Part A, Multiple Choice (1 mark each, 25 marks total)
1A researcher records the height and arm span of 40 students to explore how the two measurements are related. What type of data is this?L4
A Univariate data
B Bivariate data
C Categorical data only
D Time-series data
B, Bivariate data. Bivariate data records two measurements (height and arm span) for each individual, so it is displayed on a scatterplot to explore the association.
2A study measures hours of sleep and next-day reaction time. Reaction time is expected to respond to how much sleep a person gets. On a scatterplot, which axis should reaction time go on?L4
A The x-axis, because it is the independent variable
B Either axis, because the choice does not matter
C The y-axis, because it is the dependent (response) variable
D The x-axis, because it is measured second
C, The y-axis. Reaction time is the response that may depend on sleep, so it is the dependent variable and goes on the y-axis. Hours of sleep (the predictor) goes on the x-axis.
3On a scatterplot of daily temperature (x, °C) against ice-cream sales (y, units), the point $(31, 470)$ appears. What does this point represent?L4
A On a day that reached $31$°C, $470$ ice-creams were sold
B On a day that reached $470$°C, $31$ ice-creams were sold
C The average temperature was $31$°C over $470$ days
D Ice-cream sales caused the temperature to reach $31$°C
A. Each point is one individual's pair $(x, y)$. Here $x = 31$ is the temperature and $y = 470$ is the ice-creams sold, so the point represents a single day that reached $31$°C with $470$ ice-creams sold.
4A scatterplot of number of absences (x) against final grade (y) shows points falling from the upper-left to the lower-right, tightly grouped around a line. How is this correlation best described?L5
A Weak positive linear correlation
B Strong positive linear correlation
C Weak negative linear correlation
D Strong negative linear correlation
D, Strong negative linear correlation. Points falling from upper-left to lower-right show a negative direction; tightly grouped around a line shows strong strength. A full description always states both.
5Which of the following is a complete and correct description of correlation for an HSC response?L5
A "There is a positive correlation."
B "The line goes up."
C "There is a moderate positive linear correlation between the two variables."
D "The correlation is strong."
C. A complete description states both strength (moderate) and direction (positive), plus "linear correlation." Options A and D each give only one component, and B is not a statistical description.
6Two variables show a clear upward trend on a scatterplot, but the points are widely scattered around the line. How should this be described?L5
A Strong positive linear correlation
B Weak positive linear correlation
C Strong negative linear correlation
D No correlation
B, Weak positive linear correlation. An upward trend is positive, and wide scatter around the line means the relationship is weak. A trend still exists, so it is not "no correlation."
7Pearson's correlation coefficient $r$ can take which range of values?L6
A $0 \le r \le 1$
B $0 \le r \le 100$
C Any real number
D $-1 \le r \le 1$
D, $-1 \le r \le 1$. Pearson's $r$ always lies between $-1$ (perfect negative linear) and $+1$ (perfect positive linear), with $0$ meaning no linear correlation.
8Which of the following $r$ values indicates the strongest linear relationship?L6
A $r = -0.93$
B $r = 0.78$
C $r = 0.55$
D $r = -0.20$
A, $r = -0.93$. Strength depends on magnitude $|r|$, not sign. $|-0.93| = 0.93$ is the closest to $1$, so it is the strongest. The negative sign only tells us the direction.
9For a dataset relating daily screen time (x) and hours of sleep (y), $r = -0.88$. Which interpretation is correct?L6
A As screen time increases, sleep hours tend to increase strongly
B Screen time has no effect on sleep
C There is a strong negative linear correlation: as screen time increases, sleep hours tend to decrease
D There is a weak positive linear correlation between the two variables
C. The sign is negative (variables move in opposite directions) and $|r| = 0.88 \ge 0.8$ is strong. So as screen time increases, sleep hours tend to decrease, a strong negative linear correlation.
10A researcher finds $r = 0.04$ between two variables and concludes "these variables have no relationship at all." Why is this conclusion flawed?L6
A $r = 0.04$ actually indicates a strong relationship
B $r$ near $0$ rules out only a linear relationship; a curved (non-linear) relationship could still exist
C $r$ can never equal a value as small as $0.04$
D $r = 0.04$ proves the variables are negatively correlated
B. $r$ only measures linear association. A value near $0$ means no linear relationship, but the variables could still have a strong curved (non-linear) relationship that $r$ cannot detect.
11Ice-cream sales and the number of drowning incidents both rise in summer, giving $r = 0.85$. What is the most likely explanation?L7
A Eating ice-cream directly causes people to drown
B Drowning incidents cause an increase in ice-cream sales
C The high $r$ value proves ice-cream sales cause drownings
D A confounding variable, hot weather, drives both ice-cream sales and swimming activity
D. Hot weather is a confounding (lurking) variable that independently increases both ice-cream sales and time spent swimming (hence drownings). The correlation is spurious; neither variable causes the other.
12A study reports $r = 0.91$ between the number of firefighters sent to a fire and the damage caused. Which statement is correct?L7
A A strong correlation does not prove causation; the size of the fire is a confounding variable driving both
B Sending more firefighters causes more damage
C The damage causes more firefighters to be present before the fire starts
D Because $r = 0.91$, causation is established
A. No matter how high $r$ is, correlation alone never proves causation. The size of the fire is a confounding variable: bigger fires need more firefighters and cause more damage.
13Which of the following is defined as a third variable that influences both x and y, creating an apparent but not causal correlation?L7
A A dependent variable
B An outlier
C A confounding (lurking) variable
D A residual
C, A confounding (lurking) variable. It is a hidden third factor that influences both variables being studied, producing a spurious correlation between them.
14Every line of best fit drawn by eye must pass through which point?L8
A The origin $(0, 0)$
B The mean point $(\bar{x}, \bar{y})$
C The first data point in the set
D The point with the largest y value
B, The mean point $(\bar{x}, \bar{y})$. The line of best fit passes through the mean of the x values and the mean of the y values, which balances the scatter above and below the line.
15A line of best fit crosses the y-axis at $12$ and also passes through the point $(8, 44)$. What is the equation of the line?L8
16A line of best fit for study hours (x) and score (y) is $y = 30 + 6x$. What score does the line predict for a student who studies $5$ hours?L8
A $60$
B $36$
C $30$
D $180$
A, $60$. Substitute $x = 5$: $y = 30 + 6(5) = 30 + 30 = 60$. The predicted score is $60$.
17The HSC writes the least-squares regression line as $y = a + bx$. What does $b$ represent?L9
A The correlation coefficient
B The predicted value of y when $x = 0$
C The gradient, the change in y for each one-unit increase in x
D The mean of the y values
C, The gradient. In $y = a + bx$, $b$ is the coefficient of $x$ (the gradient), giving the change in $y$ per one-unit increase in $x$. The constant $a$ is the y-intercept, the predicted $y$ when $x = 0$.
18The regression equation for advertising spend (x, in $\$1000$s) and monthly sales (y, in $\$1000$s) is $y = 15 + 4x$. What is the correct interpretation of $b = 4$?L9
A Monthly sales are $\$4000$ when nothing is spent on advertising
B The correlation between advertising and sales is $4$
C Total sales equal $4$ times the advertising spend
D For each additional $\$1000$ spent on advertising, predicted monthly sales increase by $\$4000$
D. The gradient $b = 4$ is the rate of change: for each $1$-unit ($\$1000$) increase in advertising spend, predicted sales rise by $4$ units ($\$4000$). Option A describes $a$, not $b$.
19The regression line for temperature (x, °C) and electricity demand (y, MWh) is $y = 420 - 8.5x$. What is the predicted demand at $20$°C?L9
20A regression equation is built from data where x ranges from $10$ to $50$. A student predicts y for $x = 32$. What kind of prediction is this?L10
A Interpolation, because $32$ is within the data range $10$ to $50$
B Extrapolation, because $32$ was not one of the original data values
C Extrapolation, because $32$ is near the middle of the range
D Neither, predictions are only possible for the original x values
A, Interpolation. Because $32$ lies between the smallest ($10$) and largest ($50$) observed x values, this is interpolation, which is generally reliable. The value does not need to be an original data point.
21A regression $y = 42 + 3.5x$ was built from data for $1$ to $8$ hours of study. Predicting for $x = 100$ hours gives $y = 392\%$. Why is this prediction unreliable?L10
A The arithmetic is incorrect
B $r$ must be negative for this equation
C Interpolation is always unreliable
D It is extrapolation far outside the data range, and the linear trend does not continue (a score cannot exceed $100\%$)
D. $x = 100$ is far outside the observed range of $1$ to $8$ hours, so this is extrapolation. The linear trend does not continue that far: a score of $392\%$ is impossible, so the prediction is unreliable.
22Which statement about interpolation and extrapolation is correct?L10
A Extrapolation is always more reliable than interpolation
B Interpolation predicts outside the data range
C Interpolation (within the data range) is generally more reliable than extrapolation (outside the data range)
D Both are equally reliable regardless of the data range
C. Interpolation predicts y for an x value within the observed range and is generally reliable because the line was fitted to that region. Extrapolation predicts outside the range and is less reliable.
23A regression line has equation $y = 30 - 2x$. What must be true about the sign of Pearson's $r$ for this dataset?L11
A $r$ must be positive, because the intercept is $30$
B $r$ must be negative, because the gradient is $-2$
C $r$ must equal exactly $-2$
D $r$ could be either positive or negative
B, $r$ must be negative. The gradient of the regression line and $r$ always have the same sign. A negative gradient ($-2$) means $r$ must also be negative. Note $r$ cannot equal $-2$ because $-1 \le r \le 1$.
24A 6-mark bivariate question asks students to "use the regression line to predict and comment on the result." Which element earns the final reliability mark?L11
A Stating the direction of the correlation
B Quoting the value of $r$
C Interpreting the y-intercept in context
D Stating whether the prediction is interpolation or extrapolation and commenting on its reliability
D. The final mark in a bivariate prediction question is for classifying the prediction as interpolation or extrapolation and commenting on whether it is reliable. Never skip the reliability comment.
25A student writes: "Since $r = 0.95$ between hours of study and exam marks, studying more directly causes higher marks." What is the error in this reasoning?L11
A Claiming causation from correlation; a strong $r$ shows association only, not cause and effect
B Using the wrong direction; $r = 0.95$ is negative
C Confusing the gradient with the y-intercept
D Extrapolating beyond the data range without comment
A. A high $r$ shows a strong association, but correlation never proves causation. Other factors (motivation, prior ability, teaching quality) could influence both variables. Causation needs a controlled study.
Part B, Short Answer (show all working)
1L4 & L5
A physiotherapist records the number of rehabilitation sessions attended (x) and a patient's pain score out of $10$ (y) for $8$ patients. The scatterplot shows points falling steadily from upper-left to lower-right, fairly tightly grouped around a line.
(a) Which variable belongs on the x-axis, and why?
(b) Describe what the point $(6, 2)$ represents in context.
(c) Write a full description of the correlation shown.
(a) Sessions attended goes on the x-axis. It is the independent (explanatory) variable, the number of sessions is the predictor, and the pain score is expected to respond to it.
(b) The point $(6, 2)$ represents a patient who attended $6$ rehabilitation sessions and reported a pain score of $2$ out of $10$.
(c) Points falling from upper-left to lower-right = negative direction; tightly grouped around a line = strong. So there is a strong negative linear correlation between sessions attended and pain score: as sessions increase, pain scores tend to decrease.
2L6
For a study of daily exercise (minutes, x) and resting heart rate (bpm, y), the correlation coefficient is $r = -0.82$.
(a) State the direction of the correlation.
(b) State and justify the strength of the correlation.
(c) A student claims $r = -0.82$ proves that exercise reduces resting heart rate. Explain why this claim is not justified by the correlation alone.
(a) The sign of $r$ is negative, so the direction is negative: as exercise increases, resting heart rate tends to decrease.
(b) $|r| = 0.82 \ge 0.8$, so the correlation is strong. Overall: a strong negative linear correlation between daily exercise and resting heart rate.
(c) Correlation is not causation. A strong $r$ shows only an association. A confounding variable, for example overall fitness or age, could influence both. Establishing causation would require a controlled experiment.
3L8
A line of best fit for advertising spend (x, in $\$1000$s) and sales (y, in $\$1000$s) passes through the points $(2, 35)$ and $(6, 55)$.
(a) Calculate the gradient of the line.
(b) Find the y-intercept and write the equation of the line.
(c) Use the equation to predict sales when $\$8000$ is spent on advertising.
(b) Using $(2, 35)$: $35 = 5(2) + b \Rightarrow b = 35 - 10 = 25$. Equation: $y = 5x + 25$.
(c) $\$8000$ means $x = 8$: $y = 5(8) + 25 = 40 + 25 = 65$, so predicted sales are $\$65\,000$.
4L9
The least-squares regression equation for years of experience (x) and annual salary in thousands of dollars (y) is $y = 38 + 2.8x$.
(a) Interpret $a = 38$ in context.
(b) Interpret $b = 2.8$ in context.
(c) Predict the salary of an employee with $10$ years of experience.
(a) $a = 38$ is the y-intercept: an employee with $0$ years of experience is predicted to earn $\$38\,000$ per year (a plausible starting salary, since $x = 0$ is realistic here).
(b) $b = 2.8$ is the gradient: for each additional year of experience, the predicted annual salary increases by $2.8$ thousand dollars, that is $\$2800$.
(c) $y = 38 + 2.8(10) = 38 + 28 = 66$, so the predicted salary is $\$66\,000$.
5L9 & L10
A regression study on patient age (x, years) and recovery time (y, days) uses data for patients aged $25$ to $65$. The equation is $y = 4 + 0.3x$.
(a) Predict the recovery time for a $40$-year-old. State whether this is interpolation or extrapolation and comment on reliability.
(b) Predict the recovery time for an $80$-year-old. State whether this is interpolation or extrapolation and comment on reliability.
(c) Explain, in general, why extrapolation is less reliable than interpolation.
(a) $y = 4 + 0.3(40) = 4 + 12 = 16$ days. Since $40$ is within the data range $25$ to $65$, this is interpolation, so the prediction is reliable.
(b) $y = 4 + 0.3(80) = 4 + 24 = 28$ days. Since $80 > 65$ is outside the data range, this is extrapolation, so the prediction is less reliable.
(c) The regression line is fitted only to data within the observed range. Outside that range the linear pattern may not continue, it could level off, reverse, or be affected by other factors, so extrapolated predictions are less trustworthy.
6L7
A newspaper reports: "Towns with more ice-cream vans have higher rates of sunburn ($r = 0.86$). Ice-cream vans are causing sunburn."
(a) Interpret the value $r = 0.86$.
(b) Explain why the newspaper's causal claim is not justified.
(c) Identify a plausible confounding variable and explain how it produces the correlation.
(a) $r = 0.86$ indicates a strong positive linear correlation: towns with more ice-cream vans tend to have higher sunburn rates.
(b) Correlation does not prove causation. A strong $r$ shows only an association. Ice-cream vans do not physically cause sunburn, so a third factor is more likely to explain the pattern.
(c) The confounding variable is hot, sunny weather. Sunny days bring out more ice-cream vans and lead to more time in the sun (hence more sunburn). Both variables are driven by weather, producing a spurious correlation.
7L8, L9 & L11
A dataset of maximum temperature (x, °C) and daily hot-chocolate sales (y) at a café gives the following five days:
(b) Gradient $b = -3.5$: for each $1$°C increase in maximum temperature, predicted hot-chocolate sales decrease by $3.5$ units. Because the gradient is negative, Pearson's $r$ must also be negative (gradient and $r$ always share the same sign).
(c) $y = 111 - 3.5(11) = 111 - 38.5 = 72.5$, so about $72$ or $73$ sales. Since $11$°C is within the data range $6$ to $18$°C, this is interpolation, so the prediction is reliable.
Bivariate Data Analysis Complete
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