Pearson LCCI Level 3 Certificate in Business Statistics (VRQ)

Transcription

1 Pearson LCCI Level 3 Certificate in Business Statistics (VRQ) (ASE20100) L3 SAMPLE ASSESSMENT MATERIALS For first teaching from January 2015

2 Mark Scheme Sample Assessment Materials Pearson LCCI Level 3 Certificate in Business Statistics (ASE20100)

3 LCCI Qualifications LCCI qualifications come from Pearson, the world s leading learning company. We provide a wide range of qualifications including academic, vocational, occupational and specific programmes for employers. For further information, please visit our website at Pearson: helping people progress, everywhere Our aim is to help everyone progress in their lives through education. We believe in every kind of learning, for all kinds of people, wherever they are in the world. We ve been involved in education for over 150 years, and by working across 70 countries, in 100 languages, we have built an international reputation for our commitment to high standards and raising achievement through innovation in education. Find out more about how we can help you and your students at: All the material in this publication is copyright Pearson Education Ltd 2015

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5 General Marking Guidance All candidates must receive the same treatment. Examiners must mark the first candidate in exactly the same way as they mark the last. Mark schemes should be applied positively. Candidates must be rewarded for what they have shown they can do rather than penalised for omissions. Examiners should mark according to the mark scheme not according to their perception of where the grade boundaries may lie. There is no ceiling on achievement. All marks on the mark scheme should be used appropriately. All the marks on the mark scheme are designed to be awarded. Examiners should always award full marks if deserved, i.e. if the answer matches the mark scheme. Examiners should also be prepared to award zero marks if the candidate s response is not worthy of credit according to the mark scheme. Where some judgement is required, mark schemes will provide the principles by which marks will be awarded and exemplification may be limited. When examiners are in doubt regarding the application of the mark scheme to a candidate s response, the team leader must be consulted. Crossed out work should be marked UNLESS the candidate has replaced it with an alternative response. Where marks are awarded for own figure answers, these marks can only be awarded if evidence of how the candidate arrived at their values has been provided (their workings). If candidates fail to provide their workings when instructed in the paper, it may not be possible to achieve all marks associated with the question, even if the final answer is correct. For calculation questions, full marks can be awarded where correct answer is seen with no workings shown, unless question states that candidate must provide workings. Pearson LCCI Level 3 Certificate in Business Statistics 1

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7 Write your name here Surname Other names Pearson LCCI Centre Number Candidate Number Certificate in Business Statistics Level 3 Sample assessment material for first teaching January 2015 Time: 3 hours Paper Reference ASE20100 You must have: statistical formulae sheet (enclosed), HB pencil, eraser Total Marks S48188A 2014 Pearson Education Ltd. 1/1 Instructions Use black ink or ball-point pen use a pencil to draw diagrams/graphs. Fill in the boxes at the top of this page with your name, centre number and candidate number. Answer all questions. Answer the questions in the spaces provided there may be more space than you need. Answers should be given to an appropriate degree of accuracy. Information The total mark for this paper is 100. The marks for each question are shown in brackets use this as a guide as to how much time to spend on each question. A formulae sheet is provided at the front of the question paper. Calculators may be used. Advice Read each question carefully before you start to answer it. Try to answer every question. You are advised to show your workings. Check your answers if you have time at the end. *S48188A0125* Pearson LCCI Level 3 Certificate in Business Statistics Turn over 3

8 ( Pearson LCCI Level 3 Certificate in Business Statistics Formula sheet Median for grouped data l c m+f m m ( n 2 F m 1 Where l m, c m and f m are the lower boundary, width and frequency respectively of the median class, n is the total number of observations and F m frequency corresponding to l m. 1 is the cumulative Mean for ungrouped data x x n x Mean for grouped data x f Standard deviation for ungrouped data s Standard deviation for grouped data f s f x f x 2 ( ) 2 n ( x x f f ( 2 2 Pearson measure of skewness x Coefficient of variation x s Median s Multiplication rule of probability P A Addition rule of probability P A ( if and independent B ) P ( A) P ( B ) A B ( B ) P ( A ) P ( B ) P ( A B ) Price Quantity Laspeyres index p 1 q 0 p q p 0 q 1 p q Paasche index p 1 q 1 p q p 1 q 1 p q Weighted index WI W Product moment correlation coefficient r Spearman s rank correlation coefficient r s 1 n x y x y n x x n y y d 2 1 n ( n 2 ) Least Squares regression line y a bx b n x y x y n x 2 x 2 2 *S48188A0225* 4 Pearson LCCI Level 3 Certificate in Business Statistics

9 One sample z test Mean z x µ n Two sample z test Proportion p 1 Mean z 1 x x 2 s2 1 s2 2 n 1 n 2 One Sample t test Proportion where p n 1 p 1 n 2 p 2 n 1 n 2 t x s n µ where s 1 n 2 Independent samples t test 1 1 where s Chi squared test 2 2 Test for *S48188A0325* Pearson LCCI Level 3 Certificate in Business Statistics 3 5

10 Table 1: The Normal Distribution A is the area to the left of the given value of z A O z z A z A z A z A z A *S48188A0425* 4 6 Pearson LCCI Level 3 Certifi cate in Business Statistics

11 Table 2: t Distribution t A is the value of the t statistic with ν degrees of freedom with area A to the right of it O ta A ν t t t t ν t t t t ν t t t t A t Table 3: Chi squared Distribution Table χ 2 A is the value of the χ2 statistic with ν degrees of freedom with area A to the right of it O A 2 χ A ν χ χ ν χ χ *S48188A0525* Pearson LCCI Level 3 Certifi cate in Business Statistics 5 7

12 Answer ALL questions. Write your answers in the spaces provided. 1 Town officials want residents to apply for home improvement grants. They need to find out the most effective way to inform residents of the grants. A random sample of 300 residents was contacted by letter, or telephone, informing them about the grants. The table summarises the results obtained. Outcome Contact method Apply for a grant Do not apply for a grant Telephone Letter Analysis of these outcomes, using a software package, gave a calculated chi squared (χ 2 ) value = (a) (i) Test whether there is a significant association between the method used to contact residents and whether they proceeded to apply for a grant. (3) (ii) State, in context, what the test in (a) (i) shows. *S48188A0625* 6 8 Pearson LCCI Level 3 Certificate in Business Statistics

13 (b) Construct a 95% confidence interval estimate for the overall percentage of residents who did not apply for a grant. (5) One of the town officials suggested that the accuracy of the estimate in (b) should be improved to within 2% at the 95% confidence level. (c) (i) Calculate the sample size that would be needed to achieve this level of improvement: (4) *S48188A0725* Pearson LCCI Level 3 Certificate in Business Statistics 7 9

14 (ii) Give one non-statistical reason for rejecting this change in the sample size. The town official also claims that the average value of a grant application is An analysis of a random sample of nine applications suggests that the average value of a grant is 1272 with a variance of 441. (d) (i) Test whether the town official s claim regarding the average value of a grant application is correct. (5) (ii) State, in context, what the test in (d) (i) shows. (Total for Question 1 = 20 marks) *S48188A0825* 8 10 Pearson LCCI Level 3 Certificate in Business Statistics

15 2 A bus company is considering offering its drivers an annual bonus based on the number of journeys they complete on time. To budget for the cost of these bonus payments, management analysed a random sample of arrival and departure times on one of their bus routes. The route selected was the daily service between Towns X and Y. On 80% of the journeys recorded, the buses departed Town X on time and on 20% of the journeys, they departed Town X late. Of the buses that departed Town X on time, 15% arrived in Town Y early, 65% arrived on time and the remainder arrived late. Of the buses that departed Town X late, 25% arrived in Town Y on time and 75% arrived late. (a) Draw a fully-labelled tree diagram to represent this information. (4) *S48188A0925* Pearson LCCI Level 3 Certificate in Business Statistics 9 11

16 (b) Calculate the probability that a bus journey selected at random: (i) did not arrive in Town Y late (2) (ii) departed Town X late, given that it did not arrive in Town Y late (2) (iii) departed Town X on time, given that it did not arrive in Town Y on time. (2) *S48188A01025* Pearson LCCI Level 3 Certificate in Business Statistics

17 In an effort to increase passenger numbers and revenue, the bus company is promoting the purchase of monthly discounted tickets. One of the drivers suggested that more passengers use discounted tickets on journeys from Town X to Town Y, than on the return journeys from Town Y to Town X. To test this suggestion, she surveyed a random sample of 50 passengers who travelled from Town X to Town Y last Friday and recorded 15 passengers using discounted tickets. Equivalent data based on a random sample of 45 passengers on the return journey from Town Y to Town X recorded 12 passengers using discounted tickets. (c) (i) Test whether the proportion of passengers using discounted tickets is significantly different on journeys from Town X to Town Y than on journeys from Town Y to Town X. (9) *S48188A01125* Pearson LCCI Level 3 Certificate in Business Statistics 11 13

18 (ii) State, in context, what the test in (c) (i) shows. (Total for Question 2 = 20 marks) *S48188A01225* Pearson LCCI Level 3 Certificate in Business Statistics

19 BLANK PAGE *S48188A01325* Pearson LCCI Level 3 Certificate in Business Statistics 13 15

20 3 A large supermarket chain is planning to open a new store in one of nine English regions. The decision on which region is chosen will be based on the number of houses built, median house prices and change in regional traffic flow over a 10 year period, for each region. The table shows a summary of statistics relating to housing and transport for the nine English regions. English Region Number of houses built (000) in 2013 (X) Median house price ( 000) in 2013 (Y) Percentage increase in traffic between 2003 and 2013 North East North West Yorkshire East Midlands West Midlands East London South East South West (Source: Crown copyright 2014) Initial analysis of the data above using an appropriate software package gave the following summary results: X = 107 Y = 1460 X² = Y² = XY = *S48188A01425* Pearson LCCI Level 3 Certificate in Business Statistics

21 (a) Calculate the product moment correlation coefficient for the data, using the summary results provided. (4) (b) (i) Test whether the product moment correlation coefficient calculated in (a) is significantly different from zero. (5) (ii) State, in context, what the test in (b) (i) shows. *S48188A01525* Pearson LCCI Level 3 Certificate in Business Statistics 15 17

22 The regression line for the X,Y data passes through the following pairs of coordinates (11.889, ) and (20, ). (c) Find the regression equation in the form y = a + bx. (4) The data for median house price and percentage increase in traffic from 2003 to 2013 were ranked. A spreadsheet analysis of the ranked data gave a value for d² of 197. (d) (i) Calculate Spearman s rank correlation coefficient for the two sets of data (3) (ii) State, in context, what the coefficient in (d) (i) shows. *S48188A01625* Pearson LCCI Level 3 Certificate in Business Statistics

23 (e) Calculate the percentage difference of the median house price in the South East region in 2013 compared to the London region in the same year. (2) (Total for Question 3 = 20 marks) *S48188A01725* Pearson LCCI Level 3 Certificate in Business Statistics 17 19

24 4 The manager of a nationwide chain of coffee shops is concerned about the falling profits across the business. She suggests carrying out customer-satisfaction interviews to find likely reasons for the decline in profits. Customers are categorised into four groups: pensioners, adults, teenagers and children. Based on earlier surveys the manager believes the four groups to be in roughly equal proportions. The proposal is to have face-to-face interviews with a sample of 70 pensioners, 70 adults, 70 teenagers and 70 children. (a) (i) Name this type of sampling: (ii) Give three reasons why simple random sampling from the company s coffee shops might be unsuitable. (3) As an alternative, the manager suggests that a quota sample of customers from shops in one city should be selected. (b) (i) Give one advantage of quota sampling. (ii) Give one reason why quota sampling may not be suitable. *S48188A01825* Pearson LCCI Level 3 Certificate in Business Statistics

25 (c) Describe how a sample of customers could be selected using cluster sampling. (3) The Financial Accountant prepares a statement of quarterly sales revenue for one coffee shop from 2012 to Part of the spreadsheet summary is provided. Year Quarter Sales revenue ( 000) The first centred moving average value for sales revenue is (d) Calculate the remaining moving averages for 2012 to (2) (e) Describe the following patterns in the sales data: (i) the trend Pearson LCCI Level 3 Certificate in Business Statistics Sample Assessment Materials Issue *S48188A01925* 1 April 2015 Pearson Education Limited

26 (ii) the quarterly variation. (f) Estimate the second quarter seasonal effect using the additive model. (2) (g) Estimate the sales revenue for the second quarter of (3) The coffee shop will close if the sales revenue for any quarter falls below (h) Calculate the quarter when this is likely to happen, using only your trend estimate from part (g). (2) (Total for Question 4 = 20 marks) *S48188A02025* Pearson LCCI Level 3 Certificate in Business Statistics

27 5 As part of the trainee selection process, the manager of a large engineering company asks applicants to complete a short numeracy test. The table shows the marks achieved (out of a maximum mark of 100) by the applicants who completed the test: Marks awarded (x) Frequency 0 < x < x < x < x < x < x (a) (i) Draw a histogram to represent the data, on the graph paper provided. (5) *S48188A02125* Pearson LCCI Level 3 Certificate in Business Statistics 21 23

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29 (ii) Describe the shape of the distribution shown on the histogram in (a) (i). (iii) State the modal class of the data. After the trainees had been selected, the company arranged a training course in basic numeracy. A random sample of 10 trainees was selected for this course. The marks achieved by the sample before starting the training course were recorded together with the marks achieved on an equivalent test after completing the training course. The results are summarised. Trainee A B C D E F G H I J Marks before training course (%) Marks after training course (%) (b) (i) Test whether the marks achieved after the training course show a significant improvement on the marks achieved before the training course. (9) *S48188A02325* Pearson LCCI Level 3 Certificate in Business Statistics 23 25

30 (ii) State, in context, what the test in (b)(i) shows. (iii) Give one advantage of using a paired t test, rather than an independent t test, in this case. *S48188A02425* Pearson LCCI Level 3 Certificate in Business Statistics

31 (c) (i) Define a Type 1 error. (ii) State, with a reason, if a type 1 error has occured when carrying out the test in (b)(i). (Total for Question 5 = 20 marks) TOTAL FOR PAPER = 100 MARKS *S48188A02525* Pearson LCCI Level 3 Certificate in Business Statistics 25 27

32 Abbreviations M1 Method Mark This is used to reward candidates where there is evidence of the candidate having adopted the correct method for a calculation, but where the accuracy of the answer is not necessarily being awarded a mark. A1 Accuracy Mark This is used to reward candidates who have attained the answer to a specific calculation representing a method in the question. If stated in the mark scheme, the own figure rule can be used with the accuracy mark. of Own Figure rule Accuracy marks can be awarded where the candidates answer does not match the mark scheme, though is accurate based on their valid method. cao Correct Answer Only rule Accuracy marks will only be awarded if the candidates answer is correct, and in line with the mark scheme. oe Or Equivalent rule This rule is used when the value of an answer may be presented in a variety of ways, such as fractions (simplified/non-simplified/mixed), decimals, percentages. The candidates answer must be equivalent in value to the mark scheme answer. awrt Anything Which Rounds To rule This rule is used when the candidate supplies a figure which rounds to the value determined by the mark scheme. 28 Pearson LCCI Level 3 Certificate in Business Statistics

33 Question Answer (AO2) 1/ (AO3) 2 Mark 1(a)(i) Award 1 mark (AO3) for statement and explanation of both the null and alternative hypotheses, e.g. Null hypothesis/h 0 : there is no association between method of contact and grant application outcome. Alternative hypothesis/h A /H 1 : there is an association between method of contact and grant application outcome. Award 1 mark (AO2) for either critical chi-squared value (cao) (Degrees of freedom (r-1)(c -1) = (3-1)(2-1) = 2) Critical chi-squared ( 2 ) 5.99/9.21 Award 1 mark (AO3) for accepting the null hypothesis at the relevant level, e.g. Accept the Null hypothesis at the 0.05 level. Accept the Null hypothesis at the 0.01 level. Accept the Null hypothesis at 0.05 and 0.01 levels. (3) Question Answer (AO4) 1 Mark 1(a)(ii) Award 1 mark for interpreting a conclusion, in context, at a given level. There is no association between method of contact and grant application outcome. Pearson LCCI Level 3 Certificate in Business Statistics 29

34 Question Answer (AO2) 5 Mark 1(b) 216 p = = Standard error = 0.72 ( ) 300 (A1 cao) (M1) = (A1 of) 0.72 ± (1.96)(0.0259) Award full marks for correct range, e.g. Between: and % and 77.1% 67% and 77% (M1) (A1 cao) (5) Question Answer (AO2)4 Mark 1(c)(i) Award 1 method mark for correct use of the formula for the interval estimate. Award 1 accuracy mark for substituting own figure value of p from (b) into the correct formula. (M1,A1 of) 1.96x 0.72 ( ) n < x ( 0.72)(1 0.72) < n (M1) n=1937 (accept 1936 or 2016 based on either 1.96 or 2 SE) (A1of) (4) 30 Pearson LCCI Level 3 Certificate in Business Statistics

35 Question Answer (AO3) 1 Mark 1(c)(ii) Award 1 mark for any valid response, e.g. Increased cost/time/resources Not acceptable: doubling the sample size Question Answer (AO2) 3/(AO3) 2 Mark 1(d)(i) Award 1 mark (AO3) for statement and explanation of both the null and alternative hypotheses, e.g. Null hypothesis/h 0 : there has been no significant change in the average value of the grant request Alternative hypothesis/h A /H 1 : there has been a significant change in the average value of the grant request. Award 1 mark(ao2) for either critical (t) value (cao) (A1) (Degrees of freedom (n-1) = 8) Critical (t) = 2.31/3.36 s = t = x s / n = 21/ 9 = Award 1 mark (AO3) for rejecting and accepting the null hypothesis at the relevant level, e.g. Reject the Null hypothesis at the 0.05 level but accept the Null hypothesis at the 0.01 level. (M1) (A1cao) (5) Pearson LCCI Level 3 Certificate in Business Statistics 31

36 Question Answer (AO4)1 Mark 1(d)(ii) Award 1 mark for interpreting a conclusion, in context, at both levels. There is some evidence to suggest that there has been a significant change in the average value of the grant request. Not acceptable: There is evidence there has been a significant change in the average value of the grant request. 32 Pearson LCCI Level 3 Certificate in Business Statistics

37 Question Answer (AO3) 4 Mark 2(a) Award 1 mark for 1st pair of branches. Must show the correct label or correct probability on each branch. Award 1 mark for 2nd set of branches following on from the on time branch. Must show the correct label or correct probability on each branch. Award 1 mark for 2nd pair of branches following on from the late branch. Must show the correct label or correct probability on each branch. Award 1 mark for fully labelled - description plus probability on each branch. (4) Pearson LCCI Level 3 Certificate in Business Statistics 33

38 Question Answer (AO2) 2 Mark 2(b)(i) Award 1 mark for selecting correct branches (of) and applying multiplication and addition rules correctly. (0.8 x 0.8) + (0.2 x 0.25) = 0.69 (M1) (A1 of) (2) Question Answer(AO2) 2 Mark 2(b)(ii) Award 1 mark for selecting correct branches (of) and applying conditional rules correctly. (0.2 x 0.25) /0.69 = (M1) (A1 of) (2) Question Answer (AO2) 2 Mark 2(b)(iii) Award 1 mark for selecting correct branches (of) and applying conditional rules correctly. (0.8 x 0.65) / (0.8 x 0.65)+(0.2 x 0.25) = (M1) (A1 of) (2) 34 Pearson LCCI Level 3 Certificate in Business Statistics

39 Question Answer (AO2) 7 / (AO3) 2 Mark 2(c)(i) Award 1 mark (AO3) for statement and explanation of both the null and alternative hypotheses, e.g. Null hypothesis/h 0 : there is no significant difference in the proportions using the discounted ticket on the journeys from Town X to those returning to Town X. Alternative hypothesis/h A /H 1 : there is a significant difference in the proportions using the discounted ticket on the journeys from Town X to those returning to Town X. Award 1 mark (AO2) for both (cao) p 1 = 15/50= 0.3 p 2 = 12/45 =0.267 Award 1 mark(ao2) for either critical z value (cao) critical z = 1.96/2.58 pooled p = (15+ 12)/( ) =0.284 (M1) (A1 of) z = p p p(1 p) n m 2 = Award 1 method mark for correct use of the formula for the test. Award 1 accuracy mark for substituting own figure value of pooled p into the correct formula. (M1) (A1 of) ( ) (A1 cao) Award 1 mark (AO3) for accepting the null hypothesis at the relevant level, e.g. Accept the Null hypothesis at the 0.05 level. Accept the Null hypothesis at the 0.01 level. Accept the Null hypothesis at 0.05 and 0.01 levels. (9) Pearson LCCI Level 3 Certificate in Business Statistics 35

40 Question Answer (AO4) 1 Mark 2(c)(ii) Award 1 mark for interpreting a conclusion, in context, at either level. There is no significant difference in the proportions of passengers using the discounted ticket on the journeys from Town X compared to those returning to Town X. Not acceptable: more people travel from Town X than return to Town X 36 Pearson LCCI Level 3 Certificate in Business Statistics

41 Question Answer (AO2) 4 Mark 3(a) Award 1 method mark for correct use of the formula. Award 1 accuracy mark for substituting all correct figures into the formula. 9x x1460 (9x )(9x ) (M1) (A1cao) Award 1 mark for either denominator or numerator correct. (Accept 0.58) r= (A1) (A1cao) (4) Pearson LCCI Level 3 Certificate in Business Statistics 37

42 Question Answer (AO2) 3/ (AO3) 2 Mark 3(b)(i) Award 1 mark (AO3) for statement and explanation of both the null and alternative hypotheses, e.g. Null hypothesis H 0 : no significant correlation between houses built and price. Alternative hypothesis /H A /H 1 : there is significant correlation between houses built and price. Award 1 mark (AO2) for either critical t value (cao) (Degrees of freedom (n-2) =7) critical t = 2.37/3.5 (A1) Calculated t = r (n-2) (1-r² ) Award 1 method mark (AO2) for correct use of formula. Award 1 accuracy mark (AO2) for substituting own figures correctly into the formula. = (0.576)( 7) ( ) (M1) (Accept 1.87) = x t = (A1cao) Award 1 mark (AO3) for accepting the null hypothesis at the relevant level, e.g. Accept the Null hypothesis at the 0.05 level. Accept the Null hypothesis at the 0.01 level. Accept the Null hypothesis at 0.05 and 0.01 levels. (5) 38 Pearson LCCI Level 3 Certificate in Business Statistics

43 Question Answer (AO4) 1 Mark 3(b)(ii) Award 1 mark for interpreting a conclusion, in context, at either level. There is no significant correlation between number of houses built and median house prices. Not acceptable: there is a connection between the number of houses built and median house price. Question Answer (AO2) 4 Mark 3(c) = a b (M1 cao) = a+20b (M1 cao) Subtraction giving a= b=5.052 y= x (A1 cao for both) (A1of) (4) Question Answer (AO2) 3 Mark 3(d)(i) Award 1 method mark for correct use of the formula. Award 1 accuracy mark for substituting all correct figures into the formula. (M1, A1) Sp=1- [(6x197/(9x80)] Sp= (A1cao) (3) Question Answer (AO4) 1 Mark 3(d)(ii) Award 1 mark for interpreting the value of the Spearman s rank correlation coefficient, in context, e.g. As house prices increase some evidence that the % increase in traffic reduces/some evidence that as the % increase of traffic increases, house prices decrease. Pearson LCCI Level 3 Certificate in Business Statistics 39

44 Question Answer (AO3) 2 Mark 3(e) ( /250)X100 (M1) = 18.8% (lower in the SE region) (A1cao) (2) 40 Pearson LCCI Level 3 Certificate in Business Statistics

45 Question Answer (AO1) 1 Mark 4 (a)(i) Award 1 mark for correct answer only. Stratified sampling Question Answer (AO3) 3 Mark 4(a)(ii) Award 1 mark for any valid reasons up to a maximum of 3 marks, e.g. No sample frame Cost Geographical spread Likely non response (fast-food) Non-representative (3) Question Answer (AO1) 1 Mark 4(b)(i) Award 1 mark for any valid advantage, e.g. More convenient Cheaper Quicker Not acceptable: more accurate Question Answer (AO3) 1 Mark 4(b)(ii) Award 1 mark for any valid reason, e.g. Non representative Doesn t allow for further statistical analysis Not acceptable: too large a sample Question Answer (AO3) 3 Mark 4(c) Award 1 mark for each element of description. Random selection of a region Choose shop at random from within the region Select representative quota from shop (3) Pearson LCCI Level 3 Certificate in Business Statistics 41

46 Question Answer (AO2) 2 Mark 4(d) Award 1 method mark either for a correct method seen or for a correct moving average seen (not 91.5). Award 1 accuracy mark for moving average figures all correct. Example of method mark: (M1) ( )/4 = 92 ( )/4 = 91.5 ( )/2 = (A1) (2) Question Answer (AO3) 1 Mark 4(e)(i) Award 1 mark for correct description. Trend declining Question Answer (AO3) 1 Mark 4(e)(ii) Award 1 mark for any valid response, e.g. Seasonal peaks in Q3 Seasonal troughs in Q1 42 Pearson LCCI Level 3 Certificate in Business Statistics

47 Question Answer (AO2) 2 Mark 4(f) Award 1 method mark for use of formula. Award 1 accuracy mark for correct estimate. (Y-T) for Q mean seasonal effect 9.5/2 = 4.75 (Accept 4.8) (M1) (A1) (2) Question Answer (AO2) 3 Mark 4(g) t 1 Trend growth = n t = n = (A1of) Award 1 mark for the correct use of formula. forecast (4)(0.964) = (M1) (A1 of) (3) Question Answer (AO2) 2 Mark 4(h) Award 1 method mark for use of formula. Award 1 accuracy mark for correct quarter and year. (M1) n(0.964) = 78.5 n= Q (A1 of) (2)

48 Question Answer (AO2) 4 /(AO3) 1 Mark 5(a)(i) Award 1 method mark for calculating frequency densities. Award 1 accuracy mark for correct frequency densities (cao). (M1, A1) Frequency densities Award 1 mark (AO2) for correct horizontal plots (class interval). Award 1 mark (AO2) for correct vertical plots. Award 1 mark (AO3) for correct scales/labels. Linear scale Correct labelling of horizontal axis: marks/x Correct labelling of vertical axis: frequency density Title not required Frequency density Marks awarded (5)

49 Question Answer (AO3) 1 Mark 5(a)(ii) Award 1 mark for any valid response based on own figure histogram, e.g. Skew to left Positive skew Symmetrical/normal Not acceptable : bimodal Question Answer (AO2) 1 Mark 5(a)(iii) Award 1 mark (AO2) for 25<x 40 (of) Not acceptable: the class with the largest frequency.

50 Question Answer (AO2) 7 / (AO3) 2 Mark 5(b)(i) Award 1 mark (AO3) for statement and explanation of both the null and alternative hypotheses, e.g. (A1) Null hypothesis: there is no significant difference in the marks achieved after the training. Alternative hypothesis: there is a significant improvement in the marks achieved after the training course. Award 1 mark (AO2) for either critical t value (cao) (Degrees of freedom = (n-1) =9) critical t value OTT = 1.83/2.82 d d² (d-d) ² d 6.2 d² = 1372 or ( d d)² = s = ( d d) / n 1 = / 9 = d 0 t s/ n / Award 1 mark (AO3) for rejecting and accepting the null hypothesis at the relevant level, e.g. reject the Null hypothesis at the 0.05 level but accept the Null hypothesis at the 0.01 level. (A1) (A1cao mean) (A1caoƩd²) (M1) (A1 of) (M1) (A1 cao) (9)

51 Question Answer (AO4) 1 Mark 5(b)(ii) Award 1 mark for interpreting the value of t, in context, e.g. There is some evidence that training significantly improves the marks achieved. Question Answer (AO3) 1 Mark 5(b)(iii) Award 1 mark for advantage of using paired t test, e.g. Concentrate on the effect of the training rather than individual trainee differences. Not acceptable: Independent t test uses 2 different samples. Question Answer (AO1) 1 Mark 5(c)(i) Award 1 mark for correct definition. Type 1 error: incorrectly rejecting a true null hypothesis Question Answer (AO3) 1 Mark 5(c)(ii) Award 1 mark for indication of possible occurrence at the relevant level with a valid reason, e.g. Type 1 error may have occurred at 0.05 level as Null hypothesis is rejected.

52 For more information on LCCI qualifications please visit our website: uk.pearson.com/lcciinternational Pearson Education Limited. Registered in England and Wales No Registered Office: 80 Strand, London WC2R 0RL. VAT Reg No GB