WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 2

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2 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 2

3 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 3 QUESTION PAPER

4 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 4 Candidate Name Centre Number Candidate Number LEVEL 3 CERTIFICATE IN MATHEMATICS FOR WORK AND LIFE SPECIMEN PAPER 2 hours 30 minutes ADDITIONAL MATERIALS The use of a calculator is permitted in this examination. Statistical tables will be required. INSTRUCTIONS TO CANDIDATES Write your name, centre number and candidate number in the spaces at the top of this page. Answer all questions. Write your answers in the spaces provided in this booklet. INFORMATION FOR CANDIDATES The number of marks is given in brackets at the end of each question or part-question. For Examiner s use only Question Maximum Mark Total 100 You are reminded of the need for good English and orderly, clear presentation in your answers. No certificate will be awarded to a candidate detected in any unfair practice during the examination. Mark Awarded

5 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 5 Formula list Kinematics formulae Where a is constant acceleration, u is initial velocity, v is final velocity, s is displacement from the position when t = 0 and t is time taken: v u at s ut at v u 2as 2 Statistics and probability The sums of squares and deviations S xx, S yy and S xy are given by: 2 2 S 2 yy yi y yi yi n S x x y y x y x y n xy i i i i i i A measure of linear association between two variables X and Y is given by the Pearson correlation coefficient r. Sxy For the sample (x 1,y 1 ), (x 2,y 2 ),..., (x n,y n ), it is given by r. S S Given data, the parameters and of the linear regression model may be estimated using the principle of least squares. The least squares estimate ˆ of the slope parameter ˆ Sxy is given by. S The least squares estimate ˆ of the intercept parameter is given by The least squares line is given by S x x x x n xx i i i y ˆ ˆ x. xx yy xx ˆ y ˆ x. For events A and B, P( A B) P( A) P( B) P( A B) and Bayes Theorem: P( A j P( Aj ) P( B Aj ) B) P( A ) P( B A ) i i

6 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 6 1. Below is an artwork by Chris Jordan called Three Second Meditation, It shows 9960 mail order catalogues. This is equal to the average number of pieces of junk mail that are printed, shipped, delivered, and disposed of in the US every three seconds. Chris Jordan Using the information above, calculate how many pieces of junk mail are printed, shipped, delivered and disposed of in the US in one year (not a leap year). Express your answer in standard form, correct to three significant figures. [3]

7 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE friends go out for a meal. The total bill for all the food and wine comes to The cost of the wine is of the friends did not have any wine. One of the friends suggests two options for deciding how the total bill is to be paid. Either they share the cost equally OR those that did not have wine should only pay for the food and not contribute towards the wine bill. Each of the 9 friends rounds the amount they have to pay up to the nearest whole pound and leaves the change as the tip for the waitress. Which option will give the waitress the most tips and by how much? You must show all your working. [12]

8 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 8 3. The table below is an extract from the exchange rates offered by a bank on a particular day. Currency Standard Sell Rate Buy Rate USA Dollar Turkey Lira Euro Mexican Peso Egyptian Pound Argentine Peso Australian Dollar Bahrain Dinar Barbados Dollar For a holiday in Palma, Lauren converts 300 to euros. The cashier at the currency exchange counter makes the euros up to the nearest 10 euros. Unfortunately Lauren was ill and could not go to Palma. She decided to sell her euros back to the bank. Write Lauren s loss (in pounds) as a percentage of the original amount she paid the cashier for the euros. [10]

9 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 9 4. Erin finds this interesting fact on the Internet. The diameter of the Earth is approximately 7960 miles She says: I bet you could walk a distance equivalent to the circumference of the Earth in less than a year. Is Erin s statement reasonable? Show all your workings and state any assumptions you have made. [8]

10 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE Tony, a coffee shop owner, needs to buy some new china mugs and some new glass mugs. The china mugs and glass mugs are available to buy in packs. There are 3 china mugs or 2 glass mugs in each pack. He cannot afford to buy more than 25 packs altogether He wants to buy at least 60 new mugs He wants to buy more packs of china mugs than packs of glass mugs. (a) Define two variables and formulate three constraints in terms of your variables. [5]

11 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 11 (b) Using the graph paper, plot your three inequalities from (a), indicating the feasible region. [4]. (c) Find the maximum number of china mugs that can be bought and the corresponding number of glass mugs. [2]

12 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE A shopkeeper pays 120 for an mp3 player. He wishes to put a marked price on the mp3 player so that in the forthcoming sale, when he gives a discount of 25% on the marked price, he will still make a profit of 20% on the price paid for the mp3 player. Find the marked price. [4]

13 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE An article in a newspaper reported on the alarming level of tooth-loss among the very young. A dentist reading this headline is interested to see if the number of times children clean their teeth each day is independent of age. The figures below show the frequency of teeth brushing for a group of UK children aged 12 and 15 years old. Number of children Age Frequency of teeth brushing 12 years 15 years All Less than once a day Once a day Twice or more times a day Total The dentist decides to investigate the following hypothesis: H 0 : there is no association between frequency of teeth brushing and age for 12 and 15 years old children. H 1 : There is an association between frequency of teeth brushing and age for 12 and 15 years old children. Apply an appropriate hypothesis test to the data in the above table. Report your findings to the dentist. [10]

14 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE The table below shows information on the cost of borrowing money on a short term basis called 'a pay day loan'. Interest rate Costs 1% per day (simple interest) Transmission Fee 5.50 Extension Fee Late Payment Fee Description Calculated at the end of every day, calculated on your interest bearing balance that day. Added to loan amount requested so will accrue interest. This fee also applies to new loan top ups Charged if we accept your application to change your promise date and will accrue interest. Charged and added to your outstanding balance if we haven t received full payment by 11pm on the third day after your promise date. Example Amount of credit: 150 for 18 days Transmission fee 5.50 One total repayment of (a) Show that if you borrow 150 for one day you will need to pay back on the following day. [3]

15 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 15 (b) How much interest would you pay if you agree to pay the loan back after one year (365 days)? [3]

16 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE In a certain community, 10 per cent of all adults over fifty have diabetes. A doctor correctly diagnoses 95 per cent of all persons with diabetes as having the disease, and incorrectly diagnoses 2 per cent of all persons without diabetes as having the disease. (a) Calculate the probability that the doctor will diagnose an adult over 50 (randomly chosen from the given population) as having diabetes. [4] (b) Suppose that an adult over fifty living in the given community is diagnosed by the doctor as having diabetes. What is the probability that this adult actually has the disease? [3]

17 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE Imran works for a company called Acorn Insurance. His gross salary is per year. Below are extracts from HM Revenue and Customs and details of Imran s company pension scheme: National Insurance contributions If you earn more than 153 a week and up to 805 a week, you pay 12% of the amount you earn between 153 and 805 If you earn more than 805 a week, you also pay 2% of all your earnings over 805 Source: HMRC 2014 Income tax threshold and rates Income tax threshold Basic tax rate 10,000 per year 20% on annual earnings above income tax threshold and up to 31,865 Higher tax rate 40% on annual earnings from 31,866 to 150,000 Additional tax rate 45% on annual earnings above 150,000 Source: HMRC 2014 Acorn Insurance Pension Scheme Gross salary Contribution rate Gross salary Contribution rate Up to % to % to % to % to % to % to % or more 12 5% to %

18 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 18 Using the information on the previous page, calculate Imran's weekly net salary. You must show all your working. [13]

19 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE You work for a film company. The film company have made a new film that could fit into the horror or comedy category. Also, the title of the film has not been decided. A film company executive asks your advice on which category, horror or comedy, you think would bring in the most income and whether the number of words in the film title affects the gross income. You are given the following information: The frequency tables of the Gross Income for horror and comedy films made in the USA from 2001 to Some descriptive statistics of the Gross Income for horror and comedy films made in the USA from 2001 to A scatter graph showing Gross Income against the number of words in the title of a US film between 2001 and 2012 for a random sample of 35 comedy films. The values for S xy, S xx and S yy, where x represents the number of words in the film titles and y represents the gross income. Gross income class Millions $ Comedy film frequency Horror film frequency (400) 1 0 Total Variable Gross Income Horror Gross Income Comedy Mean millions $ Standard Deviation millions $ Minimum Value millions $ Q1 Lower Quartile millions $ Q2 Median millions $ Q3 Upper quartile millions $ Maximum Value millions $ (All income values are correct to two decimal place values and are in millions of US$)

20 Gross Income millions $ WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 20 Gross Income vs Number of words Number of words S S S xy xy yy Using the information provided, write a brief statistical report that will be presented to the film company producers advising which category, horror or comedy, you believe would bring in the most income and whether the film title should be long or short. In your report you should: clearly communicate the problem and findings and include any calculations or graphical representations (where appropriate) that may support your findings and recommendations. [16]

21 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 21

22 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 22

23 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 23 MARK SCHEME

24 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 24 LEVEL 3 CERTIFICATE IN MATHEMATICS FOR WORK AND LIFE GENERAL INSTRUCTIONS for MARKING 1. The mark scheme should be applied precisely and no departure made from it. Marks should be awarded directly as indicated and no further subdivision made. When a candidate follows a method which does not correspond to the methods explicitly set out in the mark scheme, marks should be awarded in the spirit of the mark scheme. In such cases, further advice should be sought from the Team Leader or Principal Examiner. 2. Marking Abbreviations The following may be used in marking schemes or in the marking of scripts to indicate reasons for the marks awarded. cao = correct answer only MR = misread PA = premature approximation bod = benefit of doubt oe = or equivalent si = seen or implied ISW = ignore subsequent working F.T. = follow through ( indicates correct working following an error and indicates a further error has been made) Anything given in brackets in the marking scheme is expected but, not required, to gain credit. 3. Premature Approximation A candidate who approximates prematurely and then proceeds correctly to a final answer loses 1 mark as directed by the Principal Examiner. 4. Misreads When the data of a question is misread in such a way as not to alter the aim or difficulty of a question, follow through the working and allot marks for the candidates' answers as on the scheme using the new data. This is only applicable if a wrong value is used consistently throughout a solution; if the correct value appears anywhere, the solution is not classed as MR (but may, of course, still earn other marks). 5. Marking codes M' marks are awarded for any correct method applied to appropriate working, even though a numerical error may be involved. Once earned they cannot be lost. m marks are dependant method marks. They are only given if the relevant previous M mark has been earned. A' marks are given for a numerically correct stage, for a correct result or for an answer lying within a specified range. They are only given if the relevant M/m mark has been earned either explicitly or by inference from the correct answer. 'B' marks are independent of method and are usually awarded for an accurate result or statement. S marks are awarded for strategy E marks are awarded for explanation U marks are awarded for units P marks are awarded for plotting points C marks are awarded for drawing curves

25 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 25 Specimen Assessment Materials Mathematics for Work and Life ( )184.75/9 ( )20.53 ( ) 21 9 (Total amount paid=) ( )189 [( ) ( )29.9(0)]/9 (NO WINE) ( )17.21 [( )29.9(0)/6 =] ( )4.98 ( ) ( )4.98 (WITH WINE) ( )22.19 ( ) ( )23 6] (Total amount paid=) ( )192 Conclusion 2 nd option by 3 3. ( ) (euro) Rounding up to nearest 10 euro 360 (euro) 360 (euro)/ ( ) (euro)/ ( ) [( ) ( )265.31)/( ) (015513%) Mark A2 [3] B1 [12] m1 [10] Comments Page 1 Attempting 9960 seconds minutes hours days Allow one slip e.g using ( =) 336 for number of days etc. for or Award A0 for answer of ( ) or ( )20.52 with no working Award for sight of (( )189 - ( )184.75=) ( )4.25 Award for 47(p) 9 = ( )4.23 Award A0 for answer of ( ) or ( )17.2 with no working FT their ( ) their ( )17.21 Award B1 A0 for answer of ( ) or ( )22.18 with no working Award for sight of (( )192 - ( )184.75=) ( )7.25 Award for 79(p) (p) 6 = ( )7.23 F.T their ( )192 their ( )189 OR F.T their ( )7.25 their ( )4.25 FT rounding their ( ) FT their ( )360 Award A0 for answer of ( ) or ( ) with no working FT their ( )360 Award A0 for answer of ( ) with no working FT their ( ) and their ( ) Accept 13%

26 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 26 Specimen Assessment Materials Mathematics for Work and Life 4. (Circumference =) π (.4)(miles) to 25007( )(miles) Stating a reasonable value for the speed of an average person (e.g. 2 5 mph) Mark Comments Page 2 B1 Accept a value outside the range 2 mph to 5 mph with a suitable justification for their choice e.g. I think I can run 12 mph so I could probably walk 6 mph Method for checking whether it is a reasonable statement e.g. Number of hours=) Circumference their speed (Number of days=) 24 OR I can walk about 8 hours a day, so 8 3mph = 24 miles a day = days OR Circumference 365 = number of miles to walk per day Conclusion and calculations shown (e.g. comparing with total number of days/weeks/hours/miles per day) e.g. It would take 347 days if someone walked 3 mph constantly for 24 hours a days. It must take more than a year because the person will need to sleep and rest every day. It is not reasonable, it would take much more than a year as days is just under 3 years. If an average person walks about 3mph then a person couldn t reasonably walk 67 miles in a day for a year. Assumption: e.g. A person may only reasonably walk about 8 hours a day and their speed will vary I am estimating how fast I can walk The calculations assume that the person is walking at a constant speed and for 24 hours a day so it s bound to be more The circumference is approximate How accurate is the fact from the internet? M2 E2 E1 [8] for partial method E1 with conclusion but with no workings shown or partial explanation given

27 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 27 Specimen Assessment Materials Mathematics for Work and Life 5(a). Two variables defined e.g. x = number of packs of china mugs y = number of packs of glass mugs x + y 25 3x + 2y 60 or equivalent x > y Mark B2 B1 B1 B1 Comments Page 2 B1 for each correct variable B1 if number of packs omitted from both variables ISW 5(b). cont. Plotting y = 25 x Plotting y = x or equivalent Plotting y = x P1 P1 P1 FT their inequalities from 4(b) Identifying feasible region B1 Must consider 3 lines 5(c). Maximum number of packs of china mugs = 24 and Maximum number of packs of glass mugs = 1 (Number of china mugs = 24 3 =) 72 and (Number of glass mugs = 1 2 =) 2 6. ( ) ( )144 ( ) ( ) Using Chi-squared test Stating level (0.05) and one-sided test Calculating the chi-squared test statistic Using tables to find critical region Conclusion in the context of the original problem: e.g. there is evidence to reject the null hypothesis and to suggest teeth cleaning is associated with age for 12 and 15 year old children [11] [4] A3 E2 [10] FT their graph if of equivalent difficulty Accept (24,1) indicated FT their graph if of equivalent difficulty Answers must be 0 and whole numbers FT their ( )144 Award for sight of at least three expected values or chi-squared values F.T their chi-squared values A2 for sight of correct chi-squared values F.T their expected values for sight of expected values F.T. on 'their significance level' A value from incorrect degrees of freedom gets A0 E1 for reject the null hypothesis and not relating it back to the original problem

28 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 28 Specimen Assessment Materials Mathematics for Work and Life 8(a). [( )150 + ( )5.50 =] ( ) ( ) [( ) ] or ( ) ( ) (b). ( ) or equivalent ( ) Interest = ( ) ( )150 = ( ) (a). Using law of Total Probability or equivalent or equivalent (b) (40707 ) or equivalent NATIONAL INSURANCE [Weekly gross salary ( ) =] ( ) [( )805 ( )153] [( )920 ( )805] ( )80.54 TAX ( =)( ) ( ) =( )6390.(00) (( ) ( )6390.(00)=) PENSION [( ) =] OR [( ) ] ( )78.2(0) TOTAL (Weekly) 920 [( ) ( ) ( )78.2(0)] =( ) Mark B1 [6] A2 B2 B1 [7] B1 M2 A2 B1 B1 [13] Comments Page 6 Award B1 A0 for answer of ( ) or ( ) with no working Award A0 for answer of ( ) with no working FT their ( ) Accept tree diagram B1 numerator B1 denominator FT numerator and denominator from 11(a) Allow equivalent working (e.g. working in weeks, months or annually) Allow reasonable approximation at each stage Penalise once only for use of 48 weeks (12 4 weeks) for one FT their ( )920 for ( )78.24 or ( )2.30 FT their ( ) their ( )2.30 (may be seen in later workings) Accept 0.4 ( ) Accept ( )6389.6(0) Accept (( ) ( )6389.6(0)=) (0) FT their ( ) their( )6389.6(0) Award B1 for sight of ( ) or ( ) (may be seen in later workings) FT their ( )920 Accept 920 [( ) ( ) ( )78.2(0)] FT all their values for weekly gross salary, tax, NI and pension Accept ( )554.29

29 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 29 Specimen Assessment Materials Mathematics for Work and Life 11. Stating the problem Problem explained clearly for both tasks Using Statistical methods Horror/Comedy Comparing both distributions by using an appropriate graphical representation and deciding which measure of location of central tendency and spread to use. e.g. using box plots or histograms distributions are skewed or distributions are not symmetrical Mean affected by extreme values or median is not affected by extreme values Mark E1 E2 Comments Page 6 E1 for drawing an appropriate graphical representation but not comparing distributions or deciding which measure of location of central tendency and spread to use. Comparing the location of central tendency of both distributions e.g. not much difference in the medians, slightly greater for comedy films. Comparing the spread of both distributions e.g. greater spread for comedy films than horror implies prediction of income is less reliable for comedy E1 E1 Title length Using an appropriate method for finding a relationship between number of words in a comedy film title and the gross income. e.g. calculation of Pearson's product moment correlation coefficient ( = -0 38(3299) Using an appropriate statistical test and stating the null and alternative hypothesis Stating whether it s a one tail or two tailed test Stating level of significance test (0 05) Finding critical value B1 B1 B1 B1 Accept use of regression Accept sight of line of regression Gross Income = No of words F.T their hypothesis. (one-sided at 0.05 = )

30 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 30 Specimen Assessment Materials Mathematics for Work and Life Conclusions Conclusions with valid explanations for the horror/comedy problem e.g. choose comedy as this category has higher median so worth taking the risk. Choose horror with a slightly lower median but gross income more reliable/consistent Conclusions with valid explanations for the title length problem e.g. There is evidence to suggest a negative correlation between the number of words in a film and the gross income. Advice is to keep the title short. Appropriate comment regarding validity of the conclusion e.g. sample sizes are different sample size for horror especially small these conclusion are based on a sample of films, so care should be taken when giving the advice Mark E2 E2 E1 [16] Comments Page 6 E1 for partial explanation E1 for partial explanation or for just stating reject/accept null hypothesis without relating the findings in the context of the problem

31 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 31 ASSESSMENT GRID

32 WJEC EDUQAS LEVEL 3 MATHEMATICS FOR WORK AND LIFE 32 Level 3 Certificate in Mathematics for Work and Life SPECIMEN PAPER AO1: Use and apply standard techniques AO2: Select and apply mathematical and statistical methods in a range of contexts AO3: Apply skills in mathematical thinking, reasoning and communication Q. Topic AO1 AO2 AO3 Total 1 US junk mail Friends restaurant bill Currency Estimating Mugs Shopkeeper - reverse percentage Chi-squared teeth problem Payday loan Diabetes - probability Imran's salary Film problem Total AO1 AO2 AO3 Examination marks 1.44 (80% of total assessment) Controlled assessment marks 1 (20% of total assessment) marks % 37% 39% WJEC Eduqas Level 3 Mathematics for Work and Life - esams/mlj