SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Bivariate Data, SM-Bank 016

The scatterplot below shows the rainfall (in mm) and the percentage of clear days for each month of 2023. 
 

An equation of the line of best fit for this data set is

\(\textit{rainfall}\ = 131-2.68 \times\ \textit{percentage of clear days} \)

  1. Using coordinates at the graph extremities or otherwise, draw this line on the scatterplot.  (2 marks)

  2. Describe this association in terms of strength and direction.  (1 mark)

Show Answers Only

i.   
       

ii.    \(\text{Strength: moderate (data points are moderately close to the LOBF)}\)

\(\text{Direction: negative (as percentage of clear days ↑, rainfall ↓)}\)

Show Worked Solution

i.    \(\text{Calculate values at the limits of the graph:}\)

\((0, 131)\ \ \Rightarrow \ \ y \text{-intercept}\ = 131 \)

\( (30, 50.6)\ \ \Rightarrow \ \text{At}\ x=30, \ y=131-2.68 \times 30 = 50.6 \)
 

ii.    \(\text{Strength: moderate (data points are moderately close to the LOBF)}\)

\(\text{Direction: negative (as percentage of clear days ↑, rainfall ↓)}\)

Bivariate Data, SM-Bank 015

The heights (in cm) and ages (in months) of a random sample of 15 boys have been plotted in the scatterplot below.

A line of best fit has been fitted to the data.

 

  1. State the independent variable in the graph.   (1 mark)

  2. Describe this association in terms of strength and direction.   (2 marks)
  3. Determine the gradient of the line of best fit, giving your answer correct to one decimal place.   (2 marks)
Show Answers Only

i.    \(\text{Independent variable: age (months)}\)

ii.    \(\text{Association: strong and positive}\)

iii.   \(0.5\)

Show Worked Solution

i.    \(\text{Independent variable}\ \ \Rightarrow\ \ x\text{-axis variable} \)

\(\text{age (months)}\)
 

ii.    \(\text{Association:} \)

\(\text{Strength: strong (data points are tightly gathered to the LOBF)}\)

\(\text{Direction: positive (as age ↑, height ↑)}\)
 

iii.   \(\text{LOBF passes through (15, 83.25) and (35, 94)} \)

\(\text{Gradient}\ = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{94-83.25}{35-15} = 0.5375 = 0.5\ \text{(1 d.p.)} \)

Bivariate Data, SM-Bank 014

The scatterplot below displays the mean age, in years, and the mean height, in centimetres, of 648 women from seven different age groups.

In an initial analysis of the data, a line of best fit is drawn, as shown.

 

  1. Describe this association in terms of strength and direction.   (2 marks)
  2. Determine the mean height predicted for a group of 65 year old women.   (1 mark)
Show Answers Only

i.    \(\text{Association: strong and negative}\)

ii.   \(\text{Predicted height = 160.5 cm}\)

Show Worked Solution

i.    \(\text{Association:} \)

\(\text{Strength: strong (data points are tightly gathered to the LOBF)}\)

\(\text{Direction: negative (as mean age ↑, mean height ↓)}\)
 

ii.   \(\text{Mean age = 65}\ \ \Rightarrow \ \ \text{Predicted height = 160.5 cm}\)

Bivariate Data, SM-Bank 011

The scatterplot below displays the resting pulse rate, in beats per minute, and the time spent exercising, in hours per week, of 16 students.

A line of best fit has been fitted to the data.
 

  1. If a student spends 8 hours exercising per week, determine the resting pulse rate predicted by the line of best fit.   (1 mark)

  2. Provide TWO descriptions of the association between the variables time spent exercising and resting pulse rate.   (2 marks)

Show Answers Only

i.    \(\text{8 hours exercising}\ \ \Rightarrow\ \ \text{pulse rate = 59.5} \)

ii.   \(\text{Association should include two of the following:} \)

\(\text{Linear (straight line)}\)

\(\text{Negative (as time spent exercising ↑, resting pulse rate ↓)}\)

\(\text{Strong (data points are found tightly around the LOBF)}\)

Show Worked Solution

i.    \(\text{8 hours exercising}\ \ \Rightarrow\ \ \text{pulse rate = 59.5} \)
 

ii.   \(\text{Association should include two of the following:} \)

\(\text{Linear (straight line)}\)

\(\text{Negative (as time spent exercising ↑, resting pulse rate ↓)}\)

\(\text{Strong (data points are found tightly around the LOBF)}\)

Bivariate Data, SM-Bank 002

People are placed into groups to complete a puzzle. There are 9 different groups.

The table shows the number of people in each group and the amount of time, in minutes, for each group to complete the puzzle.

\begin{array} {|l|c|c|c|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \text{Number of people} \rule[-1ex]{0pt}{0pt} & 2 & 2 & 3 & 5 & 5 & 7 & 7 & 7 & 8 \\
\hline
\rule{0pt}{2.5ex} \text{Time taken (min)} \rule[-1ex]{0pt}{0pt} & 28 & 30 & 26 & 19 & 21 & 12 & 13 & 11 & 8 \\
\hline
\end{array}

  1. Complete the scatterplot by adding the last four points from the table.  (1 mark)
     
       
  2. Add a line of best fit by eye to the graph in part (a).  (1 mark)

  3. The graph in part (a) shows the association between the time to complete the puzzle and the number of people in the group.
  4. Identify the form (linear or non-linear), the direction and the strength of the association.  (2 marks)

Show Answers Only

a.

b.


 

c.    `text(Form: linear)

`text{Direction: negative}`

`text{Strength: strong}`

Show Worked Solution

a.

b.


 

c.    `text{Form: linear (i.e. straight line)}`

`text{Direction: negative}`

`text{Strength: strong}`

Statistics, STD1 S3 2020 HSC 22

A group of students sat a test at the end of term. The number of lessons each student missed during the term and their score on the test are shown on the scatterplot.
 


 

  1. Describe the strength and direction of the linear association observed in this dataset.  (2 marks)

  2. Calculate the range of the test scores for the students who missed no lessons.  (1 mark)

  3. Draw a line of the best fit in the scatterplot above.  (1 mark)
  4. Meg did not sit the test. She missed five lessons.

     

    Use the line of the best fit drawn in part (c) to estimate Meg's score on this test. (1 mark)

  5. John also did not sit the test and he missed 16 lessons.

     

    Is it appropriate to use the line of the best fit to estimate his score on the test? Briefly explain your answer. (1 mark)

Show Answers Only

a.    \(\text{Strength : strong}\)

\(\text{Direction : negative} \)

b.    \(\text{Range}\ = \text{high}-\text{low}\ = 100-80=20\)
 

c.   

d. 


 

e.    \(\text{John’s missed days are too extreme and the LOBF is not}\)

\(\text{appropriate. The model would estimate a negative score for}\)

\(\text{John which is impossible.}\)

Show Worked Solution

a.    \(\text{Strength : strong}\)

\(\text{Direction : negative} \)

♦ Mean mark (a) 45%.
♦♦ Mean mark (b) 31%.

b.    \(\text{Range}\ = \text{high}-\text{low}\ = 100-80=20\)
 

c.   

d. 


 
\(\therefore\ \text{Meg’s estimated score = 40}\)
 

e.    \(\text{John’s missed days are too extreme and the LOBF is not}\)

\(\text{appropriate. The model would estimate a negative score for}\)

\(\text{John which is impossible.}\)

♦ Mean mark (e) 38%.

Statistics, STD2 S4 2017 HSC 12 MC

Which of the data sets graphed below has the largest positive correlation coefficient value?
 

A.      B.     
C.      D.     
Show Answers Only

\(C\)

Show Worked Solution

\(\text{Largest positive correlation occurs when both variables move}\)

\(\text{in tandem. The tighter the linear relationship, the higher the}\)

\(\text{correlation.}\)

\(\Rightarrow C\)

\(\text{(Note that B is negatively correlated)}\)

Statistics, STD2 S4 2008 HSC 12 MC

A scatterplot is shown.
 

Which of the following best describes the correlation between  \(R\)  and  \(T\)?

  1. Positive
  2. Negative 
  3. Positively skewed
  4. Negatively skewed
Show Answers Only

\(A\)

Show Worked Solution

\(\text{Correlation is positive.}\)

\(\text{NB. The skew does not directly relate to correlation.}\)

\(\Rightarrow  A\)

Statistics, STD2 S4 2009 HSC 28b

The height and mass of a child are measured and recorded over its first two years. 

\begin{array} {|l|c|c|}
\hline \rule{0pt}{2.5ex} \text{Height (cm), } H \rule[-1ex]{0pt}{0pt} & \text{45} & \text{50} & \text{55} & \text{60} & \text{65} & \text{70} & \text{75} & \text{80} \\
\hline \rule{0pt}{2.5ex} \text{Mass (kg), } M \rule[-1ex]{0pt}{0pt} & \text{2.3} & \text{3.8} & \text{4.7} & \text{6.2} & \text{7.1} & \text{7.8} & \text{8.8} & \text{10.2} \\
\hline
\end{array}

This information is displayed in a scatter graph. 
 

  1. Describe the correlation between the height and mass of this child, as shown in the graph.   (1 mark)

  2. A line of best fit has been drawn on the graph.

     

    Find the equation of this line.   (2 marks)

Show Answers Only
  1. `text(The correlation between height and)`

     

    `text(mass is positive and strong.)`

  2. `M = 0.23H-8`
Show Worked Solution

i.  `text(The correlation between height and)`

♦ Mean mark 48%. 

`text(mass is positive and strong.)`

 

ii.  `text(Using)\ \ P_1(40, 1.2)\ \ text(and)\ \ P_2(80, 10.4)`

♦♦♦ Mean mark 18%. 
MARKER’S COMMENT: Many students had difficulty due to the fact the horizontal axis started at `H= text(40cm)` and not the origin.
`text(Gradient)` `= (y_2-y_1)/(x_2-x_1)`
  `= (10.4-1.2)/(80-40)`
  `= 9.2/40`
  `= 0.23`

 

`text(Line passes through)\ \ P_1(40, 1.2)`

`text(Using)\ \ \ y-y_1` `= m(x-x_1)`
`y-1.2` `= 0.23(x-40)`
`y-1.2` `= 0.23x-9.2`
`y` `= 0.23x-8`

 
`:. text(Equation of the line is)\ \ M = 0.23H-8`

Statistics, STD2 S4 2012 HSC 11 MC

Which of the following relationships would most likely show a negative correlation?

  1. The population of a town and the number of hospitals in that town. 
  2. The hours spent training for a race and the time taken to complete the race. 
  3. The price per litre of petrol and the number of people riding bicycles to work. 
  4. The number of pets per household and the number of computers per household. 
Show Answers Only

\(B\)

Show Worked Solution

\(\text{Increased hours training should reduce the time}\)

\(\text{to complete a race.}\)

\(\Rightarrow B\)

♦ Mean mark 43%.