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Bivariate Data, SM-Bank 013

The scatterplot below plots male life expectancy (male) against female life expectancy (female) in 1950 for a number of countries.

A line of best fit has been fitted to the scatterplot as shown.
 

  1. State the dependent variable in the graph.   (1 mark)
  2. Determine the age at which males and females have the same life expectancy.   (1 mark)
Show Answers Only

i.    \(\text{Male life expectancy}\)

ii.   \(\text{Life expectancy the same at 30 years of age.}\)

Show Worked Solution

i.    \(\text{Dependent variable}\ \ \Rightarrow \ \ y \text{-axis variable} \)

\(\text{Male life expectancy}\) 
 

ii.   \(\text{Life expectancy the same at 30 years of age.}\)

Bivariate Data, SM-Bank 012

A teacher analysed the class marks of 15 students who sat two tests.

The test 1 mark and test 2 mark, all whole number values, are shown in the scatterplot below.

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

  1. If a student scored 34 in the first test, what is their expected mark in the second test.   (1 mark)
  2. The line of best fit shows the predicted test 2 mark for each student based on their test 1 mark.
  3. Determine the number of students whose actual test 2 mark was within two marks of that predicted.   (1 mark)

Show Answers Only

i.    \(46\)

ii.   \(\text{5 values are within 1 grid height (measured vertically), or 2 marks,}\)

\(\text{from the LOBF.}\)

Show Worked Solution

i.    \(\text{1st test mark = 34}\ \ \Rightarrow\ \ \text{2nd test mark = 46} \)
 

ii.   \(\text{5 values are within 1 grid height (measured vertically), or 2 marks,}\)

\(\text{from the LOBF.}\)

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 010

The scatterplot below shows the wrist circumference and ankle circumference, both in centimetres, of 13 people.

A line of best fit been drawn with ankle circumference as the independent variable.
 

  1. If a person has a wrist circumference of 18.5 centimetres, estimate the ankle circumference that is predicted by the line of best fit.   (1 mark)

  2. Explain why the \(y\)-intercept of this graph has no meaning in this context.   (1 mark)

Show Answers Only

i.    \(\text{18.5 cm wrist}\ \ \Rightarrow \ \ \text{Ankle circumference ≈ 24.5 cm}\)

ii.    \(y \text{-intercept occurs when ankle circumference = 0 cm, which is}\)

\(\text{meaningless in this context.}\)

Show Worked Solution

i.    \(\text{18.5 cm wrist}\ \ \Rightarrow \ \ \text{Ankle circumference ≈ 24.5 cm}\)
 

ii.    \(y \text{-intercept occurs when ankle circumference = 0 cm, which is}\)

\(\text{meaningless in this context.}\)

Bivariate Data, SM-Bank 009

The height (in cm) and foot length (in cm) for each of eight Year 12 students were recorded and displayed in the scatterplot below.

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

  1. Determine the predicted foot size of a student who is 176 centimetres tall.   (1 mark)
  2. Calculate the gradient of the line of best fit, giving your answer correct to two decimal places.   (2 marks)
  3. What is the equation of the line of best fit?   (2 marks)
Show Answers Only

i.    \(\text{Height 176 cm}\ \ \Rightarrow\ \ \text{Foot length = 27 cm} \)

ii.    \(1.29 \)

iii.    \(\textit{height}\ = 141.2 + 1.29 \times \textit{foot length} \)

Show Worked Solution

i.    \(\text{Height 176 cm}\ \ \Rightarrow\ \ \text{Foot length = 27 cm} \)
 

ii.    \(\text{LOBF passes through (20, 167) and (34, 185):}\)

\(\text{Gradient}\ = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{185-167}{34-20} = \dfrac{18}{14} = 1.285… = 1.29 \)
 

iii.   \(\text{Equation}\ \ \Rightarrow \ \ \text{Gradient = 1.29, passes through (20, 167)} \)

\(y-y_1 \) \(=m(x-x_1) \)  
\(y-167\) \(=1.29(x-20) \)  
\(y\) \(=1.29x + 141.2\)  

 
\(\textit{height}\ = 141.2 + 1.29 \times \textit{foot length} \)

Bivariate Data, SM-Bank 007 MC

Dr Chris measures the weights (in grams) and lengths (in cm) of 12 baby pythons.

The results were recorded and plotted in the scatterplot below. The line of best fit that enables the weight of the pythons to be predicted from their length has also been plotted.
 

The line of best fit predicts that the weight, in grams, of a python of length 30 cm would be closest to

  1. `240`
  2. `252`
  3. `262`
  4. `274`
Show Answers Only

`C`

Show Worked Solution

`text{The line of best fit crosses the 30cm length (on the}`

`xtext{-axis) at approx 262.}`

`=>C`

Bivariate Data, SM-Bank 006

The lengths and diameters, in millimetres, of a sample of jellyfish selected were recorded and displayed in the scatterplot below.

A line of best fit for this data is shown.
 

  1. Determine the expected length of a jellyfish with a diameter of 12 millimetres.   (1 mark)

  2. Determine the expected diameter of a jellyfish with a length of 16 millimetres.   (1 mark)

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i.    \(\text{A diameter of 12 mm (x-axis)}\ \ \Rightarrow \ \ \text{length 14 mm}\)

ii.    \(\text{Length of 16 mm (y-axis)}\ \ \Rightarrow \ \ \text{diameter 14.4 mm (approx)}\)

Show Worked Solution

i.    \(\text{A diameter of 12 mm (x-axis)}\ \ \Rightarrow \ \ \text{length 14 mm}\)

ii.    \(\text{Length of 16 mm (y-axis)}\ \ \Rightarrow \ \ \text{diameter 14.4 mm (approx)}\)

Bivariate Data, SM-Bank 005

Each member of a group of males had his height and foot length measured and recorded. The results were graphed and a line of fit drawn.
 

  1. Why does the value of the `y`-intercept have no meaning in this situation?  (1 mark)

  2. George is 10 cm taller than his brother Harry. Use the line of fit to estimate the difference in their foot lengths.  (1 mark)

Show Answers Only
  1. `text(The y-intercept occurs when)\ x = 0.\ text(It has`
    `text(no meaning to have a height of 0 cm.)`
  2. `text(A 10 cm height difference means George should)`
    `text(have a 3 cm longer foot.)`
Show Worked Solution

i.  `text(The y-intercept occurs when)\ x = 0.\ text(It has)`

`text(no meaning to have a height of 0 cm.)`

 

ii.  `text(A 20 cm height difference results in a foot length)`

`text(difference of 6 cm.)`

`:.\ text(A 10 cm height difference means George should)`

`text(have a 3 cm longer foot.)`

Bivariate Data, SM-Bank 004

The shoe size and height of ten students were recorded.

\begin{array} {|l|c|c|}
\hline \rule{0pt}{2.5ex} \text{Shoe size} \rule[-1ex]{0pt}{0pt} & \text{6} & \text{7} & \text{7} & \text{8} & \text{8.5} & \text{9.5} & \text{10} & \text{11} & \text{12} & \text{12} \\
\hline \rule{0pt}{2.5ex} \text{Height} \rule[-1ex]{0pt}{0pt} & \text{155} & \text{150} & \text{165} & \text{175} & \text{170} & \text{170} & \text{190} & \text{185} & \text{200} & \text{195} \\
\hline
\end{array}

  1. Complete the scatter plot AND draw a line of best fit by eye.  (2 marks)
     
     
  2. Use the line of best fit to estimate the height difference between a student who wears a size 7.5 shoe and one who wears a size 9 shoe.  (1 mark)

Show Answers Only
  1. `text(See Worked Solutions.)`
  2. `text{13 cm (or close given LOBF drawn)}`
Show Worked Solution

i.    
     

ii.    `text{Shoe size 7.5 gives a height estimate of 162 cm (see graph)}`

`text{Shoe size 9 gives a height estimate of 175 cm (see graph)}`

`:.\ text(Height difference)` `= 175-162`
  `= 13\ text{cm  (or close given LOBF drawn)}`

Bivariate Data, SM-Bank 001

The graph shows a line of best fit describing the life expectancy of people born between 1900 and 2000.
 


  1. According to the graph, what is the life expectancy of a person born in 1932?  (1 mark)

  2. Determine the value of the gradient of the line of best fit.  (2 marks)

Show Answers Only
  1. \(\text{68 years}\)
  2. \(0.25\)
Show Worked Solution

i.    \(\text{68 years}\)

ii.    \(\text{Using (1900,60), (1980,80):}\)

\(\text{Gradient}\) \(= \dfrac{y_2-y_1}{x_2-x_1}\)
  \(= \dfrac{80-60}{1980-1900}\)
  \(= 0.25\)

Statistics, STD1 S3 2023 HSC 19

The scatterplot shows the number of ice-creams sold, \(y\), at a shop over a ten-day period, and the temperature recorded at 2 pm on each of these days.
 

  1. The data are modelled by the equation of the line of best fit given below.

\(y=0.936 x-8.929\), where \(x\) is the temperature.

  1. Sam used a particular temperature with this equation and predicted that 23 ice-creams would be sold.
  2. What was the temperature used by Sam, to the nearest degree?  (2 marks)

  3. In using the equation to make the prediction in part (a), was Sam interpolating or extrapolating? Justify your answer.  (2 marks)

Show Answers Only

a.    \(34^{\circ}\text{ (nearest degree)}\)

b.    \(\text{See worked solutions}\)

Show Worked Solution

a.             \(y\) \(=0.936x-8.929\)
\(23\) \(=0.936x-8.929\)
\(0.936x\) \(=23+8.929\)
\(x\) \(=\dfrac{31.921}{0.936}\)
  \(=34.112\ldots ^{\circ}\)
  \(= 34^{\circ}\text{ (nearest degree)}\)

♦♦ Mean mark (a) 31%.

b.     \(\text{Sam is extrapolating as 34°C is outside the range of data}\)

\(\text{points shown on the graph (i.e. temp between 0 and 30°C).}\)


♦♦ Mean mark (b) 33%.

Statistics, STD1 S3 2022 HSC 23

A teacher surveyed the students in her Year 8 class to investigate the relationship between the average number of hours of phone use per day and the average number of hours of sleep per day.

The results are shown on the scatterplot below.
 

  1. The data for two new students, Alinta and Birrani, are shown in the table below. Plot their results on the scatterplot.  (2 marks)

\begin{array} {|l|c|c|}
\hline
  & \textit{Average hours of} & \textit{Average hours of} \\ & \textit{phone use per day} & \textit{sleep per day} \\
\hline
\rule{0pt}{2.5ex} \text{Alinta} \rule[-1ex]{0pt}{0pt} & 4 & 8 \\
\hline
\rule{0pt}{2.5ex} \text{Birrani} \rule[-1ex]{0pt}{0pt} & 0 & 10.5 \\
\hline
\end{array}

  1. By first fitting the line of best fit by eye on the scatterplot, estimate the average number of hours of sleep per day for a student who uses the phone for an average of 2 hours per day.  (2 marks)

Show Answers Only
  1.  
  2. 9 hours (see LOBF in diagram above)
Show Worked Solution

a.     \(\text{New data points are marks with a × on the diagram below.}\)
 

b.   \(\text{9 hours (see LOBF in diagram above)}\)

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, STD1 S3 2019 HSC 27

A set of bivariate data is collected by measuring the height and arm span of eight children. The graph shows a scatterplot of these measurements.
 

  1. On the graph, draw a line of best fit by eye.  (1 mark)

  2. Robert is a child from the class who was absent when the measurements were taken. He has an arm span of 147 cm. Using your line of best fit from part (a), estimate Robert’s height.  (1 mark)

Show Answers Only
  1.   
  2. `text(Robert’s height ≈ 151.1 cm)`
Show Worked Solution

a.     
       

♦ Mean mark (a) 38%.

b.   `text(Robert’s height ≈ 151.1 cm)`

`text{(Answers can vary slightly depending on line of best fit drawn).}`

Algebra, STD2 A2 2017 HSC 3 MC

The graph shows the relationship between infant mortality rate (deaths per 1000 live births) and life expectancy at birth (in years) for different countries.
 

What is the life expectancy at birth in a country which has an infant mortality rate of 60?

  1. 68 years
  2. 69 years
  3. 86 years
  4. 88 years
Show Answers Only

\(A\)

Show Worked Solution

\(\text{When infant mortality rate is 60, life expectancy}\)

\(\text{at birth is 68 years (see below).}\)
 

\(\Rightarrow A\)