VCE General Mathematics Further

Unit 2 2014

Bivariate Data SAC

Name: _______________________________

Total: / 35marks ( %)

Conditions: 45 min, Casio Calculator and Double sided handwritten Summary Sheet permitted

Section A: Multiple Choice ( 6 x 2 = 12 marks) 1. In which of the scatterplots below would the relationship between the variables be best

described as moderate negative

A B c

D E

2. The Pearson’s product moment correlation coefficient r=0.87056

Written as a percentage, the coefficient of determination is closest to:

A 75.8%B 0.758%C 0.871%D 87.1%E 75%

3. In which one of the following situations would there most likely be a negative correlation between the two variables

A The number of days above 250 over summer holidays and the number of times a teenager goes swimming B The number of hours spent training for a 2km race and the actual time taken to run the 2 km on the dayC Student shoe sizes and their test scores on a Maths testD An adult weight and their waist measurement in cm E A child’s height and their reading age

4. A student doing research believes that there may be a mathematical correlation between a person’s total income AND the amount they spend on holidays. She collects some data from a group of adults, draws a scatterplot of the data and then constructs the least squares regression line.

Which one of these equations could be the line of best fit between these two variables? ( hint: first identify the independent variable and the dependent variable )

A annual income = 1000 + 0.02 x holiday expenditure

B annual income = 1000 − 0.02 x holiday expenditure

C holiday expenditure = 1000 + 0.02 x annual income

D holiday expenditure = 1000 − 0.02 x annual income

E holiday expenditure = − 0.02 x annual income

5. Consider this data that compares the height of 6 teenage boys with the heights of their mothers. Enter this data in your calculator and extract any relevant information

Height of mothersM

Height of sonsS

154 164159 167168 172170 180175 180176 187

Which of these statements can be made about this data?

A There is a strong negative correlation between these two variables

B The equation of the line of best fit is M=0.95 S+17.04

C 89.5% of the variation in S values can be explained by variation in M values

D 94.6% of the variation in S values can be explained by variation in M values

E When a mother is tall, this will result in her son being tall

6. For a set of bivariate data, involving the variables x and y, the coefficient of determinationr2=0.64 The least squares regression line ( line of best fit ) is given by the equation y=−0.22 x+8.75 The value of Pearson’s coefficient r for this set of data, correct to 2 decimal places, would be

A −0.22B 0.80C 8.75D −0.80E 0.41

Section B: Short Answer Section:

Show all relevant work steps. All answers to 2 decimal places1. Consider this table of bivariate data x 15 18 19 23 25 30 32 36y 148 161 125 138 149 90 127 105a) Complete a scatterplot on the grid provided, labelling and scaling each axis carefully.

. b) Find Pearson’s correlation coefficient (r ) for this datac) Describe the form, strength and shape of the correlation as shown on your scatterplot. d) Using the Coefficient of Determination, write a statement explaining the relationship between the two variables.

e) Write the equation of the least squares regression line ( i.e. the line of best fit ) for this data.

f) Using this equation, predict the y value when x=27

( 2 + 1 + 2 + 2 + 1 + 1 = 9 marks )

2. A set of data comparing two variables x and y has the following summary statisticsr=−0.9855Sx=3.335 , Sy=5.829, x=6.375∧ y=17.375 Find the equation of the least squares regression line ( i.e. the line of best fit )

( 3 marks )

3. The government statistician declares there is a moderate positive correlation between the cost of petrol at the bowser (C )and the exchange rate between the Australian dollar and the US dollar ( E ). eg on a particular day, petrol costs 147.0 cents a litre and one Aussie dollar buys 93.0 US cents

He calculates the line of best fit between the variables has the equation C=0.675 E+84.225

a) If the exchange rate on a particular day is 98.3 cents US ( to buy $1 Aus ), use this equation to estimate the cost of petrol on that day.

b) If the exchange rate on another day rises to 103.4 cents US ( to buy $1 Aus ), what might you expect to pay to fill up an 80 litre tank with petrol?

( 2 + 3 = 5 marks )

4. Badlands finished on the bottom of the ladder in a country football league in 2014. They appoint a new coach for 2015. He decides to analyse some of the team statistics of the 8 teams from 2014 in an attempt to explain the success of the better teams. He understands that their success is probably based on a wide range of factors, but he is keen to see if there are any obvious strong correlations between certain team statistics and their winning records. Two of his findings are shown in the tables below WINS compared to average tackles per game WINS compared to % of players over 180cm

List 1 List 2 Team ladderin 2014Average tacklesper game ( T)

Number of wins ( W ) Bullato 85 16

Kooribura 92 14Lascelles 96 12

Smithtown 78 105. Whakem South 80 8

Mountainvale

55 4

Ridgefield 66 3Badlands 48 1

Enter the data into your calculator, setting T as List 1, W as List 2 and P as list 3a) Comparing the data in Lists 1 and 2, find Pearson’s Correlation Coefficient between the average number of tackles per game and the number of wins during the season. Comment on the correlation between these two variables.

List 3 List 2 Team ladderin 2014 Percentage of players over 180cm (P)Number of wins ( W )

Bullato 52% 16 Kooribura 58% 14

Lascelles 24% 12 Smithtown 44% 10 Whakem South

36% 8

Mountainvale 32% 4 Ridgefield 38% 3

Badlands 36% 1

b) Comparing data in Lists 3 and 2, find Pearson’s Correlation Coefficient between the percentage of players over 180cm and the number of wins during the season. Comment on the correlation in between these two variables.

c) Using the two Correlation Coefficients you have found and the two resulting Coefficients of Determination, write a statement suggesting to the new coach which of the two team variables ( average tackles per game or percentage of players over 180cm ) appears to be more statistically significant when compared to the number of games won. Explain your answer fully.

( 2 + 2 + 2 = 6 marks )