2.05c Significance levels: one-tail and two-tail

7 A coffee shop provides free internet access for its customers. It is known that the probability that a randomly selected customer is accessing the internet is 0.35 , independently of all other customers.

1. 10 customers are selected at random.
   (A) Find the probability that exactly 5 of them are accessing the internet.
   (B) Find the probability that at least 5 of them are accessing the internet.
   (C) Find the expected number of these customers who are accessing the internet. Another coffee shop also provides free internet access. It is suspected that the probability that a randomly selected customer at this coffee shop is accessing the internet may be different from 0.35 . A random sample of 20 customers at this coffee shop is selected. Of these, 10 are accessing the internet.
2. Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether the probability for this coffee shop is different from 0.35 . Give a reason for your choice of alternative hypothesis.
3. To get a more reliable result, a much larger random sample of 200 customers is selected over a period of time, and another hypothesis test is carried out. You are given that 90 of the 200 customers were accessing the internet. You are also given that, if \(X\) has the binomial distribution with parameters \(n = 200\) and \(p = 0.35\), then \(\mathrm { P } ( X \geqslant 90 ) = 0.0022\). Using the same hypotheses and significance level which you used in part (ii), complete this test.

8 The Department of Health 'eat five a day' advice recommends that people should eat at least five portions of fruit and vegetables per day. In a particular school, \(20 \%\) of pupils eat at least five a day.

1. 15 children are selected at random.
   (A) Find the probability that exactly 3 of them eat at least five a day.
   (B) Find the probability that at least 3 of them eat at least five a day.
   (C) Find the expected number who eat at least five a day. A programme is introduced to encourage children to eat more portions of fruit and vegetables per day. At the end of this programme, the diets of a random sample of 15 children are analysed. A hypothesis test is carried out to examine whether the proportion of children in the school who eat at least five a day has increased.
2. (A) Write down suitable null and alternative hypotheses for the test.
   (B) Give a reason for your choice of the alternative hypothesis.
3. Find the critical region for the test at the \(10 \%\) significance level, showing all of your calculations. Hence complete the test, given that 7 of the 15 children eat at least five a day.

6 A manufacturer produces tiles. On average 10\% of the tiles produced are faulty. Faulty tiles occur randomly and independently. A random sample of 18 tiles is selected.

1. (A) Find the probability that there are exactly 2 faulty tiles in the sample.
   (B) Find the probability that there are more than 2 faulty tiles in the sample.
   (C) Find the expected number of faulty tiles in the sample. A cheaper way of producing the tiles is introduced. The manufacturer believes that this may increase the proportion of faulty tiles. In order to check this, a random sample of 18 tiles produced using the cheaper process is selected and a hypothesis test is carried out.
2. (A) Write down suitable null and alternative hypotheses for the test.
   (B) Explain why the alternative hypothesis has the form that it does.
3. Find the critical region for the test at the \(5 \%\) level, showing all of your calculations.
4. In fact there are 4 faulty tiles in the sample. Complete the test, stating your conclusion clearly.

5 A manufacturer produces titanium bicycle frames. The bicycle frames are tested before use and on average \(5 \%\) of them are found to be faulty. A cheaper manufacturing process is introduced and the manufacturer wishes to check whether the proportion of faulty bicycle frames has increased. A random sample of 18 bicycle frames is selected and it is found that 4 of them are faulty. Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether the proportion of faulty bicycle frames has increased.

5 A researcher is investigating whether people can identify whether a glass of water they are given is bottled water or tap water. She suspects that people do no better than they would by guessing. Twenty people are selected at random; thirteen make a correct identification. She carries out a hypothesis test.

1. Explain why the null hypothesis should be \(p = 0.5\), where \(p\) represents the probability that a randomly selected person makes a correct identification.
2. Briefly explain why she uses an alternative hypothesis of \(p > 0.5\).
3. Complete the test at the \(5 \%\) significance level.

7 A drug for treating a particular minor illness cures, on average, \(78 \%\) of patients. Twenty people with this minor illness are selected at random and treated with the drug.

1. \(( A )\) Find the probability that exactly 19 patients are cured.
   (B) Find the probability that at most 18 patients are cured. \(( C )\) Find the expected number of patients who are cured.
2. A pharmaceutical company is trialling a new drug to treat this illness. Researchers at the company hope that a higher percentage of patients will be cured when given this new drug. Twenty patients are selected at random, and given the new drug. Of these, 19 are cured. Carry out a hypothesis test at the \(1 \%\) significance level to investigate whether there is any evidence to suggest that the new drug is more effective than the old one.
3. If the researchers had chosen to carry out the hypothesis test at the \(5 \%\) significance level, what would the result have been? Justify your answer.

4 A television company believes that the proportion of adults who watched a certain programme is 0.14 . Out of a random sample of 22 adults, it is found that 2 watched the programme.

1. Carry out a significance test, at the \(10 \%\) level, to determine, on the basis of this sample, whether the television company is overestimating the proportion of adults who watched the programme.
2. The sample was selected randomly. State what properties of this method of sampling are needed to justify the use of the distribution used in your test.

6 The weight of a plastic box manufactured by a company is \(W\) grams, where \(W \sim \mathrm {~N} ( \mu , 20.25 )\). A significance test of the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 50.0\), against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 50.0\), is carried out at the \(5 \%\) significance level, based on a sample of size \(n\).

1. Given that \(n = 81\),
   1. find the critical region for the test, in terms of the sample mean \(\bar { W }\),
   2. find the probability that the test results in a Type II error when \(\mu = 50.2\).
   3. State how the probability of this Type II error would change if \(n\) were greater than 81 .

7 A motorist records the time taken, \(T\) minutes, to drive a particular stretch of road on each of 64 occasions. Her results are summarised by $$\Sigma t = 876.8 , \quad \Sigma t ^ { 2 } = 12657.28$$

1. Test, at the \(5 \%\) significance level, whether the mean time for the motorist to drive the stretch of road is greater than 13.1 minutes.
2. Explain whether it is necessary to use the Central Limit Theorem in your test.

4 The continuous random variable \(X\) has mean \(\mu\) and standard deviation 45. A significance test is to be carried out of the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 230\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 230\), at the \(1 \%\) significance level. A random sample of size 50 is obtained, and the sample mean is found to be 213.4.

1. Carry out the test.
2. Explain whether it is necessary to use the Central Limit Theorem in your test.

5 A temporary job is advertised annually. The number of applicants for the job is a random variable which is known from many years' experience to have a distribution \(\operatorname { Po } ( 12 )\). In 2010 there were 19 applicants for the job. Test, at the 10\% significance level, whether there is evidence of an increase in the mean number of applicants for the job.

9 A pharmaceutical company is developing a new drug to treat a certain disease. The company will continue to develop the drug if the proportion \(p\) of those who have the disease and show a substantial improvement after treatment is greater than 0.7 . The company carries out a test, at the \(5 \%\) significance level, on a random sample of 14 patients who suffer from the disease.

1. Find the critical region for the test.
2. Given that 12 of the 14 patients in the sample show a substantial improvement, carry out the test.
3. Find the probability that the test results in a Type II error if in fact \(p = 0.8\). RECOGNISING ACHIEVEMENT

3 An electronics company is developing a new sound system. The company claims that \(60 \%\) of potential buyers think that the system would be good value for money. In a random sample of 12 potential buyers, 4 thought that it would be good value for money. Test, at the 5\% significance level, whether the proportion claimed by the company is too high.

8 In a large company the time taken for an employee to carry out a certain task is a normally distributed random variable with mean 78.0 s and unknown variance. A new training scheme is introduced and after its introduction the times taken by a random sample of 120 employees are recorded. The mean time for the sample is 76.4 s and an unbiased estimate of the population variance is \(68.9 \mathrm {~s} ^ { 2 }\).

1. Test, at the \(1 \%\) significance level, whether the mean time taken for the task has changed.
2. It is required to redesign the test so that the probability of making a Type I error is less than 0.01 when the sample mean is 77.0 s . Calculate an estimate of the smallest sample size needed, and explain why your answer is only an estimate.

4 The proportion of commuters in a town who travel to work by train is 0.4 . Following the opening of a new station car park, a random sample of 16 commuters is obtained, and 11 of these travel to work by train. Test at the \(1 \%\) significance level whether there is evidence of an increase in the proportion of commuters in this town who travel to work by train.

5 The time \(T\) seconds needed for a computer to be ready to use, from the moment it is switched on, is a normally distributed random variable with standard deviation 5 seconds. The specification of the computer says that the population mean time should be not more than 30 seconds.

1. A test is carried out, at the \(5 \%\) significance level, of whether the specification is being met, using the mean \(\bar { t }\) of a random sample of 10 times.
   1. Find the critical region for the test, in terms of \(\bar { t }\).
   2. Given that the population mean time is in fact 35 seconds, find the probability that the test results in a Type II error.
   3. Because of system degradation and memory load, the population mean time \(\mu\) seconds increases with the number of months of use, \(m\). A formula for \(\mu\) in terms of \(m\) is \(\mu = 20 + 0.6 m\). Use this formula to find the value of \(m\) for which the probability that the test results in rejection of the null hypothesis is 0.5 .

3 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , 5 ^ { 2 } \right)\). A hypothesis test is carried out of \(\mathrm { H } _ { 0 } : \mu = 20.0\) against \(\mathrm { H } _ { 1 } : \mu < 20.0\), at the \(1 \%\) level of significance, based on the mean of a sample of size 16. Given that in fact \(\mu = 15.0\), find the probability that the test results in a Type II error.

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}

6 The amount of caffeine in a randomly selected cup of coffee dispensed by a machine has a normal distribution. The amount of caffeine in each of a random sample of 25 cups was measured. The sample mean was 110.4 mg and the unbiased estimate of the population variance was \(50.42 \mathrm { mg } ^ { 2 }\). Calculate a 90\% confidence interval for the mean amount of caffeine dispensed.

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.

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.