8CORE SAMPLE. Revision of the Core

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1 C H A P T E R 8CORE of the Core 8. Displaying, summarising and describing univariate data The following information relates to Questions to 3 The percentage investment returns of seven superannuation funds for the year are:.%,.7%,.9%,.3%,.%,.%,.% The median investment return is: A.7% B.% C.% D.% E.3% [FM 3] The mean and standard deviation of the investment returns are closest to: A =.%, s = 3.8% B =.%, s =.9% C =.%, s = 3.8% D =.%, s = 3.8% E =.%, s = 3.8% 3 The range of investment returns is: A.% B 3.% C.% D.% E 8.% [FM 3] The following information was requested from a random sample of adults: Se (male or female) Mobile phone owner (yes or no) The variables in this survey, Se and Mobile phone owner, are: A both categorical variables B both numerical variables C categorical and numerical variables respectively D numerical and categorical variables respectively E neither categorical nor numerical variables [VCAA pre ] Consider this bo-and-whisker plot. Which one of the following statements is true? A The median is. 3 7 B Less than one quarter of the observations are less than 3. C Less than one quarter of the observations are greater than. TI-Nspire & Casio ClassPad material in collaboration with 39 Brown and McMenamin

2 Essential Further Mathematics Core D All of the observations are less than. E More than half of the observations are less than 3. The stemplot opposite shows the scores obtained by 7 students in a class test. The median test score is: A B C D E 7 The range of test scores is: A B 7 C D E 3 8 The IQR is: A B 7 C 7. D 8 E 8. The following information relates to questions 9 to Agroup of VCE Mathematics students sat for a test. There were 3 students in the group. Their test scores are summarised opposite in the form of a histogram. 9 The distribution of test scores is: A positively skewed B negatively skewed C symmetric D symmetrically skewed E symmetric with a clear outlier Displayed in the form of a boplot, the distribution of test scores would look most like: A B C E D Frequency Test scores Test score The pass mark on the test was 3. The percentage of students who failed the test is closest to: A % B 9% C 3% D % E 3% [VCAA pre ] The number of students who scored between 3 and marks is: A B 7 C 9 D E 3 The modal mark lies in the interval: A B C 3 D E

3 Chapter 8 The median mark lies in the interval: A B C 3 D E The mean weight of twelve people is 7 kg; the standard deviation of the weights of these twelve people is kg. These twelve people are about to go on a rafting adventure. Before boarding the raft, they are all required to put on a life-saving vest that weights kg. The effective weight of each person is now their weight plus the weight of the life-saving vest. The effective weights of the twelve people have: A a mean of 7 kg with a standard deviation of kg B a mean of 7 kg with a standard deviation of 7 kg C a mean of 7 kg with a standard deviation of kg D a mean of 7 kg with a standard deviation of 7 kg E a mean of 7 kg with a standard deviation of kg [VCAA 3] The following information relates to questions to 7 Over the past three years, the committee of a country golf club has been recording the wet Saturday dry Saturday number of its members who play golf each 8 8 Saturday. They also noted whether it was a Number of members playing golf wet ordry day. The data is displayed opposite in the form of parallel bo plots. The IQR for the number of golfers playing golf on a wet Saturday is closest to: A B C D 7 E 8 7 From the bo plots, it can be concluded that, on wet days, the number of members playing golf is: A much the same as on a dry Saturday B generally higher and more variable than on a dry Saturday C generally lower and less variable than on a dry Saturday D generally higher and less variable than on a dry Saturday E generally lower and more variable than on a dry Saturday [VCAA pre ] 8 These back-to-back stem-and-leaf plots compare the Class A Class B marks gained by two classes, A and B, onatest. There are students in each class. Which one of the 3 3 following statements is true? A The median mark in class A is B The median mark in class B is 7. C The top mark in both classes is 89. D The lowest mark in both classes is 8. E The range of marks is the same for both classes. [VCAA pre ]

4 Essential Further Mathematics Core 9 In a large senior secondary college there are five Further Mathematics classes. At the end of the year, a teacher compares the Eamination marks for each class by constructing five parallel bo plots as shown. From the bo plots, the class with the largest interquartile range is: A Class A B Class B C Class C D Class D E Class E 8 Percentage mark For the bo plot opposite, outliers are defined as data values that are: A less than 3 or greater than 7 B less than 8 or greater than 7 C less than 8 D greater than 7 E greater than 8 The following information relates to questions to 3 The times spent waiting in a supermarket queue by customers are displayed in the histogram opposite. From the histogram, the number of customers who waited less than four minutes in this queue is: A B 9 C 9 D 7 E 3 From the histogram, the percentage of customers waiting or more minutes is: A % B % C % D % E 88% Frequency Class A Class B Class C Class D Class E [VCAA pre ] Waiting time in minutes 3 Which of the following statistics would give the best indication of a typical waiting time for this data? A the median B the interquartile range C the range D the mean E the standard deviation [VCAA pre ] The distribution of Eamination scores in a VCE subject studied by more than students is approimately normally distributed with a mean of 3 and a standard deviation of 9. From this information it can be concluded that the percentage of students in this subject with scores between and 3 is closest to: A % B % C % D 8% E 9% [VCAA pre ] The distribution of the weights of eggs produced by a chicken farm is approimately normally distributed with a mean of 8 g and a standard deviation of g. Eggs weighing 9 g or more are classified as Etra Large. The percentage of eggs that would be classified as Etra Large is closest to: A.% B.3% C.% D % E % [VCAA ]

5 Chapter 8 3 The distribution of the weight of ice-cream served in a single scoop of Danish Delight is known to be normally distributed with a mean of g and a standard deviation of g. The percentage of single scoops of Danish Delight containing less than g will be closest to: A % B.% C % D % E 9% [VCAA pre ] 7 A student s mark on a test is. The mean mark for their class is and the standard deviation is. Their standard score is: A. B. C D E. 8. Displaying, summarising and describing relationships in bivariate data Market researchers found that for VCE students, the value of the product moment correlation coefficient between the number of hours they studied and their weekly ependiture on junk food was r =.. From this, it could be concluded that: A % of the students bought lots of junk food and studied for long hours B buying more junk food helped the students study more C buying more junk food helped the students study less D the students who spent more time studying tended to buy more junk food E the students who spent more time studying tended to buy less junk food [VCAA pre ] Foralarge sample of students at Ecksville Primary School, the correlation between score on a reading test and height is found to be.8. From this survey, it is reasonable to conclude that: A learning to read better makes children grow tall B there is no relationship between height and reading ability C a child s reading ability depends only on their height D children who have high reading scores tend to be tall E all tall children are better readers than shorter children [VCAA pre ] 3 The value of r, the product moment correlation for the scatter plot shown here is closest to: A.9 B. C D. E.9 Forwhich one of the scatterplots below would the product moment correlation coefficient, r,beclosest to zero? A B C [VCAA pre ]

6 Essential Further Mathematics Core D E [VCAA pre ] The following information relates to questions to The level of internet usage (never used, sometimes used, often used) for 7 school students sampled from Years 3 to is indicated in the table below. Some of the entries in the table are missing. Level of internet Year group usage 3 7 Total never used 9 8 sometimes used 8 often used 7 Total 73 7 For this sample of students, the total number of students who never used the internet is: A B C D 7 E 7 The percentage of Year 7 students who sometimes used the internet is closest to: A % B % C 7 D 8% E 3% [VCAA ] 7 When the correlation coefficient, r,was calculated for the data displayed in the scatterplot it was found to be.39. If the point (7, ) was replaced with the point (7, ) and the correlation coefficient, r, recalculated, then the value of r would be: A unchanged B positive but closer to C negative but closer to D positive but closer to E negative but closer to [VCAA pre ] 8 A back-to-back stem plot is a useful tool for displaying the relationship between: A weight in kilograms and height in centimetres B test score and age in years C attitude to Sunday trading (agree, do not care, disagree) and se (female, male) D height in centimetres and se (female, male) E wine consumption in litres per head of population and country [VCAA pre ] 9 To eplore the relationship between mobile phone owner (yes or no) and se (male or female), it would be best to use the data collected to construct: A an appropriately percentaged table B a back-to-back stem plot C parallel boplots D a scatterplot E a histogram [VCAA pre ]

7 Chapter 8 The relationship between resting pulse rate (in beats per minute) and fitness level (below average, average, above average) is best displayed using: A a histogram B a scatterplot C a time series plot D parallel boplots E a back-to-back stem plot [VCAA pre ] 8.3 Regression and data transformation Which one of the following statements is correct? The least squares regression line of y on is such that it minimises: A the sum of the squares of the vertical deviations of the points from the line B the square of the sum of the vertical deviations of the points from the line C the sum of the squares of the horizontal deviations of the points from the line D the square of the sum of the horizontal deviations of the points from the line E the sum of the squares of the distances of the points from the line [VCAA pre ] Given that r =.7, s =.7, s y =.983 the slope of the least squares regression line y = a + b,isclosest to: A.3 B.8 C.3 D.7 E 3.3 The following data relates to questions 3 to Number of hot dogs sold Temperature ( C) 7 We wish to determine the equation of the least squares line for this data that will enable the number of hot dogs sold to be predicted from temperature. 3 The slope of the regression line will be closest to: A.3 B. C. D.3 E 7 The coefficient of determination will be closest to: A.9 B.89 C. D.89 E.9 The following information relates to questions to Eighteen students sat for a -question multiple-choice 9 test. In the scatterplot opposite, the number of errors made by each student on the test is plotted against the time they 8 7 reported studying for the test. A least squares regression line has been determined for this data and is also displayed on the scatterplot. 3 The equation for the least squares regression line is number of errors = 8.8. study time 3 7 and the coefficient of determination is.898. Study time (minutes) Using the equation of the least squares regression line, it can be estimated that, on average, a student reporting a study time of 3 minutes would make: A.3 errors B. errors C.8 errors D. errors E 3. errors [VCAA ] Number of errors

8 Essential Further Mathematics Core The value of Pearson s product moment correlation coefficient, r, for this data, correct to two decimal places, is: A.9 B.8 C.7 D.8 E.9 [VCAA ] The following information relates to questions 7 to 8 The average rainfall and temperature range at several different locations in the South Pacific region is displayed in the scatterplot opposite. Average rainfall (cm) Temperature range ( C) 7 A least squares regression line has been fitted to the data as shown. The equation of this line is closest to: A average rainfall = temperature range B average rainfall = + temperature range C average rainfall = 8.8 temperature range D average rainfall = temperature range E average rainfall = 3 temperature range [VCAA ] 8 The value of the product moment correlation coefficient, r, for the data, is r =.9. The value of the coefficient of determination is: A.9 B.87 C.87 D.9 E.93 [VCAA ] 9 The regression line as shown on the graph predicts that, on average, a location with a temperature range of C will have an average rain fall closest to: A 8 cm B cm C cm D cm E cm For the scatterplot of points shown here, the 3-median regression line of y on is derived from the line through the points: A (, ) and (9, 7) B (, 3) and (, 9) C (., 3) and (3., 9) D (, ) and (3, 8) E (, 3) and (3, 9) y [VCAA pre ]

9 Chapter 8 7 Scatterplots A, B, C, D and E below show attempts by students to fit a 3-median line to the data shown in the scatterplot. Only one attempt is correct. Which is it? A B C D Consider the scatterplot opposite. The slope of the three-median line that could be fitted to the data is closest to A B.7 C. D. E 3 A least squares regression line has been fitted to a scatterplot as shown. The equation of this line is closest to: A y =.8 B y = +.8 C y =. + D y =. E y =.8 E 8 7 y 3 y 8 [VCAA pre ] [VCAA pre ] 8 [VCAA pre ] A least squares regression line calculated in an analysis of the relationship between fuel consumption (in kilometres per litre) and the weight (in kilograms) of a car has the following equation: fuel consumption =.. weight. Using this equation we predict that cars weighing kg have a fuel consumption of: A. km/l B.7 km/l C. km/l D. km/l E. km/l [VCAA pre ]

10 8 Essential Further Mathematics Core The following information relates to questions to 7 The scatterplot opposite shows the weekly income and hours worked for 3 university students. The line shown on the scatterplot is the least squares line of best fit. Weekly income ($) Hours worked Using the graph of the least squares line, we predict that a student working hours per week would have a weekly income of about: A $ B $9 C $ D $ E $ [VCAA pre ] From the least squares line we conclude that, on average, for each additional hour per week worked, weekly income increased by about: A $ B $8 C $ D $3 E $ [VCAA pre ] 7 The student who worked 37 hours per week earned $ per week. Using the least squares line to predict this student s income would leave a residual of about: A $3 B $3 C $ D $ E $7 [VCAA pre ] The following information relates to questions 8 to 9 A student fits a least squares line to a set of bivariate data as shown in the scatterplot opposite. 8 The residual plot for this least squares line would look like: A B D Residual Residual... 3 Residual E Residual... 3 C y 3 Residual [VCAA pre ]

11 Chapter The relationship between y and revealed in the scatterplot is clearly non-linear. In an attempt to transform the data to linearity, you could use: A alog transformation B alogytransformation C an transformation D a transformation E a y transformation The relationship between the two variables y and, asshown in the scatterplot, is clearly non-linear. A student wants to linearise the scatterplot by transforming the scale on one of the aes only. Which of the following approaches would be the best y strategy? A Use either a or a y transformation. B Use either a or a y transformation. C Use either a log or a y transformation. 3 D Use either an or a log y transformation. E Use either an or a y transformation. 8. Time series 3 [VCAA pre ] The quarterly sales figures for a soft drink company and the seasonal indices are as shown. Quarter 3 Sales ($ s) 8 Seasonal inde The deseasonalised figure in $s for Quarter 3 is: A 8 B C D E 8 [VCAA 3] Which of the following statements best describes the time series represented by the graph? A B C D E This time series shows a seasonal pattern but no linear trend. This time series shows a linear trend but no seasonal pattern. This time series shows a seasonal pattern and a linear trend. This time series shows neither a seasonal pattern nor a linear trend. Sales Quarter It is impossible to tell from this information whether this time series ehibits either a seasonal pattern or a linear trend. [VCAA pre ]

12 Essential Further Mathematics Core 3 The quarterly sales figures of Girte Coat Company and their seasonal indices are given here. Quarter 3 Sales (number of coats) 37 3 Seasonal inde. w.3.3 The value of w is: A. B. C. D. E. The pattern in the time series in the graph shown is best described as: A trend B cyclical, but not seasonal C seasonal D random E average For the time series given in the table, the 3-moving mean centred at t = is: [VCAA pre ] [VCAA pre ] y A B C D 7 E 8 [VCAA pre ] Sales for a major department store are reported quarterly. The seasonal inde for the third quarter is.8. This means that sales for the third quarter are typically: A 8% below the quarterly average for the year B % below the quarterly average for the year C % above the quarterly average for the year D 8% above the quarterly average for the year E 8% below the quarterly average for the year 7 The seasonal indices for the number of lunches sold at a school canteen are as shown. If the number of Day M T W T F lunches sold on a Friday was, then the deseasonalised figure is: Inde A B C 3 D E 8 [VCAA pre ] 8 The main purpose of smoothing a time series is to reduce: A trend B seasonal variation C cyclical variation D random variation E all variation [VCAA pre ]

13 Chapter 8 9 The time series plot shown represents the average daily use of electricity in a home in successive quarters. The pattern revealed is best described as: A seasonal only B trend only C random D cyclical but not seasonal E seasonal with trend Average daily use (kwh) 8 8 Quarter [VCAA pre ] Ice-cream sales (litres) for / and the seasonal indices for an ice-cream maker are shown in the table below. Summer Autumn Winter Spring Ice-cream sold Seasonal inde When deseasonalised, the amount of ice-cream sold in winter is, to the nearest litre: A 38 B 8 87 C 9 3 D 98 E 3 [VCAA pre ] The time series plot shows the share price of two companies over a period of time. From the plot, it can be concluded that over the interval 99, the difference in share price between the two companies has shown: A a decreasing trend B an increasing trend C seasonal variation D afive-year cycle E no trend A time series for y is shown in the graph, where t represents time. If a linear trend line is fitted to this data as shown, then the equation of the line is closest to: A y =.t B y =.t C y = +.t D y =.t E y = +.t Share price ($) Year y [VCAA ] t

14 Essential Further Mathematics Core 3 The following table gives the number of births in a country hospital over an 8-year period. Year Number of births Using the -point moving median method, the smoothed value of the number of babies born in Year, is: A 9 B C 9 D E The time series plot opposite is best described as showing: A random variation only B an increasing trend only C an increasing trend with seasonality D seasonality only E a decreasing trend only The time series plot opposite shows the price (dollars per tonne) of copper ore over the period 9 to 99. Sales ($s) Price (dollars per tonne) [VCAA pre ] When smoothed, using 3-point median smoothing, the time series plot will look most like: A 8 B 8 C 8 D Price (dollars per tonne) Price (dollars per tonne) E Price (dollars per tonne) Price (dollars per tonne) Price (dollars per tonne) [VCAA pre ]

15 Chapter Etended-response questions The data in Table 8. is based on a study of dolphin behaviour. In this study, the main activities of dolphins observed in the wild were classified as travelling, feeding and socialising. The time of observation was recorded as morning, noon, afternoon or evening. Table 8. Number of dolphins observed by activity and time of observation Time of observation Activity morning noon afternoon evening travelling 3 feeding 8 socialising 38 9 a Complete the following sentences. i The number of dolphins observed feeding at noon is. ii The dolphin activity most frequently observed in the morning is. To test the assertion that dolphin activity is associated with time of day, the table was percentaged by calculating the appropriate column percentages as displayed in Table 8.. Table 8. Percentage of dolphins observed by activity and time of observation Time of observation Activity morning noon afternoon evening travelling 8 feeding socialising b Use the information in Table 8. to describe briefly any relationship that you can see between dolphin activity and the recorded time of observation. Quote appropriate percentages to support your description. In another study of animal behaviour, investigators collected information on the average hours that various animal species spend in dreaming and non-dreaming sleep. The data for a selected group of of these animals is shown in Table 8.3.

16 Essential Further Mathematics Core Table 8.3 Species Average hours of sleep non-dreaming dreaming Baboon 9..7 Golden hamster. 3. Brazilian tapir.. Chimpanzee 8.3. African giant rat.3. Cow 3..7 Chinchilla.. Cat.9 3. American mole.3. Mole rat 8.. Desert hedgehog 7..7 Big brown bat Asian elephant..8 European hedgehog.. Stem: units Leaves: tenths Non-dreaming Figure 8. Dreaming c To compare the distributions of non-dreaming sleep and dreaming sleep, a back-to-back stem plot is to be constructed. The hours of non-dreaming sleep have already been recorded in Figure 8.. Complete this back-to-back stem plot by writing in the data for dreaming sleep. d For the animals listed, the mean and standard deviation for average hours of non-dreaming sleep are provided below correct to one decimal place. Non-dreaming sleep: mean = 8. hours, standard deviation = 3. hours Determine the mean and standard deviation for average hours of dreaming sleep correct to one decimal place. e Use this information to write a brief sentence comparing the centre and spread of the two sleep distributions. f In an attempt to predict hours of dreaming sleep from.. hours of non-dreaming sleep, the least squares 3. regression line is plotted on a scatterplot as shown 3. in Figure 8... The key assumption made about the nature of the.. relationship between hours of dreaming and. non-dreaming sleep when fitting the least squares. regression line is that this relationship 8 is. Hours of non-dreaming sleep Figure 8. Hours of dreaming sleep

17 Chapter 8 g Complete the following sentences. i In this investigation, the dependent variable is. ii For this data, the value of Pearson s product moment correlation coefficient is r =.78. From this information, we can conclude that %ofthe variation in hours of dreaming sleep can be eplained by the variation in hours of non-dreaming sleep. iii The animal whose hours of dreaming sleep is least well predicted by the regression line shown on the scatterplot is the. iv For this animal, the regression line underestimates their hours of dreaming sleep by about hours. [VCAA ] Over recent years, the salaries of Patagonian cricketers have increased rapidly. The following data gives the average salaries in dollars of a large group of these cricketers over the period 99. Year Salary a This data will be used to predict future average salaries of Patagonian cricketers. In this analysis, what is the independent variable? To begin the analysis, the years have been rescaled as = to = (99 =, 99 =, and so on), and average salary rescaled in thousands of dollars as the variable y. Year Salary ($s) This rescaled data is displayed below as a time series plot. Also displayed is the least squares regression y line which has been determined for this rescaled data. The equation of the least squares regression line is y = b Using the regression equation y = to model the increase in these cricketers average salaries, we would: i say that the average salary increase for the cricketers was $ per year ii predict that their average salary in the year will be about $ From the time series plot, the increase in Patagonian cricketers salaries over time appears non-linear. This can be confirmed by constructing the corresponding residual plot as shown overleaf. (cont d.)

18 Essential Further Mathematics Core c Complete the plot by: i calculating the value of the residual for 99 ii plotting this residual as a point on the graph y The time series, along with the residual plot, shows that the growth of salaries with time is non-linear. Inspecting the time series, it would appear that it would be appropriate to use an transformation to transform the data to linearity. The original data has been reproduced in the table below and an etra row has been added for the transformed variable,. d Year () Year ( ) Salary (y) ($s) i Complete the table. ii Complete the time series plot of the transformed data above to show that the transformation has produced a more nearly linear plot. Use the grid below. (Note the first three points have been plotted.) y iii Find the equation of the least squares regression line for the transformed data. Write + the coefficients, correct to two decimal places: y = iv Use this regression equation to predict the average salary of this group of Patagonian cricketers in. [VCAA ] 3 The table below gives the heights (m) and weights (kg) of a sample of nine people. Height (m) Weight (kg) On the scatterplot on the net page, the points representing the data for seven of these people have been plotted with height on the horizontal ais and weight on the vertical ais.

19 Chapter 8 7 a Plot points representing the data for the remaining two people (shown in bold in the table) on the scatterplot. Weight (kg) Height (m) b Determine the equation of the least squares regression line that fits the data in the table. Use height as the independent variable and weight as the dependent variable. Complete the regression equation by writing the appropriate values, correct to one decimal place: weight = + height c The coefficient of determination for this data is.. Complete the following sentence: For this sample, % of the variation in the of the people can be eplained by the variation in their. Body mass inde (BMI)isdefined as BMI = weight where weight is measured in (height) kilograms and height in metres. d Determine the body mass inde of a person who weighs kg and who is.9 m tall. Write your answer correct to one decimal place. The BMI for each person in a sample of 7 males and females is recorded in the table opposite. e Write the range of the BMI data for males in the sample. f A BMI greater than is sometimes taken as an indication that a person is overweight. Use this criterion and the data given to complete the -way frequency table below. Body mass inde Males Females Body mass inde Males Females Gender Weight rating Male Female Overweight Not overweight Total 7 (cont d.)

20 8 Essential Further Mathematics Core g Does the data support the contention that, for this sample, weight rating is associated with gender? Justify your answer by quoting appropriate percentages. h The parallel boplots have been constructed to compare the distribution of BMI for males and females in this sample. female male 3 3 Body mass inde i Use the parallel boplots to identify and name two similar properties of the BMI distributions for males and females. ii Use the information in the table to determine the mean BMI for the males in this sample. Write your answer correct to one decimal place. iii The median BMI for males is.. Of the mean or median, which measure gives a better indication of the typical BMI for males? Eplain your answer. [VCAA ]