AQA GCSE Maths Foundation Skills Book Statistics 1

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1 Guidance on the use of codes for this mark scheme ethod mark A Accuracy mark ark awarded independent of method cao Correct answer only oe Or equivalent ft Follow through AQA GCSE aths Foundation Skills ook Statistics 1

2 Question Working Answer ark AO Notes Grade 1 a Three appropriate questions using for each question statistical measure for the data provided. e.g. What is the range of heights of the students? What is the median height of students? What is the mean average height of students? e.g. stem and leaf The numer of data points makes it difficult to represent. Data all to the nearest cm makes it easy. Total numer of data points and how many in each category. for stem and leaf oe for appropriate evaluation of what makes it easy/hard to represent this data for correct description of two pieces of required 1 information for appropriate grouping of time ranges for tally chart with frequencies identified Range etween and 8 minutes, so choose at least four groups for data. e.g. The classes overlap: where would you put a drink costing 0p or 60p? A drink could cost over 1. Response identifying that the pie charts represent proportions not actual numers. 6 Response explaining that Kevin may have data to show that crowds at the cricket ground always exceed If there is no time when the crowds are elow then there is no point showing this part of the graph as it will e empty. for evaluation of/reasoning for choice of time ranges. for each correct reason for correct reasoning and selection of The proportion of people uying fruit in town is greater than 1 in the village for comment that data may exist to show that the minimum crowd is for comment oe AQA GCSE aths Foundation Skills ook Statistics

3 7 a Example of a small set of data that has a mode of 6 with a valid explanation. e.g. 6, 6, 6, for any data set with a mode of 6 and an explanation that more 6s than anything else were included c Example of a small set of data that has a mode of 6 and a range of 1 with a valid explanation e.g. 1, 6, 6, 6, 6, 16 Two different small sets of data that have the same mode and range with explanation. e.g. 1,,,, 8,,,, 9 for any data set with a mode of 6 and an explanation that more 6s than anything else were included and the difference etween the greatest and least value is 1 for two correct data sets for descriing method for creating different ranges For example, shifting the greatest and least values y the same amount to keep the difference the same oe d 8 a Octoer: 16 oys and 1 girls Decemer: 18 oys and 10 girls oth months have 8 competitors Set with a mean of and a median of with explanation e.g. 1,,, 9 Octoer and Decemer 6 for correct median for data points summing to correct value (in the example, data points must sum to 16 as = 16) for accurate interpretation of data from graph 9 a Numer of oys = Numer of girls = = = 1 Total numer taking part is. So this would e their lowest month for participation. Shows the proportion not the numer is greater. For example, if there were a larger numer of men in the company, then their pie chart would represent greater numers within each section of the chart. for correct calculation of numer of oys for correct calculation of numer of girls for comment explaining low participation for comment that proportionality shown on pie charts The pie charts show proportion ut not the actual numers. e needs to use proportion in the justification or he needs to provide the numers of men and women involved in the survey. for an understanding that some further clarification is needed if pie charts are used OR a suitale suggestion of an alternative such as a ar chart (which would need some numerical data) oe AQA GCSE aths Foundation Skills ook Statistics

4 10 a Pictogram drawn with key for pictogram for key Appropriate ar chart drawn for ar chart with at least four ars correctly drawn for appropriate scale on y-axis e.g. going up in s c Choice of either with justification such as the ar chart shows the increasing trend more clearly or the pictogram is easier for readers to understand. for justification of choice 11 a c d e Internet search, secondary data Experiment, primary data Internet search (or questionnaire), secondary (or primary) data Internet search (or questionnaire), secondary (or primary) data Questionnaire, primary data for each correct description Data handling cycle: Plan the data collection Collect the data Choose the est way to process and represent the data. Interpret the data and draw conclusions Primary data: collected y you Secondary data: used y you ut collected y a third party AQA GCSE aths Foundation Skills ook Statistics

5 1 Data set of 10 points plus reasons as to why this would not make a good stemand-leaf (wide range; variance in place value) e.g. how far in kilometres ten people live from a lackpool Tower: this could have a theoretical range of km (or greater), with any values in etween, making it meaningless when selecting representation for the stem and the leaf..6, 6.7,., 10.9, 1.6, 1.0, 1.0, 7.9, 178., a Statements that identify that pie charts represents proportions, e.g. What proportion or percentage of people are in the East? What is the fraction of people in the South? What is the proaility that someone is from the North? for appropriate example where the range of data values is too varied to make a stem-and-leaf diagram have meaning for appropriate reasoning for appropriate question Total, range and mode are difficult to use as they are numerical and the pie chart shows proportions. for justification as to why numerical measures are difficult to use in this case 1 a The data is discrete/cannot take any values etween the ones shown. It should e plotted as a ar chart. for correct identification of discrete data oe for ar chart Continuous data, e.g. cost of living over time. for appropriate example oe 1 E.g. a line graph is etter for comparing trends over time as trends in the data are more ovious. for selection of line graph with appropriate reasoning (link to over time ) for further explanation (link to clarity of diagram) oe 16,,, (mean = ) 10, 10, 10, 10 (mean = 10) True plus suitale explanation e.g. n, n, n, n (mean = n) n, n, n, n (mean = n) for true with specific example to ack up explanation for more general example to demonstrate statement is true AQA GCSE aths Foundation Skills ook Statistics

6 = 6.th position So days days 1 1 A1 1 for correct method to find median value (taulated calculations) oe 1 for correct calculation to find position of median oe A1 cao Days d 0 1 Freq f Cumulative frequency Estimated mean = total profit total frequency = = Profit, p Freq,f idpoint, m F m Answer: Need to design a questionnaire and ask students what they have for reakfast. 0 a Start with,, and adjust at oth ends. Any three values that sum to 9 with a difference etween greatest and least of, e.g.,, 1 1 A1 1 for correct method to find the estimated mean (taulated calculations) oe 1 for correct final calculation with appropriate rounding A1 cao for descriptive comment aout the data needed and/or method of collection oe 1 1 A1 1 for either correct range or mean A1 cao Start with in the centre and find a pair for the ends which sum to 6 with a difference of. Any three values with a sum of 9, a difference etween greatest and least of and a median of, e.g. 1.,,. 1 A1 1 for either correct range or mean A1 cao Note: this can e done y solving a pair of linear equations a,, such that + a = 6 and a = ; =. and a = 1. 1 They could have een referring to two different averages. For example: in a class of 10 students with the following marks 10, 17, 8, 8, 8, 8, 8,, 7, 61 The median and modal mark is 8 ut the mean mark is. for correct comment regarding two different averages for example to justify AQA GCSE aths Foundation Skills ook Statistics 6

7 A scatter graph with positive correlation shows a trend with a positive gradient/ as one quantity increases the other also increases. for diagram to support explanation for comment oe a Total numer of male students = = 0.18 (to dp) 18% to nearest percent cao c Total numer of female students = 1 1 = 0.1 (to dp) 1% to nearest percent Find the mid-point of each class ultiply mid-point of class y frequency of class and add them all together. Divide the total earnings y the total frequency. cao for explanation for explanation for explanation (no credit for actual calculation as an explanation as this is specified in the question) ean for oys = 0 = 6 a Texts, T f 6 T f ean for oys is 6, so ased on the information provided and this particular statistical measure then the answer would e no. owever, the information is limited for example the sample of oys is small and you are not told how many girls are sampled. The mean could e ased on a single girl. Demonstrate the use of data to produce a time series graph to predict the life expectancy in oth countries in 0. UK: 8 8, Ukraine: anything etween 6 and 7 for calculation of mean numer of oys text messages for comparison for appreciation of sample size oe for data interpretation either y time-series graph or y spreadsheet UK may e easier to predict as historically it has een more consistent. for understanding of consistency making a trend easier to identify AQA GCSE aths Foundation Skills ook Statistics 7

8 6 a/ Scatter diagram completed with line of est fit for correct points plotted on scatter diagram for line of est fit c d en is most likely to e the student who was sick when they took maths as his score is much lower than his music score. Kris is likely to have scored around 0 marks in music as all students except en scored similar ut less in music than in maths (see trend line) for choice of student with valid reason oe for score for music ased on their trend line (ft) e Lex could have scored 8 in maths with a 78 score in music (see trend line) for score for maths ased on their trend line (ft) AQA GCSE aths Foundation Skills ook Statistics 8

9 7 Laour: 0 0 = 1 Conservative: = 7 Li Dem: = 0 Other: = 1 Angles: 1 Laour: 60 = Conservative: 7 60 = Li Dem: 0 60 = Other: 1 60 = 100 Pie chart constructed accurately and clearly. 8 Suitale example Algeraic explanation: x 1 x x x = mode x1 + x + x + x = 8 mode e.g., if the mode is the sum of the data would have to e 16 so that the mean was. One possile set could e:,,, 10. ode is, mean is 10 = 16 =.ean is twice mode, as required. 9 Comparison of different statistical ean edian ode Range measures that provide counterargument. e.g. oys have a higher mean, median. 0., 7 71 and modal score. The range is very similar. Girls scores % oys scores % for calculation of average, rounded to the nearest percentage for correct degree calculations for accurate measurement of sectors for correct laelling of sectors for specific example with suitale explanation. for general solution (could e presented algeraically or in words) for calculation of at least two averages or one average and the range to compare for calculation of further averages to support argument in favour of oys for evaluation and interpretation of averages to provide counter argument for oys AQA GCSE aths Foundation Skills ook Statistics 9

10 0 a ale Female edian ode Range ax in 7m 0s m 6s None 9m 08 s 0 m 0 s 07 m 7 s 8m s 6m 7s m 8 s 8m 0s Valid comparison of the distriutions for male and female runners. e.g. all men except one ran faster than the women (max and min values) Or there was less variance in men s running times than women s (range) e.g. median values for first comparison for further comparison such as the median values 1 a = 00 Valid couple of sentences to summarise findings suitale for a national running magazine Over 0% of voters so should e confident of winning = A1 At four different points in the concert, entrances or ars, each point to conduct the survey on approximately 100 people each, trying to ask as many males as females. for each of two summary sentences ased on their interpretations of the distriutions (ft) for multiplication of total population y proportion of sample for for explanation that his represents over half the whole population and so a majority is likely 1 for multiplication of total population y proportion of sample A1 cao for explanation of general method of sampling for specific example related to the concert crowd AQA GCSE aths Foundation Skills ook Statistics 10

11 a Define what is meant y a good train service. Could e numer of trains per hour or numer of trains that are on time in a specified period or customer satisfaction regarding the quality of various aspects of the journey such as level of cleanliness, availaility of sets etc. To find out if the statement is true or not, the first step is to see what has changed since the aunt has een travelling (see aove) for explanation of sujective nature of this enquiry for description of what might e seen as good Data to e collected would e dependent on what the aunt felt had changed (or what has actually changed).this could e data from the train company on volume of trains, amount of rolling stock at one time, numer of complaints received and aout what. This could also e data collected from a customer survey. owever, this would e real time and would e hard to gather retrospectively as this would e sujective and may not e possile, depending on time frame. for explanation of possile data to e collected with reason for appreciation of this eing a sujective enquiry over time for a correct description of a data collection process, such as constructing a questionnaire. 0 = 0 So 0 eggs in total 0 = 0 1 = = 8 1 = = 16 eggs 0 16 = eggs to find ens could lay or eggs. ( 6 ) + ( ) = = ( ) + ( 7) = = Eggs 0 1 Frequency 6 1 Egg frequency Could e eggs, frequency 6 and eggs, frequency Or could e eggs, frequency and eggs, frequency 7 for total eggs (0) for multiplication of eggs y frequency plus addition and sutraction to find how many eggs are left to find oe for one solution for a second solution for clear description or tale oe AQA GCSE aths Foundation Skills ook Statistics 11

12 6 a = patients for addition of frequencies 7 79 =.86.. Time id point 1... Frequency id-point frequency for mid-points frequencies for total minutes for division of total minutes y total patients (ft) for appropriate rounding to whole minutes Time id point.. 6. Frequency 1 9 id-point frequency Sum of mid-point frequencies = total numer of minutes = 7 Estimate mean waiting time = minutes c Estimated mean = minutes odal time = 1 0 minutes edian time: 1 0 minutes 79 1 = 0 0th person is the median. 0th person is in the 1 0 group ode for mode for median for correct, shortest, average waiting time d Over one hour (61 70) is patients 79 = So 6% of patients were over an hour = 9% 9% 1 A for calculation of 1 ( 7 ) or as a percentage A1 cao AQA GCSE aths Foundation Skills ook Statistics 1

13 7 a No, it only shows proportions for reference to proportions Estimate: Often = 90 (=%) Very often = 80 ( 0%) Rarely = 100 ( 0%) Never = 80 ( 0%) Always= 10 ( %) e.g. All percentages are estimates ased on the approximate angle of the sector of the pie chart. % of people sampled said that they often considered their health when planning a diet, with % saying that this was always the case and 0% saying very often. 0% rarely consider their health when planning a diet and 0% never do. e.g. So approximately half the sample consider their health to some degree and half do not. 8 The minutes spent waiting is halfway etween the - and 6-minute groups. These people are in that and, ut maye no one had to wait exactly minutes. 9 Divide the frequency of the class interval y the width of the class interval. 0 Kathy s mean Kathy: mean score of.7 = ( ) 9 =.7 Connie: median score of 7 Connie s mean Evie: modal score of 8 = ( ) =.6 Evie s mean = ( ) =. Dancer Scores ean edian ode Kathy 8, 7, 6,.7,,, Connie 9, 8, 7,.6 7, Evie 8, 8,,, 1. 8 for comment on the proportions of people giving the range of answers for summary for awareness that the minutes represents the mid-point of grouped data. People may have een waiting for any time etween and 6 minutes 1 1 for correct method for calculations of frequency density for calculation of all three means for identification of all three modes for identification of all three medians for correct average identified in all three cases AQA GCSE aths Foundation Skills ook Statistics 1

14 1 RangeA = = 70 Range = = eana = 86 6 ( 18) = 9 1. ean= ( 6 6 = Firm Range ean edian ode A Valid advice ased on the salary scales provided (see notes) e.g. Jasmin should join firm if she is prepared to accept that the higher final salary is not as high as A. owever, the starting salary is higher and Jasmin can progress more quickly to a higher salary with than A. There is a greater range of salaries in firm A than firm Jasmin should join firm A if she aspires to eing the oss. Alternatively she could start with firm and move to firm A after 7 salary increments for maximum earnings. 6 for calculation of ranges for calculation of means for identification of medians for identification of modes for justification of either firm A or firm for use of tale oe for comparison AQA GCSE aths Foundation Skills ook Statistics 1

15 a e.g. 0 x <; x < ; x < Estimated range (from mid-points) = 8.. for correct example (class widths do not need to e equal) e.g. Class 0 x < x < 7 7 x < 0 id-point 6 Frequency 1 Cumulative frequency 6 11 for correct example (class widths do not need to e equal) n = 11 hence 6th data point is the median Estimated median (from mid-point) = 6 c e.g. Class 0 x < 0 0 x < x < 119 id-point Frequency 1 Cumulative frequency 6 11 for correct example of estimated median for correct example of estimated range(class widths do not need to e equal) n = 11 hence 6th data point is the median Estimated median (from mid-point) =. Estimated range (from mid-points) = 7 10 = 6 d e.g. Class Frequency id-point Frequency mid-point 0 x < x < x < x < x < x < Totals for correct division: (sum of frequencies mid-point) (total frequency) for correct example of estimated mean (ft) (class widths do not need to e equal) Estimated mean = = 6 (to one decimal place) 6 AQA GCSE aths Foundation Skills ook Statistics 1

16 n 1 n 1 ( n 1) edian = value Lower quartile = value Upper quartile = value So if you have a tale with 1 values these will e the 6th, rd and 9th values. Order Value for correct formula for median for correct formula for lower quartile for correct formula for upper quartile Difference etween 1 and 7 equals 7 as required These can e adapted as long as rows are added equally in each section (see example Order Value elow) 1, for values for upper and lower quartile with a difference of 7 for explanation of difference in relation to range in this context Difference etween 1 and 7 equals 7 as required Either arold, as he had igger tomatoes, or Connie, as she had more tomatoes (depending on if you want lots of tomatoes or large tomatoes!) for choice of one person with justification for further consideration of second person AQA GCSE aths Foundation Skills ook Statistics 16

17 a The following statement is more precise: There is strong evidence to support my hypothesis. In reality it is usually not possile to prove a hypothesis just to gather evidence to support it. for selection of There is strong evidence to support my hypothesis with explanation aout practicalities of proving an hypothesis i ii ecause you may not have gathered enough evidence. ecause you haven t gathered enough evidence or you re using a null hypothesis. for explanation of having sufficiently large sample/evidence for giving same reason / an awareness of the practical difficulties of getting all other variales equal or unchanged so that they do not affect the success or otherwise of your evidence gathering oe AQA GCSE aths Foundation Skills ook Statistics 17