Teaching guidance AS and A-level Further Maths

Transcription

1 Teaching guidance AS and A-level Further Maths (7366, 7367) Statistics Download the full set of specimen papers, specifications and resources at aqa.org.uk/teachingguidance Version 1.0, August 2017

2 Our specification is published on our website (aqa.org.uk). We will let centres know in writing about any changes to the specification. We will also publish changes on our website. The definitive version of our specification will always be the one on our website and may differ from printed versions. You can download a copy of this teaching guidance from our All About Maths website (allaboutmaths.aqa.org.uk/). This is where you will find the most up-to-date version, as well as information on version control.

3 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE Contents General information - disclaimer 5 Subject content 5 SA Discrete random variables (DRVs) and expectation 6 SB Poisson distribution 21 SC Type I and Type II errors 31 SD Continuous random variables (CRVs) 35 SE Chi squared tests for association 54 Exponential distribution SF 68 SG Inference: one sample t-distribution 75 SH Confidence intervals 78 A1 Appendix 1 Mathematical notation for AS and A-level 86 qualifications in Mathematics and Further Mathematics A2 Appendix 2 Mathematical formulae and identities 95 Version 1.0 3

4 4 Version 1.0

5 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE General information - disclaimer This AS and A-level Further Mathematics teaching guidance will help you plan your teaching by further explaining how we have interpreted content of the specification and providing examples of how the content of the specification may be assessed. The teaching guidance notes do not always cover the whole content statement. The examples included in this guidance have been chosen to illustrate the level at which this content will be assessed. The wording and format used in this guidance do not always represent how questions would appear in a question paper. Not all questions in this guidance have been through the same rigorous checking process as the ones used in our question papers. Several questions have been taken from legacy specifications and therefore represent higher levels of AO1 than will be found in a suite of exam papers for these AS and A-level Further Mathematics specification. This guidance is not, in any way, intended to restrict what can be assessed in the question papers based on the specification. Questions will be set in a variety of formats including both familiar and unfamiliar contexts. All knowledge from the GCSE Mathematics specification is assumed. Subject content This Teaching guidance is designed to illustrate the detail within the content of the AS and A-level Further Mathematics specification. Half the subject content was set out the Department for Education (DfE). The remaining half was defined by AQA, based on feedback from Higher Education and teachers. Content in bold type is contained within the AS Further Mathematics qualification as well as the A- level Further Mathematics qualification. Content in standard type is contained only within the A-level Further Mathematics qualification. Version 1.0 5

6 SA Discrete random variables (DRVs) and expectation SA1 Understand DRVs with distributions given in the form of a table or function. Assessed at AS and A-level Teaching guidance Students should be able to: recognise that discrete random variables can only take specified individual values and cannot take any values in between them understand and use the fact that the sum of the probabilities of all of the possible outcomes is 1 use given formulae to evaluate the probabilities of the possible values that a discrete random variable with finite possibilities can take. Examples 1 The number of fish, X, caught by Pearl when she goes fishing can be modelled by the following discrete probability distribution: x P(X = x) k Find the value of k. 6 Version 1.0

7 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE 2 Past experience shows that, when they play home matches, Newville Athletic football team win half the matches and lose 20% of the matches. They draw the remaining home matches. The team is awarded three points for a win, one point for a draw and no points for a loss. The table below shows the probability distribution of X, the number of points awarded to Newville Athletic after a home match. Complete the table. x P(X = x) A small supermarket has a total of four checkouts, at least one of which is always staffed. The probability distribution for R, the number of checkouts that are staffed at any given time, is 2 1 P( R = r) = 3 3 k r 1 r = 1, 2, 3 r = 4 Show that k 1 = 27 4 In a computer game, players try to collect 5 treasures. The number of treasures that Isaac collects in one play of the game is represented by the discrete random variable X. The probability distribution of X is defined by 1 x = 1, 2, 3, 4 x + 2 P( X = x ) = k x = 5 0 otherwise Show that k 1 = 20 Version 1.0 7

8 SA2 Evaluate probabilities for a DRV. Assessed at AS and A-level Teaching guidance Students should be able to use tables of probabilities, or a given random variable probability density function, to evaluate probabilities. Examples 1 The number of strokes, R, taken by the members of Duffers Golf Club to complete the first hole may be modelled by the following discrete probability distribution. r P(R = r) Determine the probability that a member, selected at random, takes at least 5 strokes to complete the first hole. 2 Marian belongs to the Handchester Building Society. She frequently visits her local branch to pay instalments on her mortgage. The number of people queuing to be served when she enters the branch may be modelled by the random variable, X, with the following probability distribution. x P(X = x) Find the probability that when she enters the branch there are 2 or more people queuing to be served. 8 Version 1.0

9 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE 3 A discrete random variable, X, has the probability distribution: Calculate P( X 5 ) P x x = 1,2,3,4,5 20 x X = x = x = otherwise ( ) 4 Alex studies 5 different subjects at school each weekday, Monday to Friday. The number of pieces of homework, X, which Alex is given each day follows the distribution shown in the table. x P(X = x) Find the probability that, on a particular day, Alex is given: (a) more than 3 pieces of homework at least 1 but fewer than 4 pieces of homework. Version 1.0 9

10 SA3 Evaluate measures of average and spread for a DRV to include mean, variance, standard deviation, mode and median. Assessed at AS and A-level Teaching guidance Students should be able to: use tables of probabilities or the probability density function of a discrete random variable to find the mean, median or mode use tables of probabilities or the probability density function of a discrete random variable to find standard deviation. Examples 1 The number of strokes, R, taken by the members of Duffers Golf Club to complete the first hole may be modelled by the following discrete probability distribution. r P(R = r) Calculate E(R). 2 Imran wishes to buy a house in Cheadleville. The number of houses, X, in Cheadleville advertised for sale in a copy of the Cheshire Weekly Sentinel may be modelled by the following probability distribution. x P(X = x) Find the mean and standard deviation of X. 10 Version 1.0

11 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE 3 A fire station has four fire engines available. The number of fire engines, which leave the station in response to emergency calls on a typical night shift, may be modelled by the variable, X, with the following probability distribution. x P(X = x) Find the mean and standard deviation of X. 4 The weekly number of visits Wendell makes to the cinema may be modelled by the discrete random variable, Y, with the following probability distribution. y P(Y = y) Find: (a) (c) the mean of Y the standard deviation of Y the modal value of Y (d) the median value of Y. Version

12 SA4 Understand expectation and know the formulae: ( X) xp i i = 2 2 E E ( ) i i 2 ( X) = ( X ) ( ( X) ) Var E E 2 X = x p Assessed at AS and A-level Teaching guidance Students should: know and be able to use the formula E( X) xp i i random variable = 2 2 know and be able to use the formula E( ) i i X to find the value of the mean of a discrete = x p for a discrete random variable 2 know and be able to use the formula ( X) = ( X ) ( ( X) ) Notes for a discrete random variable. 2 Var E E to find the value of the variance Students may calculate the mean and standard deviation using the statistical functions of their calculator. The instruction to show that in a question will mean that the formula should be used and sufficient working shown to justify this. Examples 1 A bank has an ATM which customers can use to withdraw cash. Withdrawals can be made in fixed amounts, X. the table shows the amounts available and the probability distribution for X. x P(X = x) (a) Find the value of E(X). Show that the standard deviation of X is 68.0, correct to three significant figures. 12 Version 1.0

13 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE 2 An analysis of sales throughout the year suggests that the probability distribution tabulated below would form an adequate model for X, the number of tricycles sold during a randomly selected week. x P(X = x) (a) Find the mean of X. Show that E(X 2 ) = The amount charged, X, for entry to an exhibition depends on the status of the visitor. The following table shows the charges together with the probability that a visitor will have a particular status. Status Charge ( x) P(X=x) Child under Student Senior citizen Adult For entrance charges paid by visitors to the exhibition, calculate E(X 2 ). 4 A discrete random variable X has probability distribution as given in the table. x P(X = x) p p p 1 3p (a) Find an expression, in terms of p, for E(X). Show that Var(X) = 2p(7 18p). Version

14 SA5 Understand expectation of linear functions of DRVs and know the formulae: E(aX + b) = ae(x) + b Var(aX + b) = a 2 Var(X) Know the formula E(g(X)) = g(x i )p i Find the mean, variance and standard deviation for functions of a DRV such as E(5X 3 ), E(18X 3 ), Var(6X 1 ). Assessed at AS and A-level Teaching guidance Students should be able to: use known, or found, values of E(X) and Var(X) to find the values of the mean and variance of a linear function of X, using the formula E(aX + b) = a E(X) + b and Var(aX + b) = a 2 Var(X) know and use the formula E ( ) ( ) ( ) g X = g x i p i Examples 1 The Globe Express agency organises trips to the theatre. The cost, X, of these trips can be modelled by the following probability distribution: x P(X = x) (a) Calculate the mean and standard deviation of X. For special celebrity charity performances, Globe Express increases the cost of the trips to Y, where Y = 10X Determine the mean and standard deviation of Y. 14 Version 1.0

15 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE 2 A small supermarket has a total of four checkouts, at least one of which is always staffed. The probability distribution for R, the number of checkouts that are staffed at any given time, is: 2 1 P ( R = r) = 3 3 k r 1 r = 1, 2, 3 r = 4 (a) Find the probability that, at any given time, there will be at least 3 checkouts that are staffed. It is suggested that the total number of customers, C, that can be served at the checkouts per hour may be modelled by C = 27R + 5 Find: (i) E(C). (ii) the standard deviation of C. 3 The number of fish, X, caught by Pearl when she goes fishing can be modelled by the following discrete probability distribution: x P(X = x) k (a) Find the value of k. Find: (i) (ii) E(X) Var(X). (c) When Pearl sells her fish, she earns a profit, in pounds, given by Y = 5X + 2 Find: (i) E(Y) (ii) the standard deviation of Y. Version

16 Assessed at A-level only Teaching guidance Students should be able to calculate the mean, variance or standard deviation for functions of X of the form ax b, where a and b are integers. Examples 4 A discrete random variable X has the probability distribution: x x = 1,2,3,4,5 20 x P ( X = x ) = x = otherwise (a) (i) Show that 1 7 E = X 24 (ii) Hence, or otherwise, show that 1 Var = X correct to three decimal places. Calculate the mean and the variance of A, the area of rectangles that have sides of length 1 X + 3 and X. 16 Version 1.0

17 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE SA6 Know the discrete uniform distribution defined on the set {1, 2,, n}. Understand when this distribution can be used as a model. Assessed at AS and A-level Teaching guidance Students should: understand and use the fact that a discrete uniform distribution is where all the possible values of X have equal probabilities of happening know that the probability density function of a uniform discrete random variable is of the form: P 1 for i = 1,2,3,..., n = = n 0 otherwise ( X i) know that a discrete uniform distribution may be used as a model when every possible outcome of an event is equally likely to occur (eg when rolling a fair die). Examples 1 Each side of a regular, four-sided tetrahedron is in the shape of an equilateral triangle. The numbers 1, 2, 3 and 4 are written with one on each side of the four faces. When the tetrahedron is thrown at random to fall on horizontal ground, the number on the face in contact with the ground is recorded. (a) Complete the probability table below, where X is the variable representing the number on the face that is in contact with the ground. x P(X = x) (c) Name the distribution represented in this probability table. Find the probability that a randomly selected value of X is even. Version

18 3 A fair die is rolled on a horizontal table and the discrete random variable, X, is the score on the uppermost face of the die when it stops rolling. The value of X is recorded for each roll of the die. (a) Name the distribution used to represent the probability of each of the scores. The die is rolled five times. Find the probability that a score of 6 is recorded for exactly two of the rolls of the die. 4 A fair coin is tossed five times. The discrete random variable, X, is the number of heads recorded for the five tosses of the coin. Can X be represented by a discrete uniform distribution? Justify your answer. 18 Version 1.0

19 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE SA7 Proof of mean and variance of discrete uniform distribution. Assessed at AS and A-level Teaching guidance Students should be able to: show that the mean, E(X), of a discrete uniform distribution is E( X) = ( n+ 1) show that the variance, Var(X), of a discrete uniform distribution is Var ( X) = ( n 1)( n+ 1) Examples 1 A discrete uniform random variable, X, has probability density function defined by Show that E(X) = 2.5 P 1 for i = 1, 2, 3, 4 = = 4 0 otherwise ( X i) 2 A discrete uniform random variable, X, has probability density function defined by P 1 for i = 1,2,3,4,5,6 = = 6 0 otherwise ( X i) The value of E(X) is 7 2 Show that the exact value of Var(X) is Version

20 3 A discrete uniform random variable, X, has probability density defined by Show that: P 1 for i= 1,2,3,4,5 = = 5 0 otherwise ( X i) (a) E(X) = 3 Var(X) = 2. 4 A computer is programmed to generate single digit random integers (ie 0, 1, 2,,9). Assume that each integer is equally likely to occur. (a) Complete the table below, representing the possible values that can be generated, together with their corresponding probabilities. x P(X = x) Show that: (i) E(X) = 4.5 (ii) Var(X) = A discrete uniform random variable, X, can take the values 0, 1, 2, n. Prove that Var ( X ) ( 2) = n n Version 1.0

21 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE SB Poisson distribution SB1 Understand conditions for a Poisson distribution to model a situation. Understand terminology X ~ Po(λ) Assessed at AS and A-level Teaching guidance Students should be able to: state and understand that the Poisson distribution is a discrete random variable recognise, understand and use the notation X ~ Po(λ) as the discrete random variable representing the number of outcomes of a Poisson event in a given interval where λ is the average rate of occurrence of the event in a given interval state, in the context of the given situation, necessary and sufficient conditions for a discrete random variable to follow a Poisson distribution, namely: there can only be a whole number of outcomes of the event in the given interval outcomes of the event occur at a constant average rate outcomes of the event occur independently and at random outcomes of the event occur singly. appreciate that in a theoretical Poisson distribution that there is no upper limit of the number of outcomes of an event in the given interval, but that this is not likely to be true in the real world. The Poisson distribution models the real situation appreciate that for outcomes of a Poisson event to occur independently and at random there must be no queues present (eg cars on a motorway must be able to overtake freely or calls to a switchboard must be able to be dealt with instantly (not in a queuing system) for such variables to be modelled by a Poisson distribution). Examples 1 State necessary and sufficient conditions for the number of cars passing a fixed point on a free flowing motorway, in a given 10-minute interval, to be modelled by a Poisson distribution. Version

22 2 During the football season, an amateur football club holds training sessions for its first team squad on Tuesdays and Thursdays. The number of squad members who do not attend the Tuesday training session may be modelled by a Poisson distribution with mean 3.2. The number of squad members who do not attend the Thursday training session may be modelled by a Poisson distribution with mean 3.8. (a) Give a reason why, although the Poisson distribution may provide an adequate model, it cannot provide an exact model for the number of squad members who do not attend training. Give one possible reason why the number of squad members who attend on Thursday may not be independent of the number who attend on Tuesday. 3 Critically assess whether or not the following situations are likely to be able to be modelled by a Poisson distribution: (a) (c) The number of cars passing a fixed point on a free flowing motorway during a given 5-minute interval. The number of telephone calls arriving at the switchboard of a football club during the first hour of telephone sales of tickets going live for the next round of the cup competition they are in, after they have just won the previous match. The number of cars passing a point on a single carriage road, where there are road works with traffic light control, on the way in to a town between 8:00 am and 8:30 am on a Tuesday in April. 4 State which of the following isn t a necessary condition for a discrete random variable to be modelled by a Poisson distribution. Circle your answer. Events occur at a constant average rate There can only be a fixed number of trials Outcomes of the event occur independently and at random Outcomes of the event occur singly 22 Version 1.0

23 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE SB2 Know the Poisson formula and calculate Poisson probabilities using the formula or equivalent calculator function. Assessed at AS and A-level Teaching guidance Students should: x λ λ = =e and be able to x! calculate the value of this for specific values of x and λ, to include finding such a probability directly by using the statistical functions on a calculator know and be able to use the fact that if X ~ Po(λ) then P( X x) be able to find cumulative probabilities for a discrete random variable modelled by a Poisson distribution from the use of statistical functions on a calculator. Examples 1 During the football season, an amateur football club holds training sessions for its first team squad on Tuesdays and Thursdays. The number of squad members who do not attend the Tuesday session may be modelled by a Poisson distribution with mean 3.2. (a) Find the probability that, for a particular Tuesday training session: (i) (ii) 6 or more squad members do not attend. the entire first team squad does attend. The number of squad members who do not attend the Thursday training session may be modelled by a Poisson distribution with mean 3.8. Find the probability that, for a particular Thursday training session, the number of squad members who do not attend is 2 or fewer. Version

24 2 A bus to the city centre is scheduled to stop at Beech Road at 9:30 am on weekdays. The number of passengers getting on this bus at Beech Road may be modelled by a Poisson distribution with mean 8.5. (a) Find the probability that on a particular weekday morning the number of passengers getting on this bus at Beech Road is: (i) 6 or fewer (ii) more than 9. The number of passengers getting off this bus at Beech Road on weekday mornings is independent of the number getting on and may be modelled by a Poisson distribution with mean 0.5. Find the probability that on a particular weekday morning exactly 2 passengers will get off this bus at Beech Road. 3 The number of people entering a supermarket may be modelled by a Poisson distribution with mean 2.4 per minute. Find the probability that, during a particular minute: (a) 3 or fewer people enter the supermarket exactly 3 people enter the supermarket. 4 The number of telephone calls, X, received per hour for Dr Able may be modelled by a Poisson distribution with mean 6. Determine P(X = 8). 24 Version 1.0

25 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE SB3 Know mean, variance and standard deviation of a Poisson distribution. Use the result that if X ~ Po(λ) then the mean and variance of X are equal. Assessed at AS and A-level Teaching guidance Students should be able to: understand and use the fact that if X ~ Po(λ) then the mean and the variance are equal: mean E ( X ) = = λ variance Var ( X ) = = λ recall and use the fact that the variance is the square of the standard deviation use the fact that the mean and variance of a Poisson distribution are equal to determine whether a sample could be from a Poisson distribution. Examples 1 The number of telephone calls, Y, received per hour for Dr Braken may be modelled by a Poisson distribution with mean λ and standard deviation 3. (a) Write down the value of λ. Determine P( λ ) Y>. 2 The number of computers bought from the Choicebuy computer store over a 10-day period are recorded as: (a) Calculate the mean and variance of these data. State, giving a reason based on your results in part (a), whether or not a Poisson distribution provides a suitable model for these data. Version

26 3 A new information technology centre is advertising places on its one-week residential computer courses. (a) The number of places booked each week on the publishing course, X, may be modelled by a Poission distribution with a mean of 9.0. (i) State the standard deviation of X. (ii) Calculate P(6 < X < 12). The number of places booked on the database course during each of a random sample of 10 weeks is as follows: By calculating appropriate numerical measures, state with a reason, whether or not the Poisson distribution Po(12.0) could provide a suitable model for the number of places booked each week on the database course. 26 Version 1.0

27 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE SB4 Understand the distribution of the sum of independent Poisson distributions. Assessed at AS and A-level Teaching guidance Students should be able to: recall and use the fact that the sum of independent Poisson distributions is itself a Poisson Po λ Y Po λ X +Y Po λ + λ distribution, ie if X ( x ) and ( y ), then ( x y) use this knowledge to scale up or scale down a Poisson distribution, ie if Po( λx λy) ( λ ) m X Po m, where + m is a positive integer or a positive fraction. X +Y + then Examples 1 During the football season, an amateur football club holds training sessions for its first team squad on Tuesdays and Thursdays. The number of squad members who do not attend the Tuesday training session may be modelled by a Poisson distribution with mean 3.2. The number of squad members who do not attend the Thursday training session may be modelled by a Poisson distribution with mean 3.8. Find the probability that, in a particular week, the number of squad members who do not attend the Tuesday training session plus the number of squad members who do not attend the Thursday training session is less than 2. Assume that the number who do not attend on Thursday is independent of the number who do not attend on Tuesday. 2 The number of people entering a supermarket may be modelled by a Poisson distribution with mean 2.4 per minute. Find the probability that, during a five minute interval, more than 10 people enter the supermarket. 3 The number of telephone calls, X, received per hour for Dr Able may be modelled by a Poisson distribution with mean 6. The number of telephone calls, Y, received per hour for Dr Bracken may be modelled by a Poisson distribution with mean λ and standard deviation 3. (a) Assuming that X and Y are independent Poisson variables, write down the distribution of the total number of telephone calls received per hour for Dr Able and Dr Bracken. Determine the probability that a total of at most 20 telephone calls will be received during any one hour period. Version

28 4 The number of computers, A, bought during one day from the Amplebuy computer store can be modelled by a Poisson distribution with a mean of 3.5. The number of computers, B, bought during one day from the Bestbuy computer store can be modelled by a Poisson distribution with a mean of 5.0. Find the probability that a total of fewer than 10 computers is bought from these two stores on one particular day. 28 Version 1.0

29 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE SB5 Formulate hypotheses and carry out a hypothesis test of a population mean from a single observation from a Poisson distribution using direct evaluation of Poisson probabilities. Assessed at AS and A-level Teaching guidance Students should be able to: state the null hypothesis, H 0, and the alternative hypothesis, H 1, for the given context, and appreciate that the value of the mean used in these may be either for the original interval or for the interval used in context, but for later calculations use the value of λ for the interval under consideration know whether a 1-tail test or a 2-tail test is being carried out understand and use the fact that the significance level of the test is the probability corresponding to the alternative hypothesis, H 1 if a lower tail is being examined, find P(observed value or fewer), ie the p-value, from ( ) Po λ if an upper tail is being examined, find P(observed value or more), ie the p-value from ( ) Po λ compare the probability found (p-value) with the appropriate probability corresponding to H 1 decide whether to accept or reject the null hypothesis and state their conclusion, in context. Examples 1 Maev, who works in a supermarket, was recently appointed to be responsible for fresh fruit and vegetables. During the previous year, the supermarket received an average of eight s per week complaining about the quality of the fruit and vegetables sold. The number of such s may be modelled by a Poisson distribution. During the week before Maev s appointment, 16 such s were received. Examine, using exact probabilities and the 5% level, whether there is significant evidence that, immediately before Maev s appointment, the mean number of such s received exceeded eight per week. Version

30 2 A mill produces cloth in 100-metre lengths. It is common for lengths to contain faults that have to be treated before the cloth is sold. These faults are distributed over the cloth independently, at random and at a constant average rate. (a) Name a distribution that will provide a suitable model for the number of faults in a 100-metre length of cloth. A new manager states that it is unacceptable for the mean number of faults per 100-metre length of cloth to exceed 2. The next 100-metre length of cloth produced contains 3 faults. Carry out a hypothesis test, using the 10% significance level, to test the hypothesis that the mean number of faults per 100-metre length of cloth does not exceed 2. 3 Dorian is a biologist who is concerned about the effects of recent pollution on the wildlife population in a roadside meadow. He is particularly interested in the effects on sub-species of lob worm, meadow ant and ground beetle. To investigate this, Dorian uses a one metre square frame, known as a quadrat, which he randomly positions in the meadow. At each position, the numbers of these worms, ants and beetles found in the quadrat are recorded. You may assume that the number of lob worms found in a quadrat can be modelled by a Poisson distribution. (a) What does this assumption tell you about the occurrence of lob worms in this meadow? For an initial assessment of the effect of the pollution on lob worms, Dorian randomly positions his quadrat and finds that it contains 8 lob worms. Test, at the 10% significance level, whether the recent pollution has reduced the mean density of lob worms in the meadow from the value of 12 per square metre that existed before the pollution occurred. 4 The number of complaints received by a restaurant about the quality of service follows a Poisson distribution. Over recent months, the mean number of complaints had been 2.2 per day. As a result, the restaurant manager organised a training session for all serving staff. During a five-day period after this training session, a total of seven complains were received about the quality of service. Carry out a hypothesis test to investigate whether there has been a change in the mean number of complaints per day. Use a Poisson distribution and the 10% significance level. 30 Version 1.0

31 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE SC Type I and Type II errors SC1 Understand Type I and Type II errors and define in context. Calculate the probability of making Type I error from tests based on a Poisson or binomial distribution. Calculate probability of making Type I error from tests based on a normal distribution. Assessed at AS and A-level Teaching guidance Students should be able to: know and be able to use the fact that a Type I error is made when the null hypothesis is true but is rejected know and be able to use the fact that a Type II error is made when the null hypothesis is false but is accepted state the meaning of a Type I or a Type II error in context calculate the probability of making a Type I error with tests based on a Poisson or binomial distribution. Examples 1 Kofi owns a cinema. He wishes to increase attendances and so considers offering customers unlimited amounts of free popcorn and soft drinks. He estimates that the likely increase in attendances would result in his business being more profitable, provided that the mean value of the free items consumed by each customer was less than Before deciding whether to proceed, Kofi offers 60 customers entering the cinema free popcorn and soft drinks. The value of the items consumed by each of these customers has a mean of 1.33 and a standard deviation of These customers may be regarded as a random sample of all his current customers. (a) Examine whether the mean value of free popcorn and soft drinks that would be consumed by his current customers is less than Use the 5% significance level. Explain, in the context of this question, the meaning of a Type I error. Version

32 2 The mean age of people attending a large concert is claimed to be 35 years. A random sample of 100 people attending the concert was taken and their mean age was found to be 37.9 years. (a) Given that the standard deviation of the ages of the people attending the concert is 12 years, test, at the 1% significance level, the claim that the mean age is 35 years. Explain, in the context of this question, the meaning of a Type II error. 3 It is believed that online purchases are made at a small company at a rate of 12 per hour. A hypothesis test is used to test this claim using the hypotheses H 0 : λ = 12 and H 1 : λ 12 State the meaning of a Type II error in the context of this test. Assessed at A-level only Teaching guidance Students should be able to calculate the possibility of making a Type I error with tests based on a normal distribution. Examples 1 New pupils entering a large secondary school take a general knowledge test during their first week. The mean score achieved on this test is 46.7 with a standard deviation of At the beginning of the second term, pupils were asked if they would like to be considered for a team to represent the school in a general knowledge quiz. The test scores of a random sample of the pupils who did wish to be considered for the quiz team were: (a) Test, using the 5% significance level, whether the mean test score of those pupils who wished to be considered for the quiz team exceeded Assume that the sample of scores is from a normal distribution with standard deviation Interpret your conclusion in context. A further random sample of those pupils who wished to be considered for the quiz team is to be taken and the test in part (a) repeated. State, with an explanation, the probability of making a Type I error in this test if the mean test score for all pupils who wished to be considered for the quiz team is: (i) 46.7 (ii) Version 1.0

33 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE SC2 Power of a test. Calculations of P(Type II error) and power for a test for tests based on a normal, Binomial or Poisson distribution. Assessed at A-level only Teaching guidance Students should be able to: know, state and use that the power of a test is the probability of not making a Type II error find the probability of making a Type II error in tests based on a normal distribution or a Poisson distribution use the formula Power of a test = 1 P(Type II error) in tests based on a normal distribution or a Poisson distribution. Examples 1 The power of a test is found by using one of the following. Circle your answer. P(Type I error) 1 P(Type I error) P(Type II error) 1 P(Type II error) 2 A test on the rate at which customers enter a café during a 10-minute interval uses the hypotheses H 0 : λ = 12 and H 1 : λ > 12 and the 5% significance level. Find the probability of making a Type II error when the actual value of λ is 13. Version

34 3 A test on the rate at which flaws occur in a 10 metre length of cloth produced on a machine uses the hypotheses H 0 : λ = 8 and H 1 : λ 8 and the 10% significance level. Find the power of this test when the actual value of λ is 7. 4 A test on the value of the population mean of the contents, in grams (g), of a jar of coffee filled by a machine uses the hypotheses H 0 : µ = and H 1 : µ < and the 5% significance level. The population standard deviation of the contents of jars of coffee filled by this machine is 2.4 g. Random samples of five jars of coffee filled by this machine are taken at regular intervals and the sample mean of the contents is used to carry out the hypothesis test. When the actual mean contents of jars of coffee filled by this machine is g, find: (a) the probability a Type II error is made the power of the test. 34 Version 1.0

35 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE SD Continuous random variables (CRVs) SD1 Understand and use a probability density function f(x), for a continuous distribution and understand the differences between discrete and continuous distributions. Understand and use distributions of random variables that are part discrete and part continuous. Assessed at AS and A-level Teaching guidance Students should be able to: recognise that a probability density function, f(x), is used for measured (continuous) variables rather than for discrete variables (which are usually data that is counted) appreciate that ( x) f 0 for all possible values of x understand and use the fact that areas under a probability density function represent probabilities and that if f(x) is defined on the interval f x dx= 1 b a a x bthen ( ) sketch the graph of a probability density function, f(x) find the mode by differentiation when appropriate or by reference to the graph of a pdf. Notes At A-level f(x) may be defined on an unbounded interval and evaluation of improper integrals may be required. Examples 1 The random variable X has probability density function defined by 1 ( x+ 7) 1 x 5 f ( x) = 40 0 otherwise Sketch the graph of f. Version

36 2 The continuous random variable X has probability density function given by Sketch the graph of f. 1 0 x x f ( x ) = 1 x otherwise 3 A continuous random variable X has probability density function given by 2 kx 0 x 3 f ( x ) = 9k 3 x 4 0 otherwise (a) Sketch the graph of f. Show that the value of k is The probability density function of X is defined by α 0 x 1 2 f ( x) = β ( x 4) 1 x 4 0 otherwise Show that the exact values of α and β are 1 2 and 1 18 respectively. 36 Version 1.0

37 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE SD2 Find the probability of an observation lying in a specified interval. Assessed at AS and A-level Teaching guidance Students should: know and be able to use that the area under a probability density function, f(x), represents a probability know and be able to use that P( c X d) = f( x)dx c know and be able to use the fact that P(X = x) = 0, ie the probability of getting an exact value in a continuous distribution is zero be able to represent probabilities on a sketch of the pdf. d Examples 1 The time, T hours, taken by any member of a group of friends to complete a run for charity can be modelled by the following probability density function: 1 2 t 3 t f( t) = 2 t 6 t otherwise Show that the probability that a member of the group selected at random takes at least 6 hours to complete the run is 0.3. Version

38 2 The waiting time, T minutes, before being served at a local newsagent s can be modelled by a continuous random variable with probability density function: (a) Sketch the graph of f. 3 2 t 0 t < f( t) = ( t + 5) 1 t otherwise For a customer selected at random, calculate P( T 1). 3 The delay, in hours, of certain flights from Australia may be modelled by the continuous random variable T, with probability density function: (a) Sketch the graph of f. 2 t 0 t f( t) = 1 t 3 t otherwise Calculate: (i) P( T 2) (ii) P2 ( < T< 4) 38 Version 1.0

39 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE 4 The time, T hours, that the supporters of Bracken Football Club have to queue in order to obtain their cup final tickets has the following probability density function: (a) Sketch the graph of f. Write down the value of P(T = 3). 1 0 t < f( t ) = t( 6 t) 3 t otherwise (c) Find the probability that a randomly selected supporter has to queue for at least 3 hours in order to obtain tickets. Version

40 SD3 Find the median and quartiles for a given probability density function, f(x). Assessed at AS and A-level Teaching guidance Students should: know and be able to use the fact that if Q 2 is the median, then P(X Q 2 ) = 0.5 know and be able to use the fact that if Q 2 is the median, then f( x) dx = 0.5 know and be able to use the fact that if Q 1 is the lower quartile, then f( x) dx = 0.25 know and be able to use the fact that if Q 3 is the upper quartile, then f( x) dx = 0.75 Q 2 a Q 1 a Q 3 a Examples 1 The time, T hours, that the supporters of Bracken Football Club have to queue in order to obtain their cup final tickets has the following probability density function: 1 0 t< f( t ) = t( 6 t) 3 t otherwise Show that the median queuing time is 2.5 hours. 2 The continuous random variable X has probability density function defined by: 1 ( 2 x+ 1 ) 0 x 1 5 f( x ) = 1 ( 4 x) 2 1 < x otherwise Verify that the lower quartile, Q 1, satisfies 0.5 < Q 1 < Version 1.0

41 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE 3 At a cinema, the time, T minutes, that customers have to wait in order to collect their tickets has the following probability density function: Show that the median waiting time is 3 minutes. 2 t 0 t < f() t = (5 t ) 3 t otherwise 4 All consultations with an optician are by appointment. The time, T minutes, by which an appointment is delayed has the following probability density function: 1 t 54 1 f() t = t < 3 3 t < 6 1 (10 t ) 6 t < otherwise (a) Sketch the graph of f. Use your graph to find the lower quartile. Version

42 SD4 Find the mean, variance and standard deviation of a continuous random variable. Know the formulae: E( X) = xf( x) dx E( X2) = x2 f( x) dx Var( X) = E( X ) (E( X)) 2 2 Assessed at AS and A-level Teaching guidance Students should: know and be able to use the formula E(X) = xf(x) dx to find the value of the mean of a continuous random variable know and be able to use the formula E(X 2 ) = x 2 f (x) dx know and be able to use the formula Var(X) = E(X 2 ) (E(X)) 2 to find the value of the variance 2 be able to the fact that Var(X) = σ to find the value of the standard deviation. Examples 1 The delay, in hours, of certain flights from Australia may be modelled by the continuous random variable, T, with probability density function: Determine E(T). 2 t 0 t f( t) = 1 t 3 t otherwise 42 Version 1.0

43 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE 2 Engineering work on the railway network causes an increase in the journey time of commuters travelling into work each morning. The increase in journey time, T hours, is modelled by a continuous random variable with probability density function: Show that E( T ) = t( 1 t ) 0 t 1 f() t = 0 otherwise 3 A continuous random variable X has the probability density function defined by 1 ( 4 x) 0 x f( x) = 3 x otherwise Prove that the mean of X is The random variable X has probability density function: f( x) px + q 0< x < 1 = 0 otherwise where p and q are constants. (a) Show that 1 2 p+q= 1 Find the mean of X in terms of p and q. (c) The mean of X is 0.6. Show that p = 1.2 and find the value of q. (d) Find the standard deviation of X. Version

44 SD5 Understand the expectation and variance of linear functions of CRVs and know the formulae: E(aX + b) = ae(x) + b and Var(aX + b) = a 2 Var (X) Know the formula E ( ( )) g( ) f( ) d gx = x x x Find the mean, variance and standard deviation of functions of a continuous random variable such as E(5X 3 ), E (18X -3 ), Var(6X -1 ) Assessed at AS and A-level Teaching guidance Students should: know and be able to use the formulae E(aX + b) = ae(x) + b and Var(aX + b) = a2var (X), where a and b are constants, to find the values of the mean and variance of ax + b when the values of E(X) and Var(X) are known know and be able to use the formula ( ) E(2X + 3), E(18X -3 ), for example E g( X) = g( x ) f( x) dx to find the value of E(X 2 ) or know and be able to use the fact that the variance is the square of the standard deviation to find the value of the standard deviation of such functions. i Examples 1 A continuous random variable X has probability density function defined by: 1 ( 4 x) 1 x f( x) = 3 x otherwise (a) Prove that the mean of X is Find the value of E(18X + 5) 44 Version 1.0

45 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE 2 A continuous random variable, X, has probability density function defined by: 1 2 x 6 f( x) = 8 0 otherwise You are given that E(X) = 2 and that Var(X) = 16 3 Find the values of: (a) E(3X + 4) Var(3X + 4). 3 The continuous random variable X has the probability density function defined by 4 5x 0 x 1 f( x) = 0 otherwise Find the exact values of (a) E(X + 1) Var(3X + 1). 4 The continuous random variable X has the probability density function given by 2 3x 0< x 1 f( x) = 0 otherwise (a) Determine (i) 1 E X (ii) 1 Var X Hence, or otherwise, find the mean and variance of 5+ 2 X X Version

46 SD6 Understand and use a cumulative distribution function, F(x). Know the relationship between f(x) and F(x). x F( x) = f() t dt and d F f( x) = ( x) dx Assessed at A-level Teaching guidance Students should: x know and use the fact that Fx ( ) = f() tdt to find a cumulative distribution function, F(x), for a known probability density function, f(x) d know and use the fact that f( x) = F( x)to find a probability density function, f(x), for a known dx cumulative distribution function F(x) know that in a cumulative distribution function F(Q 1 ) = 0.25 where Q 1 is the lower quartile, F(Q 2 ) = 0.5 where Q 2 is the median, and F(Q 3 ) = 0.75 where Q 3 is the upper quartile. Examples 1 A continuous random variable X has the probability density function defined by: (a) Find P(X < 1). 4 2 x 0 x f( x) = ( x 3)(3 x 11) 1 x otherwise (i) Show that for 1 x 3, the cumulative distribution function, F(x), is given by F( x) = ( x 10x + 33x 16) 20 (ii) Hence verify that the median value of X lies between 1.13 and Version 1.0

47 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE 2 The continuous random variable X has probability density function defined by 1 ( 2 x+ 1 ) 0 x f( x) = (4 x ) 1 < x otherwise (a) Sketch the graph of f. (i) Show that the cumulative distribution function, F(x), for 0 x 1 is 1 F( x) = xx ( + 1) 5 (ii) Hence write down the value of P(X 1) (iii) (iv) Find the value of x for which P(X x) = 17 Find the lower quartile of the distribution The time, t, in weeks, that a patient must wait to be given an appointment in Holmsoon Hospital may be modelled by a random variable, T, having the cumulative distribution function: 0 t < 0 3 t F() t = t 6 1 t > 6 (a) Find, to the nearest day, the time within which 90% of patients will have been given an appointment. Find the probability density function of T for all values of t. Version

48 4 A continuous random variable, X, has the probability density function defined by 2 x 0 x 1 1 f( x) = (5 2 x ) 1 x otherwise (a) Sketch the graph of f on the axes below. y O x (i) Find the cumulative distribution function, F, for 0 x 1. (ii) Hence, or otherwise, find the value of the lower quartile of X. (c) (i) Show that the cumulative distribution function for 1 x 2 is defined by Fx ( ) = ( x x 3 ) 3 (ii) Hence, or otherwise, find the value of the upper quartile of X. 48 Version 1.0

49 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE SD7 Understand the rectangular distribution f(x) where 1 a x b f( x) = b a 0 otherwise Know the conditions for the rectangular distribution to be used as a model. Calculate probabilities from a rectangular distribution. Know proofs of mean, variance and standard deviation for a rectangular distribution. Assessed at A-level only Teaching guidance Students should: be able to identify a rectangular distribution as having a probability density function, f(x), that doesn t contain x (ie it is a constant), or that has a cumulative distribution function, F(x), that is a linear function of x know that this can be used on the interval a x b where the probability is distributed uniformly; recognise this as the uniform distribution on [a,b] know that areas under the probability density function, f(x), correspond to probabilities be able to prove, using integration, that the mean of a rectangular distribution on the interval a x b is 1 2 ( a+b) and that the variance on this interval is ( b a ) Version

50 Examples 1 The error, X millimetres, made when the heights of prospective members of a new gym club are measured can be modelled by a rectangular distribution with the following probability density function: f( x) k 4 x 6 = 0 otherwise (a) State the value of k. Write down the value of E(X). (c) Calculate P(X > 0). (d) The height of a randomly selected prospective member is measured. Find the probability that the magnitude of the error made exceeds 3.5 millimetres. 2 (a) The continuous random variable, T, follows a rectangular distribution with probability density function given by k a t b f() t = 0 otherwise (i) Express k in terms of a and b. (ii) 1 Prove, using integration, that E( T) = ( b a) 2 The error, in minutes, made by a commuter when estimating the journey time by train into London may be modelled by the random variable, T, with probability density function 1 4 t 6 f() t = 10 0 otherwise (i) Write down the value of E(T). (ii) Calculate P(T < 3 or T > 3) 50 Version 1.0

51 AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE 3 (a) The continuous random variable, X, follows a rectangular distribution with probability density function defined by: 1 0 x b f( x) = b 0 otherwise (i) (ii) Write down E(X). Prove, using integration, that 1 Var( X)= b 12 2 At an athletics meeting, the error, in seconds, made in recording the time taken to complete the 10,000 metres race may be modelled by the random variable, T, having the probability density function: Calculate P( T > 0.02). f() t t 0.1 = 0 otherwise 4 (a) A random variable, X, has probability density function defined by: k a< x<b f( x) = 0 otherwise (i) Show that k 1 = b a (ii) Prove, using integration, that E( X) = 1 ( a+b) 2 The error, X grams, made when a shopkeeper weighs out loose sweets can be modelled by a rectangular distribution with the following probability density function: f( x) k 2< x< 4 = 0 otherwise (i) Write down the mean, µ, of X. (ii) Evaluate the standard deviation, σ, of X. (iii) Hence find P X 2 µ < σ Version