars with black coats are known as black panthers. Due to deforestation a population of jaguars has become isolated in part of the Amazon basin. Researchers believe that the percentage of black panthers may not be \(6 \%\) in this population.

1. Find the minimum sample size needed to conduct a two-tailed test to determine whether there is any evidence at the \(5 \%\) level to suggest that the percentage of black panthers is not \(6 \%\). A research team identifies 70 possible sites for monitoring the jaguars remotely. 30 of these sites are randomly selected and cameras are installed. 83 different jaguars are filmed during the evidence gathering period. The team finds that 10 of the jaguars are black panthers.
2. Conduct a hypothesis test to determine whether the information gathered by the research team provides any evidence at the \(5 \%\) level to suggest that the percentage of black panthers in this population is not \(6 \%\).

12 A survey conducted in 2021 showed that 10\% of British adults were vegetarians. A dietitian believes that the proportion of British adults who are vegetarians may have changed, so decides to conduct a hypothesis test at the \(5 \%\) level of significance. In a random sample of 112 adults, the dietitian finds that there are 19 vegetarians. Carry out the hypothesis test to determine whether there is any evidence to support the dietitian's belief.