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

3 At a doctor's surgery, records show that \(20 \%\) of patients who make an appointment fail to turn up. During afternoon surgery the doctor has time to see 16 patients. There are 16 appointments to see the doctor one afternoon.

1. Find the probability that all 16 patients turn up.
2. Find the probability that more than 3 patients do not turn up. To improve efficiency, the doctor decides to make more than 16 appointments for afternoon surgery, although there will still only be enough time to see 16 patients. There must be a probability of at least 0.9 that the doctor will have enough time to see all the patients who turn up.
3. The doctor makes 17 appointments for afternoon surgery. Find the probability that at least one patient does not turn up. Hence show that making 17 appointments is satisfactory.
4. Now find the greatest number of appointments the doctor can make for afternoon surgery and still have a probability of at least 0.9 of having time to see all patients who turn up. A computerised appointment system is introduced at the surgery. It is decided to test, at the \(5 \%\) level, whether the proportion of patients failing to turn up for their appointments has changed. There are always 20 appointments to see the doctor at morning surgery. On a randomly chosen morning, 1 patient does not turn up.
5. Write down suitable hypotheses and carry out the test.

2 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.

3 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.

1 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.

4 When onion seeds are sown outdoors, on average two-thirds of them germinate. A gardener sows seeds in pairs, in the hope that at least one will germinate.

1. Assuming that germination of one of the seeds in a pair is independent of germination of the other seed, find the probability that, if a pair of seeds is selected at random,
   (A) both seeds germinate,
   (B) just one seed germinates,
   (C) neither seed germinates.
2. Explain why the assumption of independence is necessary in order to calculate the above probabilities. Comment on whether the assumption is likely to be valid.
3. A pair of seeds is sown. Find the expectation and variance of the number of seeds in the pair which germinate.
4. The gardener plants 200 pairs of seeds. If both seeds in a pair germinate, the gardener destroys one of the two plants so that only one is left to grow. Of the plants that remain after this, only \(85 \%\) successfully grow to form an onion. Find the expected number of onions grown from the 200 pairs of seeds. If the seeds are sown in a greenhouse, the germination rate is higher. The seed manufacturing company claims that the germination rate is \(90 \%\). The gardener suspects that the rate will not be as high as this, and carries out a trial to investigate. 18 randomly selected seeds are sown in the greenhouse and it is found that 14 germinate.
5. Write down suitable hypotheses and carry out a test at the \(5 \%\) level to determine whether there is any evidence to support the gardener's suspicions.

7 A television company believes that the proportion of households that can receive Channel C is 0.35 .

1. In a random sample of 14 households it is found that 2 can receive Channel C. Test, at the \(2.5 \%\) significance level, whether there is evidence that the proportion of households that can receive Channel C is less than 0.35.
2. On another occasion the test is carried out again, with the same hypotheses and significance level as in part (i), but using a new sample, of size \(n\). It is found that no members of the sample can receive Channel C. Find the largest value of \(n\) for which the null hypothesis is not rejected. Show all relevant working.

8 The quantity, \(X\) milligrams per litre, of silicon dioxide in a certain brand of mineral water is a random variable with distribution \(\mathrm { N } \left( \mu , 5.6 ^ { 2 } \right)\).

1. A random sample of 80 observations of \(X\) has sample mean 100.7. Test, at the \(1 \%\) significance level, the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 102\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 102\).
2. The test is redesigned so as to meet the following conditions.
   • The hypotheses are \(\mathrm { H } _ { 0 } : \mu = 102\) and \(\mathrm { H } _ { 1 } : \mu < 102\).
   • The significance level is \(1 \%\).
   • The probability of making a Type II error when \(\mu = 100\) is to be (approximately) 0.05 .
   The sample size is \(n\), and the critical region is \(\bar { X } < c\), where \(\bar { X }\) denotes the sample mean.
   1. Show that \(n\) and \(c\) satisfy (approximately) the equation \(102 - c = \frac { 13.0256 } { \sqrt { n } }\).
   2. Find another equation satisfied by \(n\) and \(c\).
   3. Hence find the values of \(n\) and \(c\).

3 The random variable \(G\) has the distribution \(\operatorname { Po } ( \lambda )\). A test is carried out of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 4.5\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \lambda \neq 4.5\), based on a single observation of \(G\). The critical region for the test is \(G \leqslant 1\) and \(G \geqslant 9\).

1. Find the significance level of the test.
2. Given that \(\lambda = 5.5\), calculate the probability that the test results in a Type II error.

5 Over a long period the number of visitors per week to a stately home was known to have the distribution \(\mathrm { N } \left( 500,100 ^ { 2 } \right)\). After higher car parking charges were introduced, a sample of four randomly chosen weeks gave a mean number of visitors per week of 435 . You should assume that the number of visitors per week is still normally distributed with variance \(100 ^ { 2 }\).

1. Test, at the \(10 \%\) significance level, whether there is evidence that the mean number of visitors per week has fallen.
2. Explain why it is necessary to assume that the distribution of the number of visitors per week (after the introduction of higher charges) is normal in order to carry out the test.

8 Consultations are taking place as to whether a site currently in use as a car park should be developed as a shopping mall. An agency acting on behalf of a firm of developers claims that at least \(65 \%\) of the local population are in favour of the development. In a survey of a random sample of 12 members of the local population, 6 are in favour of the development.

1. Carry out a test, at the \(10 \%\) significance level, to determine whether the result of the survey is consistent with the claim of the agency.
2. A local residents' group claims that no more than \(35 \%\) of the local population are in favour of the development. Without further calculations, state with a reason what can be said about the claim of the local residents' group.
3. A test is carried out, at the \(15 \%\) significance level, of the agency's claim. The test is based on a random sample of size \(2 n\), and exactly \(n\) of the sample are in favour of the development. Find the smallest possible value of \(n\) for which the outcome of the test is to reject the agency's claim.
   [0pt] [4] 4

6 A factory makes chocolates of different types. The proportion of milk chocolates made on any day is denoted by \(p\). It is desired to test the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.8\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : p < 0.8\). The test consists of choosing a random sample of 25 chocolates. \(\mathrm { H } _ { 0 }\) is rejected if the number of milk chocolates is \(k\) or fewer. The test is carried out at a significance level as close to \(5 \%\) as possible.

1. Use tables to find the value of \(k\), giving the values of any relevant probabilities.
2. The test is carried out 20 times, and each time the value of \(p\) is 0.8 . Each of the tests is independent of all the others. State the expected number of times that the test will result in rejection of the null hypothesis.
3. The test is carried out once. If in fact the value of \(p\) is 0.6 , find the probability of rejecting \(\mathrm { H } _ { 0 }\).
4. The test is carried out twice. Each time the value of \(p\) is equally likely to be 0.8 or 0.6 . Find the probability that exactly one of the two tests results in rejection of the null hypothesis.

8 In excavating an archaeological site, Roman coins are found scattered throughout the site.

1. State two assumptions needed to model the number of coins found per square metre of the site by a Poisson distribution. Assume now that the number of coins found per square metre of the site can be modelled by a Poisson distribution with mean \(\lambda\).
2. Given that \(\lambda = 0.75\), calculate the probability that exactly 3 coins are found in a region of the site of area \(7.20 \mathrm {~m} ^ { 2 }\). A test is carried out, at the \(5 \%\) significance level, of the null hypothesis \(\lambda = 0.75\), against the alternative hypothesis \(\lambda > 0.75\), in Region LVI which has area \(4 \mathrm {~m} ^ { 2 }\).
3. Determine the smallest number of coins that, if found in Region LVI, would lead to rejection of the null hypothesis, stating also the values of any relevant probabilities.
4. Given that, in fact, \(\lambda = 1.2\) in Region LVI, find the probability that the test results in a Type II error.

2

1. The random variable \(R\) has the distribution \(\mathrm { B } ( 6 , p )\). A random observation of \(R\) is found to be 6. Carry out a \(5 \%\) significance test of the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.45\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : p \neq 0.45\), showing all necessary details of your calculation.
2. The random variable \(S\) has the distribution \(\mathrm { B } ( n , p ) . \mathrm { H } _ { 0 }\) and \(\mathrm { H } _ { 1 }\) are as in part (i). A random observation of \(S\) is found to be 1 . Use tables to find the largest value of \(n\) for which \(\mathrm { H } _ { 0 }\) is not rejected. Show the values of any relevant probabilities.

7 Three independent researchers, \(A , B\) and \(C\), carry out significance tests on the power consumption of a manufacturer's domestic heaters. The power consumption, \(X\) watts, is a normally distributed random variable with mean \(\mu\) and standard deviation 60. Each researcher tests the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 4000\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu > 4000\). Researcher \(A\) uses a sample of size 50 and a significance level of \(5 \%\).

1. Find the critical region for this test, giving your answer correct to 4 significant figures. In fact the value of \(\mu\) is 4020 .
2. Calculate the probability that Researcher \(A\) makes a Type II error.
3. Researcher \(B\) uses a sample bigger than 50 and a significance level of \(5 \%\). Explain whether the probability that Researcher \(B\) makes a Type II error is less than, equal to, or greater than your answer to part (ii).
4. Researcher \(C\) uses a sample of size 50 and a significance level bigger than \(5 \%\). Explain whether the probability that Researcher \(C\) makes a Type II error is less than, equal to, or greater than your answer to part (ii).
5. State with a reason whether it is necessary to use the Central Limit Theorem at any point in this question.

5 The number of system failures per month in a large network is a random variable with the distribution \(\operatorname { Po } ( \lambda )\). A significance test of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 2.5\) is carried out by counting \(R\), the number of system failures in a period of 6 months. The result of the test is that \(\mathrm { H } _ { 0 }\) is rejected if \(R > 23\) but is not rejected if \(R \leqslant 23\).

1. State the alternative hypothesis.
2. Find the significance level of the test.
3. Given that \(\mathrm { P } ( R > 23 ) < 0.1\), use tables to find the largest possible actual value of \(\lambda\). You should show the values of any relevant probabilities.

6 In a rearrangement code, the letters of a message are rearranged so that the frequency with which any particular letter appears is the same as in the original message. In ordinary German the letter \(e\) appears \(19 \%\) of the time. A certain encoded message of 20 letters contains one letter \(e\).

1. Using an exact binomial distribution, test at the \(10 \%\) significance level whether there is evidence that the proportion of the letter \(e\) in the language from which this message is a sample is less than in German, i.e., less than \(19 \%\).
2. Give a reason why a binomial distribution might not be an appropriate model in this context.

6 In a city the proportion of inhabitants from ethnic group \(\mathbf { Z }\) is known to be \(\mathbf { 0 . 4 }\). A sample of \(\mathbf { 1 2 }\) employees of a large company in this city is obtained and it is found that 2 of them are from ethnic group \(Z\). A test is carried out, at the \(5 \%\) significance level, of whether the proportion of employees in this company from ethnic group \(Z\) is less than in the city as a whole.
[0pt]

1. State an assumption that must be made about the sample for a significance test to be valid. [1]
   [0pt]
2. Describe briefly an appropriate way of obtaining the sample. [2]
   [0pt]
3. Carry out the test. [7]
4. A manager believes that the company discriminates against ethnic group \(Z\). Explain whether carrying out the test at the 10\% significance level would be more supportive or less supportive of the manager's belief. [2]

8 The proportion of left-handed adults in a country is known to be \(15 \%\). It is suggested that for mathematicians the proportion is greater than \(15 \%\). A random sample of 12 members of a university mathematics department is taken, and it is found to include five who are left-handed.

1. Stating your hypotheses, test whether the suggestion is justified, using a significance level as close to \(5 \%\) as possible.
2. In fact the significance test cannot be carried out at a significance level of exactly \(5 \%\). State the probability of making a Type I error in the test.
3. Find the probability of making a Type II error in the test for the case when the proportion of mathematicians who are left-handed is actually \(20 \%\).
4. Determine, as accurately as the tables of cumulative binomial probabilities allow, the actual proportion of mathematicians who are left-handed for which the probability of making a Type II error in the test is 0.01 .

9 JUNE 2005
Morning
1 hour 30 minutes
Additional materials:
Answer booklet
Graph paper
MEI Examination Formulae and Tables (MF2) TIME 1 hour 30 minutes

• Write your name, centre number and candidate number in the spaces provided on the answer booklet.
• Answer all the questions.
• You are permitted to use a graphical calculator in this paper.

• The number of marks is given in brackets [ ] at the end of each question or part question.
• You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
• Final answers should be given to a degree of accuracy appropriate to the context.
• The total number of marks for this paper is 72.

1 A student is collecting data on traffic arriving at a motorway service station during weekday lunchtimes. The random variable \(X\) denotes the number of cars arriving in a randomly chosen period of ten seconds.

1. State two assumptions necessary if a Poisson distribution is to provide a suitable model for the distribution of \(X\). Comment briefly on whether these assumptions are likely to be valid. The student counts the number of arrivals, \(x\), in each of 100 ten-second periods. The data are shown in the table below. Carry out a test at the \(5 \%\) level of significance to examine whether there is any association between type of customer and type of drink. State carefully your null and alternative hypotheses.

4 The sexes and ages of a random sample of 300 runners taking part in marathons are classified as follows.

1. Carry out a test at the \(5 \%\) significance level to examine whether there is any association between age group and sex. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic.
2. Does your analysis support the suggestion that women are less likely than men to enter marathons as they get older? Justify your answer. For marathons in general, on average \(3 \%\) of runners are 'Female, 50 and over'. The random variable \(X\) represents the number of 'Female, 50 and over' runners in a random sample of size 300.
3. Use a suitable approximating distribution to find \(\mathrm { P } ( X \geqslant 12 )\).

4 A student is investigating whether there is any association between the species of shellfish that occur on a rocky shore and where they are located. A random sample of 160 shellfish is selected and the numbers of shellfish in each category are summarised in the table below.

1. Write down null and alternative hypotheses for a test to examine whether there is any association between species and location. The contributions to the test statistic for the usual \(\chi ^ { 2 }\) test are shown in the table below.

   Contribution		Location
   \cline { 3 - 5 }	Exposed	Sheltered	Pool
   \multirow{3}{*}{Species}	Limpet	0.0009	0.2585	0.4450
   \cline { 2 - 5 }	Mussel	10.3472	1.2756	4.8773
   \cline { 2 - 5 }	Other	8.0719	0.1402	7.4298

   The sum of these contributions is 32.85 .
2. Calculate the expected frequency for mussels in pools. Verify the corresponding contribution 4.8773 to the test statistic.
3. Carry out the test at the \(5 \%\) level of significance, stating your conclusion clearly.
4. For each species, comment briefly on how its distribution compares with what would be expected if there were no association.
5. If 3 of the 160 shellfish are selected at random, one from each of the 3 types of location, find the probability that all 3 of them are limpets.

1 A random variable has the distribution \(\mathrm { B } ( n , p )\). It is required to test \(\mathrm { H } _ { 0 } : p = \frac { 2 } { 3 }\) against \(\mathrm { H } _ { 1 } : p < \frac { 2 } { 3 }\) at a significance level as close to \(1 \%\) as possible, using a sample of size \(n = 8,9\) or 10 . Use tables to find which value of \(n\) gives such a test, stating the critical region for the test and the corresponding significance level.
[0pt] [4]

9 The random variable \(A\) has the distribution \(\mathrm { B } ( 30 , p )\). A test is carried out of the hypotheses \(\mathrm { H } _ { 0 } : p = 0.6\) against \(\mathrm { H } _ { 1 } : p < 0.6\). The critical region is \(A \leqslant 13\).

1. State the probability that \(\mathrm { H } _ { 0 }\) is rejected when \(p = 0.6\).
2. Find the probability that a Type II error occurs when \(p = 0.5\).
3. It is known that on average \(p = 0.5\) on one day in five, and on other days the value of \(p\) is 0.6 . On each day two tests are carried out. If the result of the first test is that \(\mathrm { H } _ { 0 }\) is rejected, the value of \(p\) is adjusted if necessary, to ensure that \(p = 0.6\) for the rest of the day. Otherwise the value of \(p\) remains the same as for the first test. Calculate the probability that the result of the second test is to reject \(\mathrm { H } _ { 0 }\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

5

1. State an advantage of using random numbers in selecting samples.
2. It is known that in analysing the digits in large sets of financial records, the probability that the leading digit is 1 is 0.25 . A random sample of 18 leading digits from a certain large set of financial records is obtained and it is found that 8 of the leading digits are 1 s . Test, at the \(5 \%\) significance level, whether the probability that the leading digit is 1 in this set of records is greater than 0.25 .

6 It is known that \(25 \%\) of students in a particular city are smokers. A random sample of 20 of the students is selected.

1. (A) Find the probability that there are exactly 4 smokers in the sample.
   (B) Find the probability that there are at least 3 but no more than 6 smokers in the sample.
   (C) Write down the expected number of smokers in the sample. A new health education programme is introduced. This programme aims to reduce the percentage of students in this city who are smokers. After the programme has been running for a year, it is decided to carry out a hypothesis test to assess the effectiveness of the programme. A random sample of 20 students is selected.
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 3 smokers in the sample. Complete the test, stating your conclusion clearly.