One-tailed test (increase or decrease)

6 The number of fruit pips in 1 cubic centimetre of raspberry jam has the distribution \(\operatorname { Po } ( \lambda )\). Under a traditional jam-making process it is known that \(\lambda = 6.3\). A new process is introduced and a random sample of 1 cubic centimetre of jam produced by the new process is found to contain 2 pips. Test, at the \(5 \%\) significance level, whether this is evidence that under the new process the average number of pips has been reduced. Find (a) \(\mathrm { E } ( X )\),
(ii) The continuous random variable \(Y\) has the probability density function $$g ( y ) = \left\{ \begin{array} { l r } \frac { 1.5 } { y ^ { 2.5 } } & y \geqslant 1 \\ 0 & \text { otherwise. } \end{array} \right.$$ Given that \(\mathrm { E } ( Y ) = 3\), show that \(\operatorname { Var } ( Y )\) is not finite.

7 A large railway network suffers points failures at an average rate of 1 every 3 days. Assume that the number of points failures can be modelled by a Poisson distribution. The network employs a new firm of engineers. After the new engineers have become established, it is found that in a randomly chosen period of 15 days there are 2 instances of points failures.

1. Test, at the \(5 \%\) significance level, whether there is evidence that the mean number of points failures has been reduced.
2. A new test is carried out over a period of 150 days. Use a suitable approximation to find the greatest number of points failures there could be in 150 days that would lead to a \(5 \%\) significance test concluding that the average number of points failures had been reduced.

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.

4 The number of floods in a certain river plain is known to have a Poisson distribution. It is known that up until 10 years ago the mean number of floods per year was 0.32 . During the last 10 years there were 6 floods. Test at the \(1 \%\) significance level whether there is evidence of an increase in the mean number of floods per year.

8 In a small town, the number of properties sold during a week in spring by a local estate agent, Keith, can be regarded as occurring independently and with constant mean \(\mu\). Data from several years have shown the value of \(\mu\) to be 3.5 . A new housing development was built on the outskirts of the town and the properties on this development were offered for sale by the builder of the development, not by the local estate agents. During the first four weeks in spring, when properties on the new development were offered for sale by the builder, Keith sold a total of 8 properties. Keith claims that the sale of new properties by the builder reduced his mean number of properties sold during a week in spring. 8

1. Investigate Keith's claim, using the \(5 \%\) level of significance.
   [0pt] [6 marks]
   8
2. For your test carried out in part (a) state, in context, the meaning of a Type II error.
   [0pt] [1 mark]
   8
3. State one advantage and one disadvantage of using a 1\% significance level rather than a 5\% level of significance in a hypothesis test.
   [0pt] [2 marks]

2. John weaves cloth by hand. Emma believes that faults are randomly distributed in John's cloth at a rate of more than 4 per 50 metres of cloth. To check her belief, Emma takes a random sample of 100 metres of the cloth and finds that it contains 14 faults.

1. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, Emma's belief. Armani also weaves cloth by hand. He knows that faults are randomly distributed in his cloth at a rate of 4 per 50 metres of cloth. Emma decides to buy a large amount of Armani's cloth to sell in pieces of length \(l\) metres. She chooses \(l\) so that the probability of no faults in a piece is exactly 0.9
2. Show that \(l = 1.3\) to 2 significant figures. Emma sells 5000 of these pieces of cloth of length 1.3 metres. She makes a profit of \(\pounds 2.50\) on each piece of cloth that does not contain any faults but a loss of \(\pounds 0.50\) on any piece that contains at least one fault.
3. Find Emma's expected profit.

4. Accidents occur randomly at a crossroads at a rate of 0.5 per month. A researcher records the number of accidents, \(X\), which occur at the crossroads in a year.

1. Find \(\mathrm { P } ( 5 \leqslant X < 7 )\) A new system is introduced at the crossroads. In the first 18 months, 4 accidents occur at the crossroads.
2. Test, at the \(5 \%\) level of significance, whether or not there is reason to believe that the new system has led to a reduction in the mean number of accidents per month. State your hypotheses clearly.

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2. The number of houses sold per week by a firm of estate agents follows a Poisson distribution with mean 2.5. The firm appoints a new salesman and wants to find out whether or not house sales increase as a result. After the appointment of the salesman, the number of house sales in a randomly chosen 4-week period is 14. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not the new salesman has increased house sales.

6. Over a long period of time, accidents happened on a stretch of road at random at a rate of 3 per month. Find the probability that

1. in a randomly chosen month, more than 4 accidents occurred,
2. in a three-month period, more than 4 accidents occurred. At a later date, a speed restriction was introduced on this stretch of road. During a randomly chosen month only one accident occurred.
3. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the claim that this speed restriction reduced the mean number of road accidents occurring per month. The speed restriction was kept on this road. Over a two-year period, 55 accidents occurred.
4. Test, at the \(5 \%\) level of significance, whether or not there is now evidence that this speed restriction reduced the mean number of road accidents occurring per month.

1. Richard regularly travels to work on a ferry. Over a long period of time, Richard has found that the ferry is late on average 2 times every week. The company buys a new ferry to improve the service. In the 4-week period after the new ferry is launched, Richard finds the ferry is late 3 times and claims the service has improved. Assuming that the number of times the ferry is late has a Poisson distribution, test Richard's claim at the \(5 \%\) level of significance. State your hypotheses clearly.
   (6)
2. A continuous random variable \(X\) has the probability density function \(\mathrm { f } ( x )\) shown in Figure 1.

\begin{figure}[h] \end{figure}

1. Show that \(\mathrm { f } ( x ) = 4 - 8 x\) for \(0 \leqslant x \leqslant 0.5\) and specify \(\mathrm { f } ( x )\) for all real values of \(x\).
2. Find the cumulative distribution function \(\mathrm { F } ( x )\).
3. Find the median of \(X\).
4. Write down the mode of \(X\).
5. State, with a reason, the skewness of \(X\).

7. (a) Explain briefly what you understand by

1. a critical region of a test statistic,
2. the level of significance of a hypothesis test.
   (b) An estate agent has been selling houses at a rate of 8 per month. She believes that the rate of sales will decrease in the next month.
3. Using a \(5 \%\) level of significance, find the critical region for a one tailed test of the hypothesis that the rate of sales will decrease from 8 per month.
4. Write down the actual significance level of the test in part (b)(i). The estate agent is surprised to find that she actually sold 13 houses in the next month. She now claims that this is evidence of an increase in the rate of sales per month.
   (c) Test the estate agent's claim at the \(5 \%\) level of significance. State your hypotheses clearly.

5. (a) Explain what you understand by a critical region of a test statistic. The number of breakdowns per day in a large fleet of hire cars has a Poisson distribution with mean \(\frac { 1 } { 7 }\).
(b) Find the probability that on a particular day there are fewer than 2 breakdowns.
(c) Find the probability that during a 14-day period there are at most 4 breakdowns. The cars are maintained at a garage. The garage introduced a weekly check to try to decrease the number of cars that break down. In a randomly selected 28-day period after the checks are introduced, only 1 hire car broke down.
(d) Test, at the \(5 \%\) level of significance, whether or not the mean number of breakdowns has decreased. State your hypotheses clearly.

2. An effect of a certain disease is that a small number of the red blood cells are deformed. Emily has this disease and the deformed blood cells occur randomly at a rate of 2.5 per ml of her blood. Following a course of treatment, a random sample of 2 ml of Emily's blood is found to contain only 1 deformed red blood cell. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test whether or not there has been a decrease in the number of deformed red blood cells in Emily's blood.

2. A traffic officer monitors the rate at which vehicles pass a fixed point on a motorway. When the rate exceeds 36 vehicles per minute he must switch on some speed restrictions to improve traffic flow.

1. Suggest a suitable model to describe the number of vehicles passing the fixed point in a 15 s interval. The traffic officer records 12 vehicles passing the fixed point in a 15 s interval.
2. Stating your hypotheses clearly, and using a \(5 \%\) level of significance, test whether or not the traffic officer has sufficient evidence to switch on the speed restrictions.
3. Using a \(5 \%\) level of significance, determine the smallest number of vehicles the traffic officer must observe in a 10 s interval in order to have sufficient evidence to switch on the speed restrictions.

5. A traffic analyst is interested in the number of heavy lorries passing a certain junction. He counts the numbers of lorries in 100 five-minute intervals, and gets the following results: Q. 5 continued on next page ... \section*{STATISTICS 2 (A) TEST PAPER 9 Page 2}

1. continued ...
   1. Show that the mean of \(X\) is 3 , and find the variance of \(X\).
   2. Give two reasons for thinking that \(X\) can be modelled by a Poisson distribution. (2 marks)
   After a new landfill site has been established nearby, a member of an environmental group notices that 18 lorries pass the junction in a period of 15 minutes. The group claims that this is evidence that the mean number of lorries per five-minute interval has increased.
2. Test whether the group's claim is valid. Work at the \(5 \%\) significance level, and state your hypotheses clearly.

4. A rugby player scores an average of 0.4 tries per match in which he plays.

1. Find the probability that he scores 2 or more tries in a match. The team's coach moves the player to a different position in the team believing he will then score more frequently. In the next five matches he scores 6 tries.
2. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence of an increase in the number of tries the player scores per match as a result of playing in a different position.
   (5 marks)

6. A teacher is monitoring attendance at lessons in her department. She believes that the number of students absent from each lesson follows a Poisson distribution and wished to test the null hypothesis that the mean is 2.5 against the alternative hypothesis that it is greater than 2.5 She visits one lesson and decides on a critical region of 6 or more students absent.

1. Find the significance level of this test.
2. State any assumptions made in carrying out this test and comment on their validity. The teacher decides to undertake a wider study by looking at a sample of all the lessons that have taken place in the department during the previous four weeks.
3. Suggest a suitable sampling frame. She finds that there have been 96 pupils absent from the 30 lessons in her sample.
4. Using a suitable approximation, test at the \(5 \%\) level of significance the null hypothesis that the mean is 2.5 students absent per lesson against the alternative hypothesis that it is greater than 2.5. You may assume that the number of absences follows a Poisson distribution.
   (6 marks)

7 The daily number of customers visiting a small arts and crafts shop may be modelled by a Poisson distribution with a mean of 24 .

1. Using a distributional approximation, estimate the probability that there was a total of at most 150 customers visiting the shop during a given 6-day period.
2. The shop offers a picture framing service. The daily number of requests, \(Y\), for this service may be assumed to have a Poisson distribution. Prior to the shop advertising this service in the local free newspaper, the mean value of \(Y\) was 2. Following the advertisement, the shop received a total of 17 requests for the service during a period of 5 days.
   1. Using a Poisson distribution, carry out a test, at the \(10 \%\) level of significance, to investigate the claim that the advertisement increased the mean daily number of requests for the shop's picture framing service.
   2. Determine the critical value of \(Y\) for your test in part (b)(i).
   3. Hence, assuming that the advertisement increased the mean value of \(Y\) to 3, determine the power of your test in part (b)(i).

7

1. The random variable \(X\) has a Poisson distribution with parameter \(\lambda\).
   1. Prove, from first principles, that \(\mathrm { E } ( X ) = \lambda\).
   2. Given that \(\mathrm { E } \left( X ^ { 2 } - X \right) = \lambda ^ { 2 }\), deduce that \(\operatorname { Var } ( X ) = \lambda\).
2. The number of faults in a 100-metre ball of nylon string may be modelled by a Poisson distribution with parameter \(\lambda\).
   1. An analysis of one ball of string, selected at random, showed 15 faults. Using an exact test, investigate the claim that \(\lambda > 10\). Use the \(5 \%\) level of significance.
   2. A subsequent analysis of a random sample of 20 balls of string showed a total of 241 faults.
      (A) Using an approximate test, re-investigate the claim that \(\lambda > 10\). Use the \(5 \%\) level of significance.
      (B) Determine the critical value of the total number of faults for the test in part (b)(ii)(A).
      (C) Given that, in fact, \(\lambda = 12\), estimate the probability of a Type II error for a test of the claim that \(\lambda > 10\) based upon a random sample of 20 balls of string and using the \(5 \%\) level of significance.
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5. (a) Define

1. a Type I error,
2. a Type II error. A small aviary, that leaves the eggs with the parent birds, rears chicks at an average rate of 5 per year. In order to increase the number of chicks reared per year it is decided to remove the eggs from the aviary as soon as they are laid and put them in an incubator. At the end of the first year of using an incubator 7 chicks had been successfully reared.
   (b) Assuming that the number of chicks reared per year follows a Poisson distribution test, at the \(5 \%\) significance level, whether or not there is evidence of an increase in the number of chicks reared per year. State your hypotheses clearly.
   (c) Calculate the probability of the Type I error for this test.
   (d) Given that the true average number of chicks reared per year when the eggs are hatched in an incubator is 8 , calculate the probability of a Type II error.

1. The number of accidents per year in Daftstown follows a Poisson distribution with mean \(\lambda\). The value of \(\lambda\) has previously been 6 but Jonty claims that since the Council increased the speed limit, the value of \(\lambda\) has increased.

Jonty records the number of accidents in Daftstown in the first year after the speed limit was increased. He plans to test, at the \(5 \%\) significance level, whether or not there is evidence of an increase in the mean number of accidents in Daftstown per year.

1. Stating your hypotheses clearly, calculate the probability of a Type I error for this test. Given that there were 9 accidents in the first year after the speed limit was increased,
2. state, giving a reason, whether or not there is evidence to support Jonty's claim.
3. Given that the value of \(\lambda\) has actually increased to 8, calculate the probability of drawing the conclusion, using this test, that the number of accidents per year in Daftstown has not increased.

1. The number of customers entering Jeff's supermarket each morning follows a Poisson distribution.

Past information shows that customers enter at an average rate of 2 every 5 minutes.
Using this information,

1. 1. find the probability that exactly 26 customers enter Jeff's supermarket during a randomly selected 1-hour period one morning,
   2. find the probability that at least 21 customers enter Jeff's supermarket during a randomly selected 1-hour period one morning. A rival supermarket is opened nearby. Following its opening, the number of customers entering Jeff's supermarket over a randomly selected 40-minute period is found to be 10
2. Test, at the 5\% significance level, whether or not there is evidence of a decrease in the rate of customers entering Jeff's supermarket. State your hypotheses clearly. A further randomly selected 20 -minute period is observed and the hypothesis test is repeated. Given that the true rate of customers entering Jeff's supermarket is now 1 every 5 minutes,
3. calculate the probability of a Type II error.