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LEVEL 3 WJEC Level 3 Certificate in STATISTICAL PROBLEM SOLVING USING SOFTWARE SPECIMEN ASSESSMENT MATERIALS - External Teaching from 2015 Level 3 Certificate in Statistical Problem Solving using software ESAMs 1 Candidate Name Centre Number Candidate Number LEVEL 3 CERTIFICATE IN STATISTICAL PROBLEM SOLVING USING SOFTWARE SPECIMEN PAPER 1 hour ADDITIONAL MATERIALS The use of a calculator is permitted in this examination. 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. For Examiner’s use only Maximum Mark Question Mark Awarded 1. 2 2. 2 3. 5 4. 14 5. 4 6. 8 Total 35 INFORMATION FOR CANDIDATES The number of marks is given in brackets at the end of each question or part-question. 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. Level 3 Certificate in Statistical Problem Solving using software ESAMs 2 1. The data below are taken from the Global Slavery Index which states that: '29·8 million people are in modern slavery globally'. The three countries with the highest number of people in modern slavery are listed in table 1 below. Country Name India China Pakistan Rank for Number of Slaves 1 2 3 Country Population Number of Slaves 1 236 686 732 1 350 695 000 179 160 111 13 956 010 2 949 243 2 127 132 Table 1: Countries with the top 3 highest number of slaves (Global Slavery Index) From the table above, India, China and Pakistan have the highest number of people in modern slavery. Explain why these three countries may not have the highest proportions of their populations in modern slavery. [2] …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… Level 3 Certificate in Statistical Problem Solving using software ESAMs 3 2. The mean mark and the standard deviation for an end of term examination were calculated for two classes, Class A and Class B. Class A had a mean mark of 35·2 marks and a standard deviation of 2·8 marks. Class B had a mean mark of 48·8 marks and a standard deviation of 5·1 marks. Considering the summary statistics above, explain which class would you prefer to teach. Give a reason for your answer. [2] …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… Level 3 Certificate in Statistical Problem Solving using software ESAMs 4 3. House prices are always in the news. For example: The average asking price of a home has risen above the one per cent stamp duty threshold of £250,000, according to new figures. Sunday Telegraph, 23 March 2014 London's house prices soar to hits new record average of £409,000. London Evening Standard, 28 February 2014 Figure 1 represents the prices of houses sold in 2013 for an area in the South West of England. Table 2 shows the descriptive statistics for the prices of houses sold in 2013 for an area in the South West of England. 15 Frequency 10 5 House Price £s in 1000s 0 150 200 250 300 350 400 450 500 550 600 Figure 1: Prices of houses sold in 2013 for an area in the South West of England Descriptives for house price Minimum £215 000 Mean £318 179 Median £288 500 Mode £250 000 Maximum £480 000 Standard Deviation £77 678 Range £265 000 Count 28 Table 2: Descriptive statistics for prices of houses sold in 2013 for a certain area in the South West of England. Level 3 Certificate in Statistical Problem Solving using software ESAMs 5 (a) Which is the most appropriate average to use for this sample of house prices? [1] ………………………………………………………………………………………………… ………………………………………………………………………………………………… (b) Explain why you consider the average you chose in part (a) to be the most appropriate. [2] ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… (c) Referring to the information on the previous page, state two possible ways in which reports on the average cost of houses can be misleading. [2] ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… Level 3 Certificate in Statistical Problem Solving using software ESAMs 6 4. “Smoking cigarettes is probably the No. 1 cause of adverse outcomes for babies," says Robert Welch, MD, chairman of the Department of Obstetrics and Gynecology at Providence Hospital in Southfield, Michigan.' Table 3 shows 4 responses from a total of 1132 obtained from mothers with newborn babies. Mother's Weight (lbs) 90 135 107 250 Mother's Height (inches) 60 67 66 66 Baby's Birth Weight (lbs) 6·88 9·31 7·69 7·88 Mother Smokes NO NO YES NO Table 3: Extract from 1132 responses from mothers with newborn babies. (a) A statistician wishes to compare the average weight of babies born to mothers who smoke with those who do not smoke. How could the statistician present the data from the 1132 mothers graphically to make the comparison? State the variables the statistician would use. [2] ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… (b) Give a reason for your choice of graphical display in part (a). [1] ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… (c) The weights of newborn babies for mothers who smoke and for the weights of newborn babies for mothers who do not smoke may be considered to be Normally distributed and have equal variances. Give two reasons why a two sample t-test is an appropriate test to compare the average birth weight of babies born to mothers who smoke with those who do not smoke. [2] ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… Level 3 Certificate in Statistical Problem Solving using software ESAMs 7 (d State clearly the null and the alternative hypothesis for a two sample t-test to compare the average birth weight of babies born to mothers who smoke with those who do not smoke. [4] ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… A two sample t-test at the 0·05 level is carried out to compare the average birth weight of babies born to mothers who smoke with those who do not smoke using statistical software. The output for this test is in table 4. Mean Variance Number of Observations Pooled Variance Degrees of freedom t Stat P(T≤t) one-tail t Critical one-tail P(T≤t) two-tail t Critical two-tail Birth Weight (lb) Non Smoker Birth Weight (lb) Smoker 7·690 7·089 1·187 1·289 688 444 1·227 1130 8·914 0·000 1·646 0·000 1·962 Table 4: Statistical software output for the two sample t-test for birth weight of babies born to mothers who smoke with those who do not smoke (e) Interpret this output and clearly state your conclusions. [3] ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… Level 3 Certificate in Statistical Problem Solving using software ESAMs 8 (f) If the statistician wanted to carry out further studies on the weights of newborn babies what other factors concerning the mothers might they wish to investigate? [2] ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… 5. A researcher wanted to investigate whether UK secondary school pupils learn how to carry out a statistical investigation more effectively by using statistical software or by plotting graphs by hand and carrying out calculations using a calculator. The study involved 52 pupils in year 11 at a secondary school in the UK. The researcher decided that one class (26 pupils) would be taught in a classroom using calculators for the statistical analysis and plotting graphs by hand. The other class (26 pupils) would carry out the analysis in a computer room using statistical software. (a) State the target population for this investigation. [1] ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… (b) Write down three weaknesses in this experiment design. [3] ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… Level 3 Certificate in Statistical Problem Solving using software ESAMs 9 6. “Brush your child’s teeth twice a day for two minutes and pay special attention to the back molars, where cavities tend to develop” is the advice from Live Science 21 March 2014. A researcher wishes to investigate whether the frequency of teeth brushing by children in the UK is related to age. The researcher was only able to collect data from 12-year-olds and 15-year-olds at one large UK secondary school. The results are shown in table 5. Frequency of teeth brushing Less than once a day Once a day Twice a day Three or more times a day Total Number of children Age 12 years 15 years 24 10 149 84 434 339 45 65 652 498 Total 34 233 773 110 1150 Table 5: The number of times children clean their teeth a day at a UK secondary school. The researcher decided to carry out a Chi-Squared test on these data. (a) Give a reason why the Chi-Squared test is appropriate. [1] ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… (b) State clearly the null and the alternative hypothesis that the researcher is testing. [4] ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… Level 3 Certificate in Statistical Problem Solving using software ESAMs 10 The p-value for the Chi-Squared test in this case is less than 0·001. Interpret the p-value at the 0·05 level of significance. [2] …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… (c) Write the conclusion in relation to the original problem. [1] ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… Level 3 Certificate in Statistical Problem Solving using software ESAMs 11 Specimen Assessment Materials Statistical Problem Solving 1. Full explanation with reference to the population not taken into account e.g. “in order to compare we should take into account the population and use the percentages/proportions or fractions. The table only shows the counts”. 2. Choice of class with clear reasoning for choice referring to the means and standard deviations e.g. “Prefer to teach Class A. Although the students have not such a high mean/average mark as Class B the spread is less and therefore the ability of the children similar” or “Prefer to teach Class B. The students in Class B have a higher mark and perhaps are more able but would be more challenging to teach as the class would be of mixed ability”. 3. (a) Median or modal group/class. 3. (b) Explanation that the distribution is positively skewed (or not symmetrical) so median is the best measure of location as it is not distorted by extreme values. OR Explanation that the distribution is positively skewed (or not symmetrical) so the modal group/class is not distorted by extreme values. 3. (c) Two statements relating to the possible misuse using the mean instead of the median or mode. e.g. “The statements from the Sunday Telegraph and the London Evening Standard could be related to selling prices not the sold price” “The mean can be used to exaggerate the sold prices of houses” “The way the average is calculated is not given; not sure which average is being used so very difficult to compare”. Comments Page 1 1 mark for sight of percentage/fraction but no reference to the population OR 1 mark for a partial explanation Mark AC 2 3.3 [2] 2 3.3 No marks if Class A or B stated with no reason. 1 mark if Class A or B is stated referring to either the mean or the spread OR 1 mark if discussion of mean and spread but no choice of class. 1.4 No marks for mode. 1.5 1 mark for mention of shape of the distribution e.g. most values around £250,000 with a few much more expensive, not symmetrical. 1 mark for stating the median is not affected by extreme values as it is the middle value OR 1 mark for stating the modal value or group is not affected by extreme values as it is the most frequent value or group. 4.1 1 mark for each appropriate statement. [2] 1 (1) 2 (2) 2 (2) [5] Level 3 Certificate in Statistical Problem Solving using software ESAMs 12 Specimen Assessment Materials Statistical Problem Solving 4. (a) Box plots or histograms plus variables used: Baby's Birth Weight (lbs) and Mother Smokes. 4. (b) Box plots to compare medians and spread, or histograms to compare the shape of the distribution and the spread 4. (c) Two clear statements on why the two sample t-test is appropriate in this case. e.g. the data are not paired; different sample sizes; the assumptions of normally distributed populations and equal variances hold (given in question); parametric test. 4. (d) Clear expressed null hypothesis including correct notation Ho or Null Hypothesis and relating to the context of the problem. e.g. Ho or Null Hypothesis: The population mean birth weight of babies born to mothers who smoke is the same as the mean birth weight of babies born to mothers who do not smoke. Clear expressed alternative hypothesis including correct notation H1 or Alternative Hypothesis and relating to the context of the problem. e.g. H1 or Alternative Hypothesis: The population mean birth weight of babies born to mothers who smoke is different to the population mean birth weight of babies born to mothers who do not smoke. 4. (e) Correct interpretation of the p-value at the 0·05 significance level e.g. since the p-value is less than 0·05 we reject the null hypotheses (Ho) and conclude there is evidence to suggest there is a difference in the means for the weights of babies born to mothers who smoke and those who do not smoke. 4. (f) Two examples of what other factors/variables could be considered e.g. Mother's weight, age, height, life style Comments Page 2 1 mark for box plot or histogram with no mention of variables used or 1 mark for variables used i.e. Baby's Birth Weight (lbs) and Mother Smokes Mark AC 2 1.4 (2) 1 1.5 (1) 2 4.2 1 mark for each correct statement (2) 2 1.1 1 mark for partial explanation for null hypothesis e.g. Ho or Null Hypothesis: Means are equal. Ho or Null Hypothesis: Mean baby weight are equal. 1 mark for correct hypothesis but H0 missing. 2 1.1 1 mark for partial explanation for alternative hypothesis e.g. H1 or Alternative Hypothesis: Means are not equal. H1 or Alternative Hypothesis: Mean baby weight are not equal. 1 mark for correct hypothesis but H1 or Alternative Hypothesis missing. Full marks cannot be awarded if population is not mentioned or implied. (4) 3 3.3 1 mark for stating the p-value is less than 0·05. (No marks for saying the p-value is equal to zero.) 1 mark for rejecting the null hypotheses (Ho) or accepting the alternative hypotheses (H1). 1 mark for concluding there is evidence to suggest there is a difference in the means for the weights of babies born to mothers who smoke and those who do not smoke. (3) 2 4.2 1 mark for each appropriate example. (2) [14] Level 3 Certificate in Statistical Problem Solving using software ESAMs 13 Specimen Assessment Materials Statistical Problem Solving 5. (a) All children in the UK. 5. (b) Three weaknesses stated e.g. No randomisation within the classes; teacher has to decide which class received each treatment; the abilities of the classes may vary; the classes may have different teachers; only one school. 6. (a) Explanation of the appropriateness of Chi-squared test e.g. “it is appropriate because counts are being used” 6. (b) Clear expressed null hypothesis including correct notation Ho or Null Hypothesis and relating to the context of the problem e.g. H0 or Null Hypothesis: the number of times a day children brush their teeth is independent of age OR H0 or Null Hypothesis: there is no association between the number of times a day children brush their teeth and age Clear expressed alternative hypothesis including correct notation H1 or Alternative Hypothesis and relating to the context of the problem e.g. H1 or Alternative Hypothesis: the number of times a day children brush their teeth is not independent of age OR H1: or Alternative Hypothesis there is an association between the number of times a day children brush their teeth and age 6. (c) Correct interpretation of the p-value at the 0.05 significance level e.g. “since the p-value is less than 0.05, there is evidence to suggest we reject the null hypothesis (Ho) (or accept the alternative hypotheses (H1))” 6. (d) For relating the conclusion to the original problem e.g. “there is evidence to suggest that age does have an effect on how often children brush their teeth a day”. Mark AC 1 (1) 3 1.3 2.2 Comments Page 3 1 mark for each weakness stated. (3) [4] 1 1.5 (1) 2 1.1 1 mark for partial explanation for null hypothesis. e.g. reference to independence or categories of number of times children clean their teeth but not both. e.g. “H0 or Null Hypothesis : the variables are independent” OR 1 mark for correct hypothesis but H0 or Null Hypothesis missing. 2 1.1 1 mark for partial explanation for alternative hypothesis e.g. reference to association/non-independence or categories of number of times children clean their teeth but not both. e.g. “H1 or Alternative Hypothesis: the variables are associated”. OR 1 mark for correct hypothesis but H1 or Alternative Hypothesis missing. Full marks cannot be awarded if population is not mentioned or implied. (4) 2 3.3 (2) 1 Must include “reject H0” or equivalent for 2 marks 1 mark for comparison of p-value with 0.05 but no conclusion OR 1 mark for “reject H0” or equivalent with no reference to 0.05 4.1 (1) [8] Level 3 Certificate in Statistical Problem Solving using software ESAMs 14 Mark and Assessment Criteria grid ESAMs LEVEL 3 Certificate In Statistical Problem Solving Using Software/HT/04 11 2014