Bivariate data analysis

Construct and interpret scatterplots; describe the direction, form and strength of associations; estimate and interpret Pearson's correlation coefficient r; fit a line of best fit and use it for prediction.

Statistical Analysis · MS-S4

Sub-topics

Scatterplots and correlation

Construct and interpret scatterplots; describe the direction, form and strength of a linear association; estimate and interpret Pearson's correlation coefficient r; understand that correlation does not imply causation.

Line of best fit and regression

Fit a line of best fit by eye and identify the least-squares regression line; interpret the gradient and y-intercept in context; use the regression equation for prediction; distinguish between interpolation and extrapolation and understand the risks of extrapolating beyond the data range.

Open Math

Bivariate data analysis

Statistical Analysis · MS-S4

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Date:

Scatterplots and correlation

Q1Straightforward

A scatterplot shows that as temperature increases, ice cream sales also increase. Which term best describes this association?

A.Negative association
B.Positive association
C.No association
D.Non-linear association

Q2Straightforward

Pearson's correlation coefficient is

r = 0

. What does this indicate?

A.A perfect positive linear relationship
B.A perfect negative linear relationship
C.No linear relationship
D.A strong non-linear relationship

Q3Straightforward

Four datasets have correlation coefficients

r_1 = 0.45

r_2 = -0.82

r_3 = 0.31

, and

r_4 = -0.60

. Which dataset has the **strongest** linear association?

A.Dataset 1 ( $r_1 = 0.45$ )
B.Dataset 2 ( $r_2 = -0.82$ )
C.Dataset 3 ( $r_3 = 0.31$ )
D.Dataset 4 ( $r_4 = -0.60$ )

Q4Straightforward

A scatterplot has all its points lying exactly on a straight line with a downward slope. What is the value of

r

A. $r = 1$
B. $r = -1$
C. $r = 0$
D. $r = -0.5$

Q5Moderate

A study finds a strong positive correlation (

r = 0.91

) between the number of fire trucks sent to a fire and the amount of property damage caused. A journalist concludes that fire trucks cause property damage. What is the most accurate response to this conclusion?

A.The value of $r$ is not high enough to draw any conclusion
B.Correlation does not imply causation — both variables increase with the size of the fire
C.The study should have used a larger sample before any conclusion is drawn
D.A negative correlation would be needed to conclude that fire trucks reduce damage

Q6Moderate

A dataset of 20 students shows a correlation of

r = -0.74

between hours of social media use per day (

x

) and test scores (

y

). Which statement is best supported by this result?

A.Social media use causes lower test scores
B.There is a moderate to strong negative linear association between social media use and test scores
C.There is no relationship between social media use and test scores
D.Students with high test scores spend more time on social media

Q7Moderate

A scatterplot shows points scattered widely around an upward-sloping trend line. Which description best fits this scatterplot?

A.Strong positive linear association
B.Weak positive linear association
C.Strong negative linear association
D.Perfect positive linear association

Q8Moderate

Two bivariate datasets are analysed. Dataset A has correlation coefficient

r_A = 0.68

, and Dataset B has

r_B = -0.85

. By how much does the absolute value of

r_B

exceed the absolute value of

r_A

? Give your answer to 2 decimal places.

Q9Moderate

Which of the following pairs of variables is most likely to show a **negative** linear association?

A.Height and weight of adults
B.Hours of study per day and exam score
C.Speed of a car and travel time for a fixed distance
D.Number of pages in a book and total reading time

Q10Challenging

A global study finds that countries with more televisions per capita have longer average life expectancies (

r = 0.88

). A politician uses this to argue that distributing televisions will increase life expectancy. What is the most likely explanation for the correlation?

A.Televisions directly increase life expectancy through health programming
B.Both television ownership and life expectancy are associated with national wealth (GDP per capita)
C.Longer-lived people have more time to acquire televisions
D.The study must be wrong because $r = 0.88$ is not meaningful

Q11Challenging

Three datasets have correlation coefficients

r_1 = 0.56

r_2 = -0.89

, and

r_3 = -0.42

. The dataset with the **strongest** linear association has

|r| =

___. Give your answer to 2 decimal places.

Q12Challenging

A scatterplot shows a clear U-shaped (curved) pattern between two variables. When Pearson's

r

is calculated, it is found to be close to 0. Which conclusion is correct?

A.There is no relationship between the two variables
B.There is a strong relationship, but it is non-linear, so $r \approx 0$ is misleading
C.The data must have been collected incorrectly
D.A value of $r \approx 0$ always means the variables are independent

Line of best fit and regression

Q13Straightforward

A regression line for the relationship between hours of study (

x

) and exam mark (

y

) is

y = 8x + 22

. Predict the exam mark for a student who studies for 6 hours.

Q14Straightforward

A regression line has equation

y = -2.5x + 85

, where

x

is the number of staff absent and

y

is daily production output (units). Predict the output when 8 staff are absent.

Q15Straightforward

A regression line for weekly advertising spend (

x

, in $) and weekly revenue (

y

, in $) is

\hat{y} = 12x + 800

. What does the gradient

12

represent?

A.The predicted revenue when advertising spend is zero
B.For every additional $1 spent on advertising, predicted revenue increases by $12
C.The maximum possible weekly revenue
D.The number of advertising campaigns per week

Q16Straightforward

In the regression equation

\hat{y} = 12x + 800

, where

x

is weekly advertising spend and

y

is weekly revenue (both in $), what does the

y

-intercept

800

represent?

A.The advertising spend that gives the maximum revenue
B.The predicted weekly revenue when there is no advertising spend ( $x = 0$ )
C.The minimum revenue the business can earn
D.The number of customers per week

Q17Moderate

A regression line is

y = 4.5x + 15

, where

x

is temperature (°C) and

y

is daily ice cream sales. A shop owner wants to sell exactly

78

ice creams. What temperature does the model predict this will occur at? Give your answer to the nearest whole number.

Q18Moderate

A regression line for arm span (

y

, cm) versus height (

x

, cm) is

\hat{y} = 0.98x + 4.2

. A student is

165

cm tall and has an arm span of

166.5

cm. What is the residual (actual minus predicted) for this student? Give your answer to 1 decimal place.

Q19Moderate

A regression line was fitted to data for students aged 12 to 17, relating weekly reading hours (

x

) to vocabulary score (

y

). Which prediction is more reliable, and why?

A.Predicting vocabulary for a student who reads 6 hours per week, because this is within the data range (interpolation)
B.Predicting vocabulary for an adult who reads 6 hours per week, because adults read more carefully
C.Predicting vocabulary for a student who reads 50 hours per week, because more data is always better
D.Both predictions are equally reliable because they use the same equation

Q20Moderate

Two points that lie on a regression line are

(2, 14)

and

(6, 26)

. Find the gradient of the regression line.

Q21Moderate

The least-squares regression line for a dataset is

\hat{y} = 5.4x + 32

, where

x

is hours studied per day and

y

is exam score (out of 100). Which statement correctly interprets the gradient

5.4

A.A student with no study time is predicted to score $5.4$
B.For each additional hour of study per day, the predicted exam score increases by $5.4$ marks
C.The maximum possible exam score is $5.4$ more than the minimum
D. $5.4\%$ of students studied for more than one hour per day

Q22Challenging

A regression line for years of experience (

x

) and annual salary in $1000s (

y

) is

\hat{y} = 3.2x + 28.4

. A worker with 12 years of experience earns $72\,000. Find the residual (actual minus predicted salary) in thousands of dollars, to 1 decimal place.

Q23Challenging

A regression model for house price (

y

, in $1000s) versus floor area (

x

, in m²) was built using data for houses between 80 m² and 250 m². Two predictions are made: Prediction A is for a 160 m² house; Prediction B is for a 400 m² house. Which statement is correct?

A.Prediction A is more reliable because 160 m² is within the data range (interpolation)
B.Prediction B is more reliable because the house is larger and worth more
C.Both predictions are equally reliable because they use the same regression equation
D.Prediction A is less reliable because it is closer to the middle of the data range

Q24Challenging

A regression line for vehicle mass (

x

, tonnes) and fuel consumption (

y

, L/100 km) is

\hat{y} = 6.8x + 2.1

. A car weighs 1.5 tonnes and actually uses 12.6 L/100 km. Find the residual to 1 decimal place.

Worked solutions and answers at openmath.au/year-12/standard-2/bivariate-data-analysis