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GEOMETRY, FUR2 2011 VCAA 2

Ship A and Ship B can both be seen from a lighthouse.

Ship A is 5 kilometres from the lighthouse, on a bearing of 028°.

Ship B is 5 kilometres from Ship A, on a bearing of 130°.

This information is shown in Figure 1 below.

GEOMETRY, FUR2 2011 VCAA 2

  1. Two angles, and , are shown in Figure 1 above.

    1. Determine the size of the angle in degrees.  (1 mark)
    2. Determine the size of the angle in degrees.  (1 mark)
  2. Determine the bearing of the lighthouse from Ship A.  (1 mark)
  3. Determine the bearing of Ship B from the lighthouse.  (1 mark)

 

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Show Worked Solution

a.i. 

VCAA GEO FUR2 2011 2i

 

 

   ii.   
   

 

b.  

 

c.  

MARKER’S COMMENT: True bearings are required and an answer of 79° is incorrect.

 

MATRICES, FUR2 2012 VCAA 2

Rosa uses the following six-digit pin number for her bank account: 216342. 

With her knowledge of matrices, she decides to use matrix multiplication to disguise this pin number.

First she writes the six digits in the 2 × 3 matrix .

Next she creates a new matrix by forming the matrix product, ,

where

    1. Determine the matrix .   (1 mark)

    2. From the matrix , Rosa is able to write down a six-digit number that disguises her original pin number. She uses the same pattern that she used to create matrix from the digits 216342.

       

      Write down the new six-digit number that Rosa uses to disguise her pin number.   (1 mark)

  2. Show how the original matrix can be regenerated from matrix .   (1 mark) 
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a.i.  
   
   

 

a.ii.

 

b.  
 
 

MATRICES, FUR2 2014 VCAA 2

There are three candidates in the election: Ms Aboud (), Mr Broad () and Mr Choi ().

The election campaign will run for six months, from the start of January until the election at the end of June.

A survey of voters found that voting preference can change from month to month leading up to the election.

The transition diagram below shows the percentage of voters who are expected to change their preferred candidate from month to month.
 

MATRICES, FUR2 2014 VCAA 2

    1. Of the voters who prefer Mr Choi this month, what percentage are expected to prefer Ms Aboud next month?   (1 mark)

    2. Of the voters who prefer Ms Aboud this month, what percentage are expected to change their preferred candidate next month?   (1 mark)

In January, 12 000 voters are expected in the city. The number of voters in the city is expected to remain constant until the election is held in June.

The state matrix that indicates the number of voters who are expected to have a preference for each candidate in January, , is given below.

  1. How many voters are expected to change their preference to Mr Broad in February?   (1 mark)

The information in the transition diagram has been used to write the transition matrix, , shown below.

    1. Evaluate the matrix and write it down in the space below.
    2. Write the elements, correct to the nearest whole number.   (1 mark)

    4. What information does matrix contain?   (1 mark)

  2. Using matrix , how many votes would the winner of the election in June be expected to receive?
  3. Write your answer, correct to the nearest whole number.   (1 mark)

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    1.  


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a.i.   
 

a.ii.   
 

b.  

 

c.i.   
   

 

c.ii.   

 

 

MARKER’S COMMENT: A common error used . Know why this is incorrect.
d.   
   

 

MATRICES, FUR2 2015 VCAA 2

The ability level of students is assessed regularly and classified as beginner (), intermediate () or advanced ().

After each assessment, students either stay at their current level or progress to a higher level.

Students cannot be assessed at a level that is lower than their current level.

The expected number of students at each level after each assessment can be determined using the transition matrix, , shown below.

  1. The element in the third row and third column of matrix is the number 1.
  2. Explain what this tells you about the advanced-level students.   (1 mark)

Let matrix be a state matrix that lists the number of students at beginner, intermediate and advanced levels after assessments.

The number of students in the school, immediately before the first assessment of the year, is shown in matrix below.

    1. Write down the matrix that contains the expected number of students at each level after one assessment.
    2. Write the elements of this matrix correct to the nearest whole number.   (1 mark)

    3. How many intermediate-level students have become advanced-level students after one assessment?   (1 mark)

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Show Worked Solution

a.   

 

b.i.   
   
   

 

b.ii.  

CORE, FUR2 2006 VCAA 1

Table 1 shows the heights (in cm) of three groups of randomly chosen boys aged 18 months, 27 months and 36 months respectively.

Core, FUR2 2006 VCAA 11

  1. Complete Table 2 by calculating the standard deviation of the heights of the 18-month-old boys.

     

    Write your answer correct to one decimal place.   (1 mark)

     

    Core, FUR2 2006 VCAA 12

A 27-month-old boy has a height of 83.1 cm.

  1. Calculate his standardised height ( score) relative to this sample of 27-month-old boys.
  2. Write your answer correct to one decimal place.   (1 mark)

The heights of the 36-month-old boys are normally distributed.

A 36-month-old boy has a standardised height of 2.

  1. Approximately what percentage of 36-month-old boys will be shorter than this child?   (1 mark)

Using the data from Table 1, boxplots have been constructed to display the distributions of heights of 36-month-old and 27-month-old boys as shown below. 

     Core, FUR2 2006 VCAA 13

  1. Complete the display by constructing and drawing a boxplot that shows the distribution of heights for the 18-month-old boys.   (2 marks)

  2. Use the appropriate boxplot to determine the median height (in centimetres) of the 27-month-old boys.   (1 mark)

The three parallel boxplots suggest that height and age (18 months, 27 months, 36 months) are positively related.

  1. Explain why, giving reference to an appropriate statistic.   (1 mark)

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  3.  
  4.  
    Core, FUR2 2006 VCAA 13 Asnwer
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a.   

 

b.   
   
   
   

 

♦♦ MARKER’S COMMENT: Attention required here as this standard question was “very poorly answered”.

c.  

CORE, FUR2 2006 VCAA Answer 111

 

d.   

Core, FUR2 2006 VCAA 13 Asnwer

e.   

MARKER’S COMMENT: A boxplot statistic was required, so mean values were not relevant.

 

f.  

CORE, FUR2 2006 VCAA 3

The heights (in cm) and ages (in months) of the 15 boys are shown in the scatterplot below.
 

Core, FUR2 2006 VCAA 3
 

  1. Fit a three median line to the scatterplot. Circle the three points you used to determine this three median line.  (2 marks)
  2. Determine the equation of the three median line. Write the equation in terms of the variables height and age and give the slope and intercept correct to one decimal place.  (2 marks)
  3. Explain why the three median line might model the relationship between height and age better than the least squares regression line.  (1 mark)
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  1.   
    Core, FUR2 2006 VCAA 3 Answer

Show Worked Solution
a.    Core, FUR2 2006 VCAA 3 Answer
♦ Average mean mark for all parts 36%.

b.  

 
 

 

 

 

 
 

 

 

c.   

CORE, FUR2 2006 VCAA 2

The heights (in cm) and ages (in months) of a random sample of 15 boys have been plotted in the scatterplot below. The least squares regression line has been fitted to the data.
 


The equation of the least squares regression line is 

The correlation coefficient is 

  1. Complete the following sentence.

     

    On average, the height of a boy increases by _______ cm for each one-month increase in age.   (1 mark)

  2.  i. Evaluate the coefficient of determination.
  3.     Write your answer, as a percentage, correct to one decimal place.   (1 mark)

  4. ii. Interpret the coefficient of determination in terms of the variables height and age.   (1 mark)

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a.   

b.i.   

b.ii.

Show Worked Solution

a.   

♦ Average mean mark for all parts 44%.
MARKER’S COMMENT: b.i. errors included not converting to % and rounding as a decimal before converting and answering 60%.

 

b.i.   
   
   
MARKER’S COMMENT: Any reference to causation in b.ii. was marked incorrect.

 

b.ii.  

CORE, FUR2 2007 VCAA 3

The table below displays the mean surface temperature (in °C) and the mean duration of warm spell (in days) in Australia for 13 years selected at random from the period 1960 to 2005.
 

CORE, FUR2 2007 VCAA 31
 

This data set has been used to construct the scatterplot below. The scatterplot is incomplete.

  1. Complete the scatterplot below by plotting the bold data values given in the table above. Mark the point with a cross (×).  (1 mark)

          

     

  2. Mean surface temperature is the explanatory variable.

     

    1. Determine the equation of the least squares regression line for this set of data. Write the equation in terms of the variables mean duration of warm spell and mean surface temperature. Write the value of the coefficients correct to one decimal place.  (2 marks)

    2. Plot the least squares regression line on Scatterplot 1.  (1 mark)

The residual plot below was constructed to test the assumption of linearity for the relationship between the variables mean duration of warm spell and the mean surface temperature.

CORE, FUR2 2007 VCAA 33

  1. Explain why this residual plot supports the assumption of linearity for this relationship.  (1 mark)

  2. Write down the percentage of variation in the mean duration of a warm spell that is explained by the variation in mean surface temperature. Write your answer correct to the nearest per cent.  (1 mark)

  3. Describe the relationship between the mean duration of a warm spell and the mean surface temperature in terms of strength, direction and form.  (2 marks)

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  1.  
    CORE, FUR2 2007 VCAA 3 Answer



Show Worked Solution
a.    CORE, FUR2 2007 VCAA 3 Answer

 

b.i.   

 

MARKER’S COMMENT: A consistent error in these type of questions is taking 2 points that are too close together!

b.ii.   

 

 

 

c.   


 

d.  


 

e.  

CORE, FUR2 2007 VCAA 2

The mean surface temperature (in °C) of Australia for the period 1960 to 2005 is displayed in the time series plot below.

2007 2-1

  1. In what year was the lowest mean surface temperature recorded?   (1 mark)

The least squares method is used to fit a trend line to the time series plot.

  1.   i. The equation of this trend line is found to be
  2.          mean surface temperature = – 12.361 + 0.013 × year
  3.      Use the trend line to predict the mean surface temperature (in °C) for 2010.
  4.      Write your answer correct to two decimal places.   (1 mark)

  5.  ii. The actual mean surface temperature in the year 2000 was 13.55°C.
  6.      Determine the residual value (in °C) when the trend line is used to predict the mean surface temperature for this year.
  7.      Write your answer correct to two decimal places.   (1 mark)

  8. iii. By how many degrees does the trend line predict Australia's mean surface temperature will rise each year?
  9.      Write your answer correct to three decimal places.   (1 mark)

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Show Worked Solution

a.   
 

b.i.   


 

b.ii.   

MARKER’S COMMENT: A common error was to omit the negative sign.

 

 
 

 

b.iii.   

 

 

CORE, FUR2 2008 VCAA 4

The arm spans (in cm) and heights (in cm) for a group of 13 boys have been measured. The results are displayed in the table below.
 

CORE, FUR2 2008 VCAA 4 

The aim is to find a linear equation that allows arm span to be predicted from height.

  1. What will be the explanatory variable in the equation?   (1 mark)

  2. Assuming a linear association, determine the equation of the least squares regression line that enables arm span to be predicted from height. Write this equation in terms of the variables arm span and height. Give the coefficients correct to two decimal places.   (2 marks)

  3. Using the equation that you have determined in part b., interpret the slope of the least squares regression line in terms of the variables height and arm span.   (1 mark)

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  3.  

Show Worked Solution

a.  

♦ Mean mark sub 50% (exact data not available).
MARKER’S COMMENT: Many students did not understand the term co-efficients as it applies to the regression equation.

 

b.   

 

c.   

CORE, FUR2 2008 VCAA 3

The arm spans (in cm) were also recorded for each of the Years 6, 8 and 10 girls in the larger survey. The results are summarised in the three parallel box plots displayed below.
 

CORE, FUR2 2008 VCAA 3

  1. Complete the following sentence.
  2. The middle 50% of Year 6 students have an arm span between _______ and _______ cm.   (1 mark)
  3. The three parallel box plots suggest that arm span and year level are associated.
  4. Explain why.   (1 mark)

  5. The arm span of 110 cm of a Year 10 girl is shown as an outlier on the box plot. This value is an error. Her real arm span is 140 cm. If the error is corrected, would this girl’s arm span still show as an outlier on the box plot? Give reasons for your answer showing an appropriate calculation.   (2 marks)

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Show Worked Solution

a.  
 

b.   

♦ Sub 50% mean.
MARKER’S COMMENT: Use specific metrics! Stating “arm span increases” did not receive a mark.

 

c.   

MARKER’S COMMENT: The final comparison here, “Since 140 < 145” is worth a full mark.

CORE, FUR2 2009 VCAA 4

  1. Table 2 shows the seasonal indices for rainfall in summer, autumn and winter. Complete the table by calculating the seasonal index for spring.   (1 mark)

     

    CORE, FUR2 2009 VCAA 4

  2. In 2008, a total of 188 mm of rain fell during summer.

     

    Using the appropriate seasonal index in Table 2, determine the deseasonalised value for the summer rainfall in 2008. Write your answer correct to the nearest millimetre.   (1 mark)

  3. What does a seasonal index of 1.05 tell us about the rainfall in autumn?   (1 mark)

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Show Worked Solution

a.  

 

b.  

♦ Part (c) was “poorly answered” (no exact data).
MARKER’S COMMENT: A common error was to say rainfall was above average monthly rainfall.

 

c.   

 

CORE, FUR2 2009 VCAA 3

The scatterplot below shows the rainfall (in mm) and the percentage of clear days for each month of 2008. 
 

An equation of the least squares regression line for this data set is

rainfall = 131 – 2.68 × percentage of clear days

  1. Draw this line on the scatterplot.   (1 mark)

  2. Use the equation of the least squares regression line to predict the rainfall for a month with 35% of clear days. Write your answer in mm correct to one decimal place.   (1 mark)

  3. The coefficient of determination for this data set is 0.8081.
  4.  i. Interpret the coefficient of determination in terms of the variables rainfall and percentage of clear days.   (1 mark)

  5. ii. Determine the value of Pearson’s product moment correlation coefficient. Write your answer correct to three decimal places.   (2 marks)

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  1.  
    CORE, FUR2 2009 VCAA 3 Answer


Show Worked Solution
a.    CORE, FUR2 2009 VCAA 3 Answer

 

b.   
   
MARKER’S COMMENT: Any answers that suggested causation with terms like “is due to” etc.., did not receive a mark.

 

c.i.   

 

 

 

MARKER’S COMMENT: Many students failed to include the negative sign as indicated by the negative slope of the graph.
c.ii.   
   
   

CORE, FUR2 2009 VCAA 2

The time series plot below shows the rainfall (in mm) for each month during 2008.
 

CORE, FUR2 2009 VCAA 2
 

  1. Which month had the highest rainfall?   (1 mark)

  2. Use three-median smoothing to smooth the time series. Plot the smoothed time series on the plot above.
  3. Mark each smoothed data point with a cross (×).   (2 marks)

  4. Describe the general pattern in rainfall that is revealed by the smoothed time series plot.   (1 mark)

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  2.  
    CORE, FUR2 2009 VCAA 2 Answer
     


  4.  
Show Worked Solution

a.  

 

b.    CORE, FUR2 2009 VCAA 2 Answer
MARKER’S COMMENT: Locate medians graphically by inspection. Explaining a general pattern with more than one trend proved challenging.

c.   

 

MATRICES, FUR2 2015 VCAA 1

Students in a music school are classified according to three ability levels: beginner (), intermediate () or advanced ().

Matrix , shown below, lists the number of students at each level in the school for a particular week.

  1. How many students in total are in the music school that week?   (1 mark)

The music school has four teachers, David (), Edith (), Flavio () and Geoff ().

Each teacher will teach a proportion of the students from each level, as shown in matrix below.

The matrix product, , can be used to find the number of students from each level taught by each teacher.

  1.  i. Complete matrix , shown below, by writing the missing elements in the shaded boxes.   (1 mark) 

 Matrices, FUR2 2015 VCAA 1

  1. ii. How many intermediate students does Edith teach?  (1 mark)

The music school pays the teachers $15 per week for each beginner student, $25 per week for each intermediate student and $40 per week for each advanced student.

These amounts are shown in matrix below.

The amount paid to each teacher each week can be found using a matrix calculation.

  1.  i. Write down a matrix calculation in terms of and that results in a matrix that lists the amount paid to each teacher each week.   (1 mark)

  2. ii. How much is paid to Geoff each week?   (1 mark)

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Show Worked Solution

a.   
 

b.i.  
   
   

 
b.ii.   

 
c.i.   

 
c.ii.   

CORE, FUR2 2010 VCAA 3

Table 2 shows the Australian gross domestic product (GDP) per person, in dollars, at five yearly intervals for the period 1980 to 2005.
 

CORE, FUR2 2010 VCAA 31 

  1. Complete the time series plot above by plotting the GDP for the years 2000 and 2005.   (1 mark)

  2. Briefly describe the general trend in the data.   (1 mark)

In Table 3, the variable year has been rescaled using 1980 = 0, 1985 = 5 and so on. The new variable is time.

CORE, FUR2 2010 VCAA 32

  1. Use the variables time and GDP to write down the equation of the least squares regression line that can be used to predict GDP from time. Take time as the independent variable.   (2 marks)

  2. In the year 2007, the GDP was $34 900. Find the error in the prediction if the least squares regression line calculated in part c. is used to predict GDP in 2007.   (2 marks)

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  1.  
    CORE, FUR2 2010 VCAA 31 Answer
Show Worked Solution
a.    CORE, FUR2 2010 VCAA 31 Answer

 

b.  

 

c.   

MARKER’S COMMENT: Students are expected to use the variables in the diagram rather than just and .


 

d.  

CORE, FUR2 2010 VCAA 2

In the scatterplot below, average annual female income, in dollars, is plotted against average annual male income, in dollars, for 16 countries. A least squares regression line is fitted to the data.
 


 

The equation of the least squares regression line for predicting female income from male income is

female income = 13 000 + 0.35 × male income

  1. What is the explanatory variable?  (1 mark)

  2. Complete the following statement by filling in the missing information.

     

    From the least squares regression line equation it can be concluded that, for these countries, on average, female income increases by for each $1000 increase in male income.  (1 mark)

    1. Use the least squares regression line equation to predict the average annual female income (in dollars) in a country where the average annual male income is $15 000.  (1 mark)

    2. The prediction made in part c.i. is not likely to be reliable.

       

      Explain why.  (1 mark)


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  4.  

Show Worked Solution

a.  
 

b.   


 

c.i.  

♦♦ This part was poorly answered (exact data unavailable).
MARKER’S COMMENT: Many students offered “real world” explanations which did not gain a mark here.

 
c.ii.   

   

   

CORE, FUR2 2010 VCAA 1

Table 1 shows the percentage of women ministers in the parliaments of 22 countries in 2008.
 

CORE, FUR2 2010 VCAA 11
 

  1. What proportion of these 22 countries have a higher percentage of women ministers in their parliament than Australia?  (1 mark)

  2. Determine the median, range and interquartile range of this data.  (2 marks)

The ordered stemplot below displays the distribution of the percentage of women ministers in parliament for 21 of these countries. The value of Canada is missing.
 

    CORE, FUR2 2010 VCAA 12
 

  1. Complete the stemplot above by adding the value for Canada.  (1 mark)

  2. Both the median and the mean appropriate measures of centre for this distribution.
  3. Explain why.  (1 mark)

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  4.  

     

Show Worked Solution

a.  

b.  

 
 

 

 

 

 

 

c.   

♦♦ Part (d) was “poorly answered”.
MARKER’S COMMENT: The use of “symmetric” gained a mark while “evenly distributed” was deemed too vague.

 

d.   

CORE, FUR2 2011 VCAA 3

The following time series plot shows the average age of women at first marriage in a particular country during the period 1915 to 1970.
 

CORE, FUR2 2011 VCAA 31

  1. Use this plot to describe, in general terms, the way in which the average age of women at first marriage in this country has changed during the period 1915 to 1970.   (1 mark)

During the period 1986 to 2006, the average age of men at first marriage in a particular country indicated an increasing linear trend, as shown in the time plot below.

CORE, FUR2 2011 VCAA 32

A three-median line could be used to model this trend.

  1. On the graph above
  2.  i. clearly mark with a cross (×) the three points that would be used to fit a three-median line to this time series plot.   (2 marks)

  3. ii. draw in the three-median line.   (1 mark)

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  2.  
    CORE, FUR2 2011 VCAA 3 Answer
Show Worked Solution

a.  

 

b.i. & ii.

CORE, FUR2 2011 VCAA 3 Answer

CORE, FUR2 2011 VCAA 2

Table 1 shows information about a particular country. It shows the percentage of women by age at first marriage, for the years 1986, 1996 and 2006.

CORE, FUR2 2011 VCAA 2

  1. Of the women who first married in 1986, what percentage were aged 20 to 29 years inclusive?   (1 mark)

  2. Does the information in Table 1 support the opinion that, for the years 1986, 1996 and 2006, the age of women at first marriage was associated with years of marriage?
  3. Justify your answer by quoting appropriate percentages. It is sufficient to consider one age group only when justifying your answer.   (2 marks) 
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a.  


 

b.  

CORE, FUR2 2011 VCAA 1

The stemplot in Figure 1 shows the distribution of the average age, in years, at which women first marry in 17 countries.
 

CORE, FUR2 2011 VCAA 11
 

  1. For these countries, determine
    1. the lowest average age of women at first marriage  (1 mark)

    2. the median average age of women at first marriage  (1 mark)

The stemplot in Figure 2 shows the distribution of the average age, in years, at which men first marry in 17 countries.
 

CORE, FUR2 2011 VCAA 12

  1. For these countries, determine the interquartile range (IQR) for the average age of men at first marriage.  (1 mark)

  2. If the data values displayed in Figure 2 were used to construct a boxplot with outliers, then the country for which the average age of men at first marriage is 26.0 years would be shown as an outlier.
  3. Explain why is this so. Show an appropriate calculation to support your explanation.  (2 marks)

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Show Worked Solution

a.i.  

 

a.ii.  

   

 

b.  

 
 

 

c.  

MARKER’S COMMENT: Many students correctly calculated 28.25 but then failed to discuss how 26.0 related to it.

 

 

CORE, FUR2 2012 VCAA 4

The wind speeds (in km/h) that were recorded at the weather station at 9.00 am and 3.00 pm respectively on 18 days in November are given in the table below. A scatterplot has been constructed from this data set.
 

CORE, FUR2 2012 VCAA 41
 

Let the wind speed at 9.00 am be represented by the variable ws9.00am and the wind speed at 3.00 pm be represented by the variable ws3.00pm. 

The relationship between ws9.00am and ws3.00pm shown in the scatterplot above is nonlinear. 

A squared transformation can be applied to the variable ws3.00pm to linearise the data in the scatterplot.

  1. Apply the squared transformation to the variable ws3.00pm and determine the equation of the least squares regression line that allows (ws3.00pm)² to be predicted from ws9.00am.

     

    In the boxes provided, write the coefficients for this equation, correct to one decimal place.   (2 marks)

    2012 4-1

  2. Use this equation to predict the wind speed at 3.00 pm on a day when the wind speed at 9.00 am is 24 km/h.

     

    Write your answer, correct to the nearest whole number.   (1 mark)

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Show Worked Solution

a.  

MARKER’S COMMENT: Many students were unable to deal with the squared transformation.


 

b.  

CORE, FUR2 2012 VCAA 3

A weather station records the wind speed and the wind direction each day at 9.00 am. 

The wind speed is recorded, correct to the nearest whole number. 

The parallel boxplots below have been constructed from data that was collected on the 214 days from June to December in 2011.
 

CORE, FUR2 2012 VCAA 3

  1. Complete the following statements.
  2. The wind direction with the lowest recorded wind speed was ________.
  3. The wind direction with the largest range of recorded wind speeds was _______.   (1 mark)

  4. The wind blew from the south on eight days.
  5. Reading from the parallel boxplots above we know that, for these eight wind speeds, the
first quartile
median
third quartile
  1. Given that the eight wind speeds were recorded to the nearest whole number, write down the eight wind speeds.   (1 mark) 
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Show Worked Solution

a.  
 

b.  

♦♦ MARKER’S COMMENT: Although specific data isn’t available, “few” students answered this part correctly.


  

CORE, FUR2 2012 VCAA 2

The maximum temperature and the minimum temperature at this weather station on each of the 30 days in November 2011 are displayed in the scatterplot below.

CORE, FUR2 2012 VCAA 2

The correlation coefficient for this data set is  . 

The equation of the least squares regression line for this data set is

maximum temperature = × minimum temperature

  1. Draw this least squares regression line on the scatterplot above.   (1 mark)

  2. Interpret the vertical intercept of the least squares regression line in terms of maximum temperature and minimum temperature.   (1 mark)

  3. Describe the relationship between the maximum temperature and the minimum temperature in terms of strength and direction.   (1 mark)

  4. Interpret the slope of the least squares regression line in terms of maximum temperature and minimum temperature.   (1 mark)

  5. Determine the percentage of variation in the maximum temperature that may be explained by the variation in the minimum temperature.
  6. Write your answer, correct to the nearest percentage.   (1 mark)

On the day that the minimum temperature was 11.1 °C, the actual maximum temperature was 12.2 °C.

  1. Determine the residual value for this day if the least squares regression line is used to predict the maximum temperature.
  2. Write your answer, correct to the nearest degree.   (2 marks)

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Show Worked Solution

a.  

 CORE, FUR2 2012 VCAA 2 Answer 

b.  

 

♦ Parts (i) to (vi) have an average mean mark of 41%.

c.   

 

d.  

 

 

 

e.   
   
   

 

f.  

MARKER’S COMMENT: Students had particular difficulty with this part, with many using the incorrect calculation of  12.2 – 11.1 = 1.1.
 
 
 

CORE, FUR2 2012 VCAA 1

The dot plot below displays the maximum daily temperature (in °C) recorded at a weather station on each of the 30 days in November 2011. 
  

CORE, FUR2 2012 VCAA 1

  1. From this dot plot, determine

     

  2.  i. the median maximum daily temperature, correct to the nearest degree  (1 mark)
  3. ii. the percentage of days on which the maximum temperature was less than 16 °C.
  4.     Write your answer, correct to one decimal place.  (1 mark)

Records show that the minimum daily temperature for November at this weather station is approximately normally distributed with a mean of 9.5 °C and a standard deviation of 2.25 °C.

  1. Determine the percentage of days in November that are expected to have a minimum daily temperature less than 14 °C at this weather station.
  2. Write your answer, correct to one decimal place.  (1 mark)

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a.i.  

.


 

a.ii.   


 

b.   
   
   

 

CORE, FUR2 2012 VCAA 1 Answer 
 

CORE, FUR2 2013 VCAA 4

The time series plot below shows the average pay rate, in dollars per hour, for workers in a particular country for the years 1991 to 2005.
 

CORE, FUR2 2013 VCAA 4

A three median line will be used to model the increasing trend in the average pay rate shown in this time series.

The explanatory variable to be used is year.

  1. Three medians will be used to draw the three median line.

     

    1. On the time series plot above, mark the location of each of the three medians with a cross ().  (2 marks)
    2. Draw the three median line on the time series plot.  (1 mark)
  2. Calculate the slope of the three median line.

     

    Write your answer, correct to one decimal place.  (1 mark)

  3. Interpret the slope of the three median line in terms of the relationship between the variables average pay rate and year.  (1 mark)
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  1. i. & ii.

  3.  

Show Worked Solution

a.i. & ii.   

♦ Mean mark 45%.
MARKER’S COMMENT: Few students found the correct middle point.

 

CORE, FUR2 2013 VCAA 4 Answer

♦♦ Parts (ii) and (iii) produced an overall mean mark of 32%.

 

b.   
   
   
   

 

MARKER’S COMMENT: Answering “an increase of 0.8 each year” received no marks (refer to actul units).

 

c.   

 

CORE, FUR2 2013 VCAA 3

The development index and the average pay rate for workers, in dollars per hour, for a selection of 25 countries are displayed in the scatterplot below.

CORE, FUR2 2013 VCAA 31   

The table below contains the values of some statistics that have been calculated for this data.

     CORE, FUR2 2013 VCAA 32

  1. Determine the standardised value of the development index ( score) for a country with a development index of 91.
  2. Write your answer, correct to one decimal place.   (1 mark)

  3. Use the information in the table to show that the equation of the least squares regression line for a country’s development index, , in terms of its average pay rate, , is given by
  4.           (2 marks)

  5. The country with an average pay rate of $14.30 per hour has a development index of 83.
  6. Determine the residual value when the least squares regression line given in part (b) is used to predict this country’s development index.
  7. Write your answer, correct to one decimal place.   (2 marks)

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MARKER’S COMMENT: Many students made a careless error by using the Average pay rate (-variable).
a.   
   
   
   

 

♦ Parts (b) and (c) had a combined mean mark of 48%.
b.   
   
   

 

 
 

 

 

c.   
   

 

 
 

CORE, FUR2 2013 VCAA 2

The development index for each country is a whole number between 0 and 100.

The dot plot below displays the values of the development index for each of the 28 countries that has a high development index.
 

CORE, FUR2 2013 VCAA 21 
 

  1. Using the information in the dot plot, determine each of the following.  (1 mark)

     

     

    Core, FUR2 2013 VCAA 2_2   
     

  2. Write down an appropriate calculation and use it to explain why the country with a development index of 70 is an outlier for this group of countries.  (2 marks)

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a.   

 

b.  

 

 

 

CORE, FUR2 2014 VCAA 4

The scatterplot below shows the population density, in people per square kilometre, and the area, in square kilometres, of 38 inner suburbs of the same city.

Core, FUR2 2015 VCAA 41

For this scatterplot,

  1. Describe the association between the variables population density and area for these suburbs in terms of strength, direction and form.   (1 mark)

  2. The mean and standard deviation of the variables population density and area for these 38 inner suburbs are shown in the table below.
     
    Core, FUR2 2015 VCAA 42
  3.   i. One of these suburbs has a population density of 3082 people per square kilometre.
  4.     Determine the standard -score of this suburb’s population density.
  5.     Write your answer, correct to one decimal place.   (1 mark)

Assume the areas of these inner suburbs are approximately normally distributed.

  1.  ii. How many of these 38 suburbs are expected to have an area that is two standard deviations or more above the mean?
  2.     Write your answer, correct to the nearest whole number.   (1 mark)

  3. iii. How many of these 38 inner suburbs actually have an area that is two standard deviations or more above the mean?   (1 mark)

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♦♦ Mean mark 30%.
MARKER’S COMMENT: The most common error was to describe the association as positive.
a.   
   

 

 

b.i   
   
   
   

 

b.ii.   

 

b.iii.   

♦♦ Very few students answered this part correctly.

²

 

CORE, FUR2 2014 VCAA 2

The scatterplot below shows the population and area (in square kilometres) of a sample of inner suburbs of a large city.
 

Core, FUR2 2015 VCAA 2

The equation of the least squares regression line for the data in the scatterplot is

population = 5330 + 2680 × area

  1. Write down the response variable.   (1 mark)

  2. Draw the least squares regression line on the scatterplot above.   (1 mark)

  3. Interpret the slope of this least squares regression line in terms of the variables area and population.  (2 marks)

  4. Wiston is an inner suburb. It has an area of 4 km² and a population of 6690.
  5. The correlation coefficient, , is equal to 0.668
  6.  i. Calculate the residual when the least squares regression line is used to predict the population of Wiston from its area.  (1 mark)

  7. ii. What percentage of the variation in the population of the suburbs is explained by the variaton in area.
  8.     Write your answer, correct to one decimal place.  (1 mark)

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  2.  
          Core, FUR2 2015 VCAA 2 Answer

  3. ²
Show Worked Solution

a.  
 

♦ Mean mark 36%.
MARKER’S COMMENT: Use the equation to draw the line and use points at the extremities.
b.   

Core, FUR2 2015 VCAA 2 Answer

 

c.   

♦ Mean mark 41% (part (iii)).

²
 

d.i. 


 

♦ Part (iv) in total had a mean mark 42%.
d.ii.   
 
   

 

CORE, FUR2 2014 VCAA 1

The segmented bar chart below shows the age distribution of people in three countries, Australia, India and Japan, for the year 2010.
 

Core, FUR2 2015 VCAA 1

  1. Write down the percentage of people in Australia who were aged 0 – 14 years in 2010.
  2. Write your answer, correct to the nearest percentage.  (1 mark)

  3. In 2010, the population of Japan was 128 000 000.
  4. How many people in Japan were aged 65 years and over in 2010?  (1 mark)

  5. From the graph above, it appears that there is no association between the percentage of people in the 15 – 64 age group and the country in which they live.
  6. Explain why, quoting appropriate percentages to support your explanation.  (1 mark)

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  3.  
     
     

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a.   

 

b.  

 

c.   

♦ Mean mark 41%.

CORE, FUR2 2015 VCAA 5

The time series plot below displays the life expectancy, in years, of people living in Australia and the United Kingdom (UK) for each year from 1920 to 2010.
 

Core, FUR2 2015 VCAA 51

  1. By how much did life expectancy in Australia increase during the period 1920 to 2010?
  2. Write your answer correct to the nearest year.   (1 mark)

  3. In 1975, the life expectancies in Australia and the UK were very similar.
  4. From 1975, the gap between the life expectancies in the two countries increased, with people in Australia having a longer life expectancy than people in the UK.
  5. To investigate the difference in life expectancies, least squares regression lines were fitted to the data for both Australia and the UK for the period 1975 to 2010.
  6. The results are shown below. 

  Core, FUR2 2015 VCAA 52

  1. The equations of the least squares regression lines are as follows.
 

 

  1. Use these equations to predict the difference between the life expectancies of Australia and the UK in 2030.
  2. Give your answer correct to the nearest year.   (2 marks)

  3. Explain why this prediction may be of limited reliability.   (1 mark)

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    2.  

       

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a.   

 

b.i.   
   
     
 
   

 

 
♦ Mean mark 45%.
MARKER’S COMMENT: Relate answers directly to the limitations of the given statistical data rather than future events in (b)(ii).

 

b.ii.   

 

 

CORE, FUR2 2015 VCAA 4

The table below shows male life expectancy (male) and female life expectancy (female) for a number of countries in 2013. The scatterplot has been constructed from this data.
 

Core, FUR2 2015 VCAA 4

  1. Use the scatterplot to describe the association between male life expectancy and female life expectancy in terms of strength, direction and form.   (1 mark)

  2. Determine the equation of a least squares regression line that can be used to predict male life expectancy from female life expectancy for the year 2013.

     

    Complete the equation for the least squares regression line below by writing the intercept and slope in the space provided.

     

    Write these values correct to two decimal places.  (1 mark)

male = ______________ + ______________ ×  female

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a.   

♦ Mean mark 49%.
MARKER’S COMMENT: Common errors included using the 1st column as the independent () variable and poor rounding.

 

b.  

Probability, 2UG SM-Bank 01 MC

A circle has 5 equally spaced points on its circumference, as shown.

How many different triangles using 3 of these points as vertices can be drawn?  (3 marks)

2UG SM-bank 01 mc

A.  

B.  

C.  

D.  

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STRATEGY: The number of triangles is divided by because the selections of any 3 vertices are unordered.

Financial Maths, 2ADV M1 SM-Bank 10 MC

The graph above shows the first six terms of a sequence.

This sequence could be

  1. an arithmetic sequence that sums to one.
  2. an arithmetic sequence with a common difference of one.
  3. a geometric sequence with an infinite sum of one.
  4. a geometric sequence with a common ratio of one. 
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CORE, FUR2 2015 VCAA 3

The scatterplot below plots male life expectancy (male) against female life expectancy (female) in 1950 for a number of countries. A least squares regression line has been fitted to the scatterplot as shown.
 


 

The slope of this least squares regression line is 0.88

  1. Interpret the slope in terms of the variables male life expectancy and female life expectancy.  (1 mark)

The equation of this least squares regression line is

male = 3.6 + 0.88 × female

  1. In a particular country in 1950, female life expectancy was 35 years.

     

    Use the equation to predict male life expectancy for that country.  (1 mark)

  2. The coefficient of determination is 0.95

     

    Interpret the coefficient of determination in terms of male life expectancy and female life expectancy.  (1 mark)

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a.   

♦ Mean mark 40%.
MARKER’S COMMENT: Many students did not describe the slope, despite being specifically asked about it!

 

b.   

 

c.   

MARKER’S COMMENT: A common error: use of as the basis of the interpretation.

 

CORE, FUR2 2015 VCAA 1

The histogram below shows the distribution of life expectancy of people for 183 countries.
 

CORE, FUR2 2015 VCAA 1 

  1. For this distribution, the modal interval is  ___________  (1 mark)

  2. In how many of these countries is life expectancy less than 55 years?  (1 mark)

  3. In what percentage of these 183 countries is life expectancy between 75 and 80 years?
  4. Write your answer correct to one decimal place.  (1 mark)

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a.   

MARKER’S COMMENT: 40% of students answered this basic question incorrectly.

 

b.   

 

c.   

MARKER’S COMMENT: Answers of 16% or an unrounded figure received no marks.

NETWORKS, FUR1 2015 VCAA 4 MC

The arrows on the diagram below show the direction of the flow of waste through a series of pipelines from a factory to a waste dump.

The numbers along the edges show the number of megalitres of waste per week that can flow through each section of pipeline.
  

NETWORKS, FUR1 2015 VCAA 4 MC

 
The minimum cut is shown as a dotted line.

The capacity of this cut, in megalitres of waste per week, is

A.     

B.   

C.   

D.   

E.   

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MATRICES, FUR1 2015 VCAA 6 MC

A carpenter can make four coffee tables and seven stools in a total of 33 hours.

The carpenter can make two coffee tables and three pencil boxes in a total of 12 hours.

The carpenter can make five stools and one pencil box in a total of 10 hours.

The time, in hours, that it takes to make one coffee table is closest to

A.   2

B.   3

C.   4

D.   5

E.   6

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MATRICES, FUR1 2015 VCAA 5 MC

Wendy buys one type of flower each day.

She chooses from tulips (), roses (), carnations (), irises () and daisies ().

The type of flower she buys on one day depends on the type of flower she bought the previous day, according to a transition matrix.

Today, Wendy bought tulips.

The transition matrix that, starting tomorrow, ensures Wendy buys flowers in alphabetical order (, , , , ) is

MATRICES, FUR1 2015 VCAA 5 MCab

MATRICES, FUR1 2015 VCAA 5 MCcd

MATRICES, FUR1 2015 VCAA 5 MCe 

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MATRICES, FUR1 2015 VCAA 4 MC

The numbers of adult and child tickets purchased for five performances of a stage show are shown in the table below.

MATRICES, FUR1 2015 VCAA 4 MC

Which one of the following matrix calculations can be used to determine both the total number of adult tickets and the total number of child tickets purchased for all five performances?

A.  

B.  

C.  

D.  

E.  

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GRAPHS, FUR1 2015 VCAA 9 MC

The shaded area in the graph below shows the feasible region for a linear programming problem.

GRAPHS, FUR1 2015 VCAA 9 MC

The maximum value of the objective function    for this problem occurs at point

A.   

B.   

C.   

D.   

E.   

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GRAPHS, FUR1 2015 VCAA 8 MC

To raise funds, a club plans to sell lunches at a weekend market.

The club will pay $190 to rent a stall.

Each lunch will cost $12 to prepare and will be sold for $35.

To make a profit of at least $1000, the minimum number of lunches that must be sold is

A.   

B.   

C.   

D.   

E.   

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GRAPHS, FUR1 2015 VCAA 7 MC

The graph below shows the cost, , of printing wedding invitations.

GRAPHS, FUR1 2015 VCAA 7 MC

A function that can be used to model this is

The value of is

A.     

B.   

C.   

D.   

E.   

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GRAPHS, FUR1 2015 VCAA 6 MC

The graph below shows the braking distance, in metres, of a car at different speeds, in kilometres per hour.

The coordinates of a point on the graph are also shown.

GRAPHS, FUR1 2015 VCAA 6 MC

The relationship between braking distance and speed can be modelled by an equation of the form

²

Using this model, the braking distance, in metres, when the speed is 60 km/h is

A.   

B.   

C.   

D.   

E.   

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Financial Maths, 2ADV M1 SM-Bank 8 MC

There are 3000 tickets available for a concert.

On the first day of ticket sales, 200 tickets are sold.

On the second day, 250 tickets are sold.

On the third day, 300 tickets are sold.

This pattern of ticket sales continues until all 3000 tickets are sold.

How many days does it take for all of the tickets to be sold?

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Statistics, STD2 S5 SM-Bank 1 MC

The head circumference (in cm) of a population of infant boys is normally distributed with a mean of 49.5 cm and a standard deviation of 1.5 cm.

Four hundred of these boys are selected at random and each boy’s head circumference is measured.

The number of these boys with a head circumference of less than 48.0 cm is closest to 

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GRAPHS, FUR1 2010 VCAA 6 MC

The graphs of the linear relations

are shown below.

graphs 2010 VCAA 6mc

A point that satisfies both the inequalities

  is

A.   

B.   

C.   

D.  

E.  

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graphs 2010 VCAA 6mci

GRAPHS, FUR1 2010 VCAA 5 MC

The cost in dollars, , of making pottery mugs is given by the equation

A loss will result from selling

A.   60 mugs at $9.00 each.

B.   70 mugs at $8.50 each.

C.   80 mugs at $7.50 each.

D.   90 mugs at $8.00 each.

E.   100 mugs at $9.50 each.

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GRAPHS, FUR1 2010 VCAA 4 MC

The manager of an office is ordering finger food for an office party.

Hot items cost $2.15 each and cold items cost $1.50 each.

Let be the number of hot items ordered.
Let be the number of cold items ordered.

The manager can spend no more than $5 for each of the 200 employees.

An inequality that can be used to represent this constraint is

A.  

B.  

C.  

D.   

E.   

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GRAPHS, FUR1 2010 VCAA 1-2 MC

The volume of water that is stored in a tank over a 24-hour period is shown in the graph below.

Part 1

What is the difference in the volume of water (in litres) in the tank between 8 am and 6 pm?

A.     

B.   

C.   

D.  

E.   

 

Part 2

The rate of increase in the volume of water in the tank (in litres/hour) between 8 am and 10 am is

A.     

B.     

C.     

D.   

E.   

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GRAPHS, FUR1 2011 VCAA 7-9 MC

Craig plays sport and computer games every Saturday.

Let be the number of hours that he spends playing sport.
Let be the number of hours that he spends playing computer games.

Craig has placed some constraints on the amount of time that he spends playing sport and computer games.

These constraints define the feasible region shown shaded in the graph below. The equations of the lines that define the boundaries of the feasible regions are also shown.

Part 1

One of the constraints that defines the feasible region is

A.  

B.  

C.  

D.  

E.  

 

Part 2

By spending Saturday playing sport and computer games, Craig believes he can improve his health.

Let be the health rating Craig achieves by spending a day playing sport and computer games.

The value of is determined by using the rule  .

For the feasible region shown in the graph above, the maximum value of occurs at

A.   point 

B.   point 

C.   point 

D.   point 

E.   point 

 

Part 3

By spending Saturday playing sport and computer games, Craig believes he can improve his mental alertness.

Let be the mental alertness rating Craig achieves by spending a day playing sport and computer games.

For the feasible region shown in the graph above, the maximum value of occurs at any point that lies on the line that joins points and is the feasible region.

The rule for could be

A.  

B.  

C.  

D.  

E.  

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GRAPHS, FUR1 2011 VCAA 5 MC

The cost, , of making kilograms of chocolate fudge is given by  .

The revenue, , from selling kilograms of chocolate fudge is given by  .

A particular quantity of chocolate fudge was made and sold. It resulted in a loss of $20.

The number of kilograms of chocolate fudge made and sold was

A.     

B.     

C.     

D.   

E.   

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GRAPHS, FUR1 2011 VCAA 4 MC

The fare, , to travel a distance on kilometres in a taxi is given by the rule

To travel a distance of 20 kilometres, the taxi fare is $18.20

To travel a distance of 30 kilometres, the taxi fare is $25.70

The charge per kilometre, , is

A.   

B.   

C.   

D.   

E.   

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