Each of the scatterplots shows the same number of data points.
Which scatterplot has the most appropriate line of best fit?
Statistics, STD1 2026 HSC 13
A coach was interested in whether there was a relationship between the number of hours spent practising kicking a ball and the number of goals scored in a competition.
The coach collected data from 10 players. A graph of the dataset is shown.
- Describe the bivariate dataset in terms of its form, direction and strength. (3 marks)
- Form: ..............................................................
- Direction: ........................................................
- Strength: .........................................................
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- The line of best fit has equation \(y=2.8+0.4x\).
- Using this equation, what is the expected number of goals scored by a player who spent 5 hours practising kicking a ball? (1 mark)
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Statistics, STD1 S3 2025 HSC 15
A researcher is using the statistical investigation process to investigate a possible relationship between average number of minutes per day a person spends watching television, and the average number of minutes per day the person spends exercising.
- State the statistical question being posed. (1 mark)
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Participants were asked to record the number of minutes they spent watching television each day and the number of minutes they spent exercising each day. The averages for each participant were recorded and graphed, and a line of best fit was included.
- From the graph, identify the dependent variable. (1 mark)
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- Describe the bivariate dataset in terms of its form and direction. (2 marks)
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- The points \((0, 70)\) and \((60, 10)\) lie on the line of best fit. By first plotting these points on the graph, find the gradient and the \(y\)-intercept of the line of best fit. (3 marks)
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- Explain why it is NOT appropriate to extrapolate the line of best fit to predict the average number of minutes of exercise per day for someone who watches an average of 2 hours of television per day. (1 mark)
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Show Worked Solution
a. \(\text{How does the average daily time spent watching television}\)
\(\text{relate to the average daily time spent exercising?}\)
b. \(\text{Dependent variable:}\)
\(\text{Average minutes per day exercising, or }y.\)
♦♦♦ Mean mark (b) 21%.
c. \(\text{Form: Linear}\)
\(\text{Direction: Negative}\)
♦♦ Mean mark (c) 45%.
d.
\(y-\text{intercept = 70}\)
\(\text{Gradient}=\dfrac{\text{rise}}{\text{run}}=\dfrac{-60}{60}=-1\)
♦♦♦ Mean mark (d) 30%.
e. \(\text{The extrapolation of the graph past 70 minutes produces}\)
\(\text{negative average minutes per day exercising (impossible).}\)
♦♦♦♦ Mean mark (e) 14%.
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. \(y=0.936 x-8.929\), where \(x\) is the temperature. --- 4 WORK AREA LINES (style=lined) --- --- 3 WORK AREA LINES (style=lined) ---
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.
- 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}
- 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)
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a.
b. \(\text{9 hours of sleep per day}\)
Show Worked Solution
a.
b. \(\text{Using the line of best fit added to the diagram in part (a):}\)
\(\text{2 hours of phone use per day}\ \ \Rightarrow\ \ \text{9 hours of sleep per day}\)
Statistics, STD1 S3 2022 HSC 16
A concert organiser is interested in the relationship between the distance from the stage, in metres, and the loudness of the sound measured in decibels.
The data the concert organiser collected is shown on the graph.
- Is the relationship between distance and loudness linear or non-linear? (1 mark)
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- Based on this dataset, at approximately what distance from the stage would the sound be at 90 decibels? (1 mark)
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Statistics, STD1 S3 2021 HSC 18
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} \textit{Number of people} \rule[-1ex]{0pt}{0pt} & 2 & 2 & 3 & 5 & 5 & 7 & 7 & 7 & 8 \\
\hline
\rule{0pt}{2.5ex} \textit{Time taken (min)} \rule[-1ex]{0pt}{0pt} & 28 & 30 & 26 & 19 & 21 & 12 & 13 & 11 & 8 \\
\hline
\end{array}
- Complete the scatterplot by adding the last four points from the table. (2 marks)
- Add a line of best fit by eye to the graph in part (a). (1 mark)
- The graph in part (a) shows the association between the time to complete the puzzle and the number of people in the group.
- Identify the form (linear or non-linear), the direction and the strength of the association. (3 marks)
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- Calculate the mean of the time taken to complete the puzzle for the three groups of size 7 observed in the dataset. (1 mark)
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a.
b.
c. `text(The association is linear, negative and strong.)`
d. `12\ text(minutes)`
Show Worked Solution
a.
b.
c. `text(Form: linear)`
♦ Mean mark (c) 50%.
`text{Direction: negative}`
`text{Strength: strong}`
d. `text{Mean time (7 people)}= (12 + 13 + 11)/3= 12\ text(minutes)`
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.
- Describe the strength and direction of the linear association observed in this dataset. (2 marks)
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- Calculate the range of the test scores for the students who missed no lessons. (1 mark)
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- Draw a line of the best fit in the scatterplot above. (1 mark)
- 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)
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- 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)
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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%.
♦♦ 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.
- On the graph, draw a line of best fit by eye. (1 mark)
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- 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)
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a.
b. `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?
- 68 years
- 69 years
- 86 years
- 88 years
Show Worked Solution
\(\text{When infant mortality rate is 60, life expectancy}\)
\(\text{at birth is 68 years (see below).}\)
\(\Rightarrow A\)