Bivariate Statistics - IB Math AI Question Bank

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Bivariate Statistics

Practice exam-style IB Math AI questions for Bivariate Statistics, aligned with the syllabus and grouped by topic.

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Question 1

SL • Paper 1

Easy

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SL • Paper 1

Easy

Calculator Permitted

A scatter diagram shows the relationship between the age, $x$ months, and height, $y$ cm, of a sample of young plants. The mean point is $(12,18)$ . A line of best fit through the mean point has equation $y=0.75x+c$ .

Describe the correlation shown by the scatter diagram.

[1]

Find the value of $c$ .

[2]

Use the line of best fit to estimate the height of a plant aged $15$ months.

[2]

State why the correlation does not, by itself, prove that increasing the age of a plant causes its height to increase.

[1]

Question 2

SL • Paper 1

Easy

Calculator Permitted

SL • Paper 1

Easy

Calculator Permitted

A school uses the regression equation

y=-2.35x+84.6

to model a student's predicted test score, $y$ , from the number of lessons missed, $x$ . The observed values of $x$ range from $2$ to $15$ .

Interpret the slope and the vertical intercept in context.

[2]

Calculate the predicted score for a student who misses $12$ lessons.

[2]

Explain why the interpretation of the vertical intercept should be treated with caution.

[1]

Question 3

SL • Paper 1

Medium

Calculator Permitted

SL • Paper 1

Medium

Calculator Permitted

A cycling coach records the weekly training time, $x$ hours, and the increase in endurance score, $y$ , for six cyclists. The results are shown in the table.

Variable	1	2	3	4	5	6
x [h]	1	2	3	4	5	6
y	3	5	5	7	8	12

Find Pearson's product-moment correlation coefficient, $r$ .

[2]

Find the equation of the regression line of $y$ on $x$ .

[2]

Use the regression line to estimate the increase in endurance score for a cyclist who trains for $7$ hours per week.

[2]

State whether the estimate in part (c) is an interpolation or an extrapolation.

[1]

Question 4

SL • Paper 1

Medium

Calculator Permitted

SL • Paper 1

Medium

Calculator Permitted

For a sample of $12$ coastal locations, Pearson's product-moment correlation coefficient between distance from the sea and average winter temperature is $r=-0.684$ . For this sample size, evidence of linear correlation is accepted when $|r|>0.632$ .

Determine whether the data meet the stated criterion for linear correlation. Give a reason for your answer.

[2]

Explain why this result is not sufficient to conclude that greater distance from the sea causes lower winter temperatures.

[2]

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Question 5

SL • Paper 1

Medium

Calculator Permitted

SL • Paper 1

Medium

Calculator Permitted

A scatter diagram shows the relationship between the concentration of a chemical, $x$ , and the growth rate of a microorganism, $y$ . The points follow an increasing curved pattern, apart from one extreme observation.

State whether Pearson's coefficient or Spearman's coefficient is more appropriate for describing the association. Justify your answer.

[2]

Using the displayed data, technology gives $r\approx0.963$ and $r_s=1.000$ . Interpret the difference between these two values.

[2]

Explain why a value of $r$ close to zero would not necessarily imply that the variables are unrelated.

[1]

Question 6

SL • Paper 1

Medium

Calculator Permitted

SL • Paper 1

Medium

Calculator Permitted

A technician records the age, $x$ years, and battery capacity, $y$ ampere-hours, for five rechargeable batteries. The data are shown in the table.

Measurement	Battery 1	Battery 2	Battery 3	Battery 4	Battery 5
Age $x$ [years]	2	4	6	8	10
Capacity $y$ [Ah]	18.0	15.2	12.4	9.6	6.8

Find the equation of the regression line of $y$ on $x$ .

[2]

Estimate the capacity of a battery that is $7.5$ years old.

[2]

State whether the estimate in part (b) is an interpolation or an extrapolation.

[1]

In general, explain why rearranging a regression equation of $y$ on $x$ to predict battery age from a known capacity may be unreliable.

[1]

Question 7

SL • Paper 1

Medium

Calculator Permitted

SL • Paper 1

Medium

Calculator Permitted

For data from ten towns, the relationship between population density and public-transport use gives $r=0.962$ and $r_s=0.952$ . After one unusually large city is added, the coefficients become $r=0.418$ and $r_s=0.903$ .

Calculate the change in each correlation coefficient when the city is added.

[2]

Explain why the value of $r_s$ changes less than the value of $r$ .

[1]

State which coefficient gives stronger evidence of a positive monotonic relationship after the city is added.

[1]

Question 8

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

An engineer records the load, $x$ kilonewtons, and the resulting deflection, $y$ millimetres, of a flexible component. The data are shown in the table and follow a quadratic pattern.

Load x [kN]	Deflection y [mm]
0	4
1	7
2	14
3	25
4	40
5	59

Find a quadratic regression model for $y$ in terms of $x$ .

[2]

Use the model to predict the deflection when the load is $6$ kilonewtons.

[2]

Technology gives $R^2=1$ . Interpret this value in context.

[2]

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Question 9

HL • Paper 1

Medium

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HL • Paper 1

Medium

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Four observed response values are compared with predictions from a linear model and a quadratic model. The observed and predicted values are shown in the table.

x	Observed response	Linear model prediction	Quadratic model prediction
1	10	8	9.5
2	8	10	8.5
3	6	8	5.5
4	4	2	4.5

Calculate the residuals for the linear model and hence find its sum of square residuals.

[2]

Find the sum of square residuals for the quadratic model.

[1]

Based on these results, identify the model that fits the observations more closely and state one limitation of making a final model choice using only these sums.

[2]

Question 10

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

A non-linear regression model for the fuel consumption of delivery vehicles has coefficient of determination $R^2=0.873$ . Its residual plot shows that the spread of the residuals increases as the predicted fuel consumption increases.

Interpret the value $R^2=0.873$ in context.

[2]

Explain why the residual plot gives a reason to question the suitability of the model.

[2]

Question 11

SL • Paper 1

Medium

Calculator Permitted

SL • Paper 1

Medium

Calculator Permitted

Six designs are assessed by two judges. Judge A gives each design a numerical score, while Judge B ranks the designs from $1$ to $6$ . Two designs receive the same score from Judge A. The results are shown in the table.

Judge	Design 1	Design 2	Design 3	Design 4	Design 5	Design 6
Judge A score	7	5	9	5	2	8
Judge B rank	4	3	6	2	1	5

Write down the ranks assigned to the designs by Judge A, using rank $1$ for the lowest score.

[2]

Calculate Spearman's rank correlation coefficient, $r_s$ .

[2]

Interpret the value of $r_s$ in this context.

[1]

Question 12

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

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The mass, $M$ grams, of a chemical remaining after $t$ days is recorded for the first five days. The data follow an exponential model and are shown in the table.

t [days]	M [g]
0	120
1	96
2	76.8
3	61.44
4	49.152

Find an exponential regression model of the form $M=Hq^t$ .

[2]

Determine the time at which the model predicts that $60$ grams remain.

[2]

Use the model to estimate the mass remaining after $12$ days, and state one reason why this estimate may be unreliable.

[3]

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Question 13

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

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A power model $y=Jx^p$ is proposed for the relationship between the mass, $x$ , and energy consumption, $y$ , of a group of machines. Linear regression of the transformed data gives

\ln(y)=1.10+1.50\ln(x)

Determine the corresponding power model for $y$ in terms of $x$ .

[2]

Use the model to estimate $y$ when $x=12$ .

[2]

Determine the factor by which the model predicts $y$ will change when $x$ is doubled.

[2]

Question 14

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The average depth, $d$ metres, of water at a harbour is modelled by

d=3.20\sin(0.524t-1.05)+12.4

where $t$ is the number of hours after midnight and angles are measured in radians.

Determine the period of the model.

[2]

Find the maximum depth predicted by the model.

[1]

Determine the first time after midnight at which the maximum depth occurs.

[2]

Interpret the value $12.4$ in the model.

[1]

Question 15

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

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The number of bacteria, $B$ thousand, in a culture is modelled using an exponential regression. A linear regression of the transformed data gives

\ln(B)=0.182t+2.31

where $t$ is the time in hours.

Determine an exponential model of the form $B=Hq^t$ .

[2]

Use the displayed transformed model to estimate the number of bacteria after $8$ hours.

[2]

Interpret the value of $q$ in context.

[1]

Question 16

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

Linear, quadratic and cubic regressions are fitted to the same environmental dataset. Their coefficients of determination and residual plots are summarized below. The residual plots shown are schematic illustrations of the patterns and are not intended to represent the exact numerical residuals. The linear residuals show a curved pattern. The quadratic and cubic residuals are both randomly scattered about zero. The corresponding values of $R^2$ are $0.941$ , $0.989$ and $0.992$ , respectively.

Explain why the linear model should be rejected despite its relatively high value of $R^2$ .

[2]

Suggest why the quadratic model may be preferred to the cubic model.

[2]

State one additional contextual factor that should be considered before using the selected model for prediction.

[1]

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Question 17

SL • Paper 2

Medium

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SL • Paper 2

Medium

Calculator Permitted

A café investigates how the time, $x$ minutes, for which a drink is left to cool affects its temperature, $y$ in $^\circ\text{C}$ . Six observations are shown in the table.

Cooling time $x$ [min]	Drink temperature $y$ [ $^\circ\text{C}$ ]
2	78.98
4	69.26
6	60.54
8	56.82
10	48.10
12	38.38

I.

Calculate Pearson's product-moment correlation coefficient, $r$ .

[2]

II.

Describe the correlation in context.

[2]

Determine the equation of the regression line of $y$ on $x$ .

[2]

I.

Use the regression line to estimate the temperature after $7$ minutes.

[2]

II.

The actual temperature after $7$ minutes is $61.0^\circ\text{C}$ . Calculate the residual and interpret its sign.

[2]

Question 18

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A property analyst records the distance, $x$ kilometres, of eight apartments from a city centre and their monthly rent, $y$ euros.

Distance $x$ / km	Monthly rent $y$ / euros
1	1401.4
3	1364.2
5	1327.0
7	1289.8
9	1252.6
11	1215.4
13	1178.2
15	1141.0

I.

Find the regression line of $y$ on $x$ .

[2]

II.

Interpret the slope of the regression line.

[2]

Estimate the monthly rent of an apartment $9.5$ km from the city centre.

[2]

A penthouse is $24$ km from the city centre. An agent uses the regression line to predict its rent. Explain why this prediction may be unreliable.

[3]

Question 19

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A solar-energy company records daily cloud cover, $x$ %, and the electrical energy, $y$ kWh, generated by a test panel on nine days.

Cloud cover x [%]	Energy output y [kWh]
15	23.452
25	19.839
35	16.717
45	14.086
50	12.952
55	11.946
65	10.297
75	9.139
85	8.472

I.

Calculate Pearson's product-moment correlation coefficient.

[2]

II.

Find the regression line of $y$ on $x$ .

[2]

Use the regression line to estimate the energy generated on a day with $60\%$ cloud cover.

[2]

The company plans to use the model for a different panel installed in another country. State three factors that should be considered when evaluating the validity of this use.

[3]

Question 20

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A museum studies the relationship between weekly advertising expenditure, $x$ hundreds of dollars, and weekly attendance, $y$ hundreds of visitors. The mean point is $(6.4,18.7)$ . A line of best fit drawn by a manager has gradient $1.75$ .

I.

Find the equation of the manager's line of best fit, given that it passes through the mean point.

[3]

II.

State one graphical feature, other than passing through the mean point, that a reasonable line of best fit should have.

[1]

Use the line to estimate attendance when the museum spends $850 on advertising.

[2]

The museum director concludes that spending an additional $100$ on advertising will cause attendance to increase by exactly $175$ visitors. Give three reasons why this conclusion is not justified.

[3]

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Question 21

SL • Paper 2

Hard

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SL • Paper 2

Hard

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Eight hiking routes are assessed using a difficulty score, $D$ , and an enjoyment score, $E$ . Two routes have the same difficulty score. Rank $1$ is assigned to the lowest score in each variable.

Route	Difficulty score D	Enjoyment score E
1	12	44
2	15	40
3	15	46
4	18	48
5	21	54
6	23	52
7	26	60
8	28	58

I.

Write down the ranks for the two routes that have equal difficulty scores.

[2]

II.

Complete both rank columns and calculate Spearman's rank correlation coefficient, $r_s$ .

[3]

Interpret the value of $r_s$ in context.

[2]

A tourism company claims that making a route more difficult will cause hikers to enjoy it more. Evaluate this claim.

[3]

Question 22

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

A marine biologist records water depth, $x$ , and light intensity, $y$ , at several locations. The scatter diagram shows a decreasing curved pattern and one possible outlier. Technology gives the correlation coefficients shown below.

I.

For all observations, technology gives $r\approx -0.909$ and $r_s\approx -0.930$ . Explain the difference between these values.

[2]

II.

State which coefficient is more appropriate for these data and justify your answer.

[2]

After the possible outlier is removed, technology gives $r\approx -0.949$ and $r_s=-1.000$ . Calculate the change in the magnitude of each coefficient.

[2]

Discuss whether the biologist should remove the possible outlier before reporting a correlation coefficient.

[3]

Question 23

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

For ten libraries, a researcher records annual spending on children's events, $x$ thousand dollars, and the number of child visits, $y$ thousand. Technology gives the regression line of $y$ on $x$ as $y=2.84x+11.6$ and the regression line of $x$ on $y$ as $x=0.318y-2.71$ .

I.

Use the appropriate regression equation to estimate child visits when spending is $18$ thousand dollars.

[2]

II.

Use the appropriate regression equation to estimate the spending associated with $70$ thousand child visits.

[2]

A student rearranges $y=2.84x+11.6$ to predict spending when there are $70$ thousand visits. Find this estimate.

[2]

Explain why the estimates in parts (a)(ii) and (b) are different, and state which should be used to predict spending.

[4]

Question 24

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

A researcher compares average commuting time, $x$ minutes, and a self-reported stress score, $y$ , for twelve workplaces. One workplace has an unusually long commuting time.

I.

For the first eleven workplaces, $r=0.904$ and $r_s=0.908$ . Describe the association.

[2]

II.

After the unusual workplace is added, $r=0.250$ and $r_s=0.726$ . Calculate the change in each coefficient.

[2]

Explain why the unusual workplace has a greater effect on $r$ than on $r_s$ .

[3]

Suggest how the researcher should report the effect of the unusual workplace.

[3]

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Question 25

HL • Paper 2

Hard

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HL • Paper 2

Hard

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The height, $h$ metres, of a water jet is recorded at different horizontal distances, $x$ metres, from a fountain nozzle. The data follow a quadratic pattern.

x [m]	h [m]
0.0	2.0
1.0	9.5
2.0	14.0
3.0	15.5
4.0	14.0
5.0	9.5

I.

Use quadratic regression to find a model for $h$ in terms of $x$ .

[3]

II.

Write down the coefficient of determination and interpret it.

[2]

Determine the maximum height predicted by the model and the horizontal distance at which it occurs.

[3]

The model predicts that the jet reaches ground level when $h=0$ . Determine the positive value of $x$ for which this occurs, and comment on the reliability of this prediction.

[3]

Question 26

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

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The area, $A$ hectares, covered by an invasive aquatic plant is measured at the end of each year, where $t$ is the number of years after monitoring begins.

t [years]	A [ha]
0	12.0
1	16.2
2	21.87
3	29.5245
4	39.858075
5	53.80840125
6	72.64134169
7	98.06581128
8	132.38884523

I.

Find an exponential regression model of the form $A=Hq^t$ .

[2]

II.

Interpret $H$ and $q$ in context.

[2]

Determine the first time at which the model predicts that the covered area exceeds $100$ hectares.

[3]

Determine the first end-of-year measurement at which the model predicts that the covered area exceeds $100$ hectares.

[3]

Question 27

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

An environmental scientist investigates the relationship between the percentage of paved land, $x$ (%), and the mean night-time temperature increase, $y$ degrees Celsius, for six districts. The paired data are shown in the table. The observed values of $x$ range from $1$ to $6$ .

Percentage of paved land, $x$ / %	Mean night-time temperature increase / $^\circ\text{C}$
1	3.814
2	5.729
3	6.743
4	10.257
5	11.271
6	13.186

I.

Determine Pearson's product-moment correlation coefficient, $r$ .

[2]

II.

Find the regression line of $y$ on $x$ in the form $y=ax+b$ .

[2]

Use the unrounded regression equation to estimate the temperature increase when $x=4.5$ , and classify this prediction.

[3]

For six observations, use the criterion $|r|>0.811$ to assess the significance of the correlation. Evaluate the scientist's claim that paving land causes higher night-time temperatures.

[3]

Question 28

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A marine biologist records nutrient concentration, $x$ , and algae coverage, $y$ , at ten coastal sites. The data show an increasing curved pattern. One site has an unusually high nutrient concentration.

Site	A	B	C	D	E	F	G	H	I	J
Nutrient concentration, $x$ / mg L $^{-1}$	1.0	1.4	1.9	2.3	2.8	3.4	3.9	4.5	5.2	12.5
Algae coverage, $y$ / %	6	9	13	13	18	24	31	40	52	54

I.

Determine Pearson's coefficient, $r$ , and Spearman's rank coefficient, $r_s$ , for all ten sites.

[2]

II.

Interpret the two coefficients in context.

[2]

After removing the unusual site, the coefficients are $r=0.977$ and $r_s=0.996$ . Compare the effect of removing the site on the two coefficients.

[3]

Determine which coefficient should be used to summarize the association, and evaluate whether the unusual site should automatically be removed.

[3]

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Question 29

HL • Paper 3

Hard

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HL • Paper 3

Hard

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The mass, $M$ grams, of a biodegradable material is measured at equal time intervals, $t$ weeks. An exponential model of the form $M=Hq^t$ is proposed.

I.

Use exponential regression to find a model for $M$ in terms of $t$ .

[2]

II.

Interpret the value $0.860$ in context.

[2]

Determine when the model predicts that the mass will first be $50$ grams.

[3]

At $t=4$ , the observed mass is $68.0$ grams. Calculate the residual and explain what its sign indicates.

[3]

Question 30

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The water depth, $d$ metres, in a lagoon is recorded every hour. Sine regression gives
$d=2.40\sin(0.785t-1.18)+5.60$
where $t$ is the number of hours after midnight.

I.

Determine the period of the model.

[2]

II.

State the maximum and minimum depths predicted by the model.

[2]

Determine the first time after midnight at which the model predicts a maximum depth.

[3]

The coefficient of determination is $R^2=0.998$ . Interpret this value and state one reason why it does not guarantee accurate depth predictions.

[3]

Question 31

HL • Paper 3

Hard

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HL • Paper 3

Hard

Calculator Permitted

A zoologist investigates how daily energy use, $E$ megajoules, scales with body mass, $m$ kilograms, for several mammal species. Regression of the transformed data gives
$\ln E=0.875+1.32\ln m$

I.

Determine the corresponding power model $E=Jm^p$ .

[2]

II.

Interpret the exponent $1.32$ in terms of proportional change.

[2]

Hence determine the factor by which predicted energy use changes when body mass is doubled.

[3]

A blue whale has a mass far greater than any mammal in the sample. Evaluate the use of the model to predict its energy use.

[3]

Question 32

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A manufacturer measures the maximum output, $P$ percent of its original value, of a solar module after $m$ months of accelerated ageing. Exponential regression gives
$P=98.5(0.972)^m$

I.

Interpret the values $98.5$ and $0.972$ in context.

[2]

II.

Determine the predicted output after $24$ months.

[2]

Determine at what whole number of months the model first predicts that the module's output will fall below $80\%$ .

[3]

The data were collected under accelerated laboratory conditions for only $12$ months. Evaluate the manufacturer's use of the model to predict outdoor performance after $10$ years.

[3]

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Question 33

HL • Paper 3

Hard

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HL • Paper 3

Hard

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The sound intensity $I$ , in watts per square metre, from a machine is measured at distance $d$ metres. A power regression gives
$I=82.0d^{-1.87}$

Symbol	Meaning	Unit
$I$	Sound intensity	[ $\mathrm{W\,m^{-2}}$ ]
$d$	Distance from machine	[m]

I.

Express the model as a linear relationship between $\ln I$ and $\ln d$ .

[2]

II.

Interpret the negative exponent in context.

[2]

Hence determine the percentage decrease in predicted intensity when the distance is doubled.

[3]

Theory for an unobstructed point source predicts an exponent of $-2$ . Evaluate whether the fitted exponent $-1.87$ alone disproves this theory.

[3]

Question 34

HL • Paper 3

Hard

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HL • Paper 3

Hard

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The number of visitors, $V$ thousands, arriving at an island each month is modelled by
$V=310\sin(0.524t-0.800)+620$
where $t$ is the number of months after the start of January.

I.

Determine the period and explain its meaning in context.

[2]

II.

State the maximum predicted number of visitors and interpret the vertical shift.

[2]

Determine the first value of $t>0$ for which the model predicts the maximum number of visitors.

[3]

A new travel regulation is introduced after the data-collection period. Evaluate the use of this model to forecast visitor numbers for the following five years.

[3]

Question 35

HL • Paper 3

Hard

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HL • Paper 3

Hard

Calculator Permitted

A university records an admissions score, $x$ , and first-year examination score, $y$ , for a sample of students. Technology gives the regression lines
$y=0.680x+21.0$
and
$x=1.15y-18.0$

I.

Use the appropriate regression line to predict the first-year score of a student whose admissions score is $70$ .

[2]

II.

Use the appropriate regression line to predict the admissions score of a student whose first-year score is $70$ .

[2]

A student rearranges $x=1.15y-18.0$ to predict $y$ when $x=70$ . Calculate this prediction and explain why the method is inappropriate.

[3]

Evaluate the claim that increasing a student's admissions score would cause their first-year score to increase.

[3]

Question 36

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A biologist models the relationship between body length, $x$ cm, and average daily food requirement, $F$ grams, for six species of lizard using a power function.

Body length / cm	Food requirement / g
8.987	4.159
12.131	5.641
16.375	9.891
24.428	19.688
32.974	31.530
44.511	65.155

I.

Use power regression to find a model of the form $F=Jx^p$ .

[3]

II.

Estimate the daily food requirement of a lizard with body length $35$ cm.

[2]

Determine the factor by which the model predicts $F$ changes when body length is multiplied by $1.5$ .

[3]

A linear model has $R^2=0.941$ , while the power model has $R^2=0.975$ ; both values are calculated on the original $F$ scale. State three further considerations, in addition to $R^2$ , that should be used when selecting a model.

[3]

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Question 37

HL • Paper 2

Hard

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HL • Paper 2

Hard

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During a controlled reaction, the concentration, $C$ millimoles per litre, of an intermediate chemical is measured at times $t$ minutes. A cubic model is proposed.

t [min]	C [mmol L^-1]
0	10.0
1	12.4
2	12.4
3	11.2
4	10.0
5	10.0
6	12.4

I.

Use cubic regression to find a model for $C$ in terms of $t$ .

[3]

II.

Use the model to estimate $C$ when $t=2.5$ .

[2]

Determine the times of the two stationary points of the model.

[4]

The reaction is observed only for $0\le t\le6$ . Evaluate the use of this cubic model to predict concentration at $t=20$ .

[3]

Question 38

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The monthly average electricity demand, $E$ gigawatt-hours, for a region is recorded over one year. The month number is $m$ , with January represented by $m=1$ . A sine regression model is appropriate.

I.

Use sine regression to find a model for $E$ in terms of $m$ .

[3]

II.

Interpret the amplitude and vertical shift.

[2]

Determine the period of the model and hence identify the month in which maximum demand is predicted.

[4]

The actual demand in June is $25.1$ gigawatt-hours. Find the residual and interpret it.

[2]

Question 39

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Linear, quadratic and cubic models are fitted to data relating the age, $x$ years, of a road surface to its maintenance cost, $y$ thousand dollars. Their coefficients of determination and residual plots are shown.

I.

The linear model has $R^2=0.924$ . Interpret this value.

[2]

II.

Explain why the linear model should not be selected despite its high value of $R^2$ .

[2]

The quadratic and cubic models have $R^2=0.983$ and $R^2=0.985$ , respectively. Suggest which model should be selected.

[3]

State three limitations of using the selected model to estimate maintenance costs for a newly developed type of road surface.

[3]

Question 40

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A manufacturer investigates the relationship between operating speed, $x$ revolutions per minute, and vibration level, $y$ millimetres per second, for a machine. A linear model and a quadratic model are fitted to the same observations.

x [rev/min]	Observed y [mm/s]	Linear model prediction [mm/s]	Quadratic model prediction [mm/s]
1200	6.60	7.20	6.32
1600	8.40	7.65	8.60
2000	9.00	8.10	8.84
2400	8.10	8.55	8.26
2800	8.40	9.00	8.28

I.

For one observation, the actual vibration level is $8.40$ and the linear-model prediction is $7.65$ . Calculate the residual and interpret its sign.

[2]

II.

The linear residuals are $-0.60,0.75,0.90,-0.45,-0.60$ . Calculate the sum of square residuals.

[3]

The quadratic model has $SS_{\mathrm{res}}=0.184$ . Compare the in-sample fit of the two models.

[3]

Evaluate whether the quadratic model should automatically be used to predict vibration at much higher operating speeds.

[4]

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Question 41

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

An engineer models the vertical displacement, $y$ millimetres, of a flexible walkway under a load $x$ kilonewtons. Linear, quadratic and cubic regressions are fitted to the observations.

I.

The linear model has $SS_{\mathrm{res}}=84.6$ and the quadratic model has $SS_{\mathrm{res}}=11.8$ . State which model gives the closer in-sample fit and justify your answer.

[2]

II.

The total sum of squares is $169.0$ . Calculate $R^2$ for the quadratic model.

[2]

The cubic model has $R^2=0.944$ , while the quadratic model has $R^2=0.930$ . The residual plots for both models show random scatter about zero. Explain why the quadratic model could still be preferred.

[3]

The largest observed load is $8$ kilonewtons. Evaluate the use of either model to predict displacement under a load of $15$ kilonewtons.

[3]

Question 42

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A transport analyst records the mean journey time, $T$ minutes, for different traffic flows, $x$ hundreds of vehicles per hour. A cubic regression gives
$T=0.084x^3-1.26x^2+6.80x+9.40$

I.

Calculate the predicted journey time when $x=8$ .

[2]

II.

The observed journey time at $x=8$ is $29.0$ minutes. Calculate the residual.

[2]

The residual plot has residuals close to zero for low traffic flows but increasingly large positive residuals for high traffic flows. Explain what this suggests about the model.

[3]

Although the model has $R^2=0.933$ , evaluate whether it should be used to plan emergency-vehicle journey times at high traffic flows.

[3]

Question 43

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

An agronomist models crop yield, $Y$ tonnes per hectare, against fertilizer application, $x$ tens of kilograms per hectare. Quadratic regression gives
$Y=-0.420x^2+5.10x+18.0$

I.

Determine the fertilizer application that maximizes the predicted yield.

[2]

II.

Find the maximum yield predicted by the model.

[2]

Determine the two fertilizer applications for which the model predicts a yield of $30$ tonnes per hectare.

[3]

The observed fertilizer applications lie between $20$ and $100$ kilograms per hectare. Explain why the model should not be used to conclude that sufficiently large applications will eventually produce negative crop yields.

[3]

Question 44

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A researcher models the concentration, $C$ milligrams per litre, of a medicine in blood at time $t$ hours after administration. Exponential and cubic models are fitted to the same observations.

I.

The exponential model has $SS_{\mathrm{res}}=18.4$ and $SS_{\mathrm{tot}}=460$ . Calculate its coefficient of determination.

[2]

II.

Interpret this coefficient of determination.

[2]

The cubic model has $R^2=0.982$ . Its residual plot shows a cluster of positive residuals at late times, while the exponential residuals are randomly scattered about zero. Compare the models.

[3]

The cubic model eventually predicts negative concentrations. Deduce which model is more defensible for predictions shortly beyond the observed interval.

[3]

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Question 45

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

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A geographer compares river discharge, $x$ , with sediment concentration, $y$ , at twelve monitoring stations. One station was measured during a rare flood.

I.

For all stations, technology gives $r=0.498$ and $r_s=0.921$ . Interpret the difference between these values.

[2]

II.

State which coefficient is more appropriate for describing the overall association and justify your answer.

[2]

When the flood observation is removed, $r$ becomes $0.953$ and $r_s$ becomes $0.993$ . Determine and compare the increases in the coefficients.

[3]

Evaluate whether the flood observation should be excluded from a model intended to predict sediment levels during future floods.

[3]

Question 46

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A digital platform records the number of active users, $U$ thousand, during the first eight weeks after launching a new service. Exponential and quadratic regression models are fitted to the data.

I.

An exponential regression on the displayed data gives the model $U=18.9256(1.32913)^t$ . Calculate the predicted number of active users at $t=10$ .

[2]

II.

The quadratic regression model is $U=1.71667t^2+3.94286t+18.6583$ . Calculate its prediction at $t=10$ and find the positive difference between the two predictions.

[3]

The exponential and quadratic regression models have $R^2\approx0.993$ and $R^2\approx0.9999$ , respectively. Explain why these values do not establish that the quadratic model will give the better week- $10$ prediction.

[3]

The platform has a maximum capacity of $500$ thousand active users. Evaluate both models for long-term forecasting.

[4]

Question 47

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A data analyst compares linear, quadratic and cubic models for predicting the daily electricity demand of a factory. Daily electricity demand is measured in $\text{kWh}$ . The models are fitted using a training dataset and assessed on the same holdout set of days. The values in the separate-data column are sums of squared prediction errors calculated on this holdout set; smaller values indicate better predictive performance.

Model	Training $SS_{\mathrm{tot}}$ / $(\text{kWh})^2$	Training $SS_{\mathrm{res}}$ / $(\text{kWh})^2$	Training $R^2$	Separate-data sum of squared errors / $(\text{kWh})^2$
Linear	2400	312	—	92
Quadratic	2400	120	—	58
Cubic	2400	—	0.980	170

I.

For the linear model, $SS_{\mathrm{tot}}=2400$ and $SS_{\mathrm{res}}=312$ . Calculate $R^2$ .

[2]

II.

The quadratic model has $SS_{\mathrm{res}}=120$ . Calculate its $R^2$ using the same value of $SS_{\mathrm{tot}}$ .

[2]

The sums of squared prediction errors on the same separate (holdout) days are $92$ for the linear model, $58$ for the quadratic model and $170$ for the cubic model. The cubic model has training $R^2=0.980$ . Evaluate these results.

[3]

Recommend a model for future prediction and explain why choosing only the model with the highest training $R^2$ is unreliable.

[3]

Question 48

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For paired quantitative variables $x$ and $y$ , the regression line of $y$ on $x$ is
$y=3.20x+7.00$
and Pearson's correlation coefficient is $r=0.840$ . New variables are defined by
$u=2x-5\quad\text{and}\quad v=10-4y$

Obs	x	y	u	v
1	-1.000	5.134	-7.000	-10.537
2	-0.500	5.400	-6.000	-11.600
3	0.000	4.332	-5.000	-7.326
4	0.500	8.600	-4.000	-24.400
5	1.000	11.534	-3.000	-36.137

I.

Express $x$ and $y$ in terms of $u$ and $v$ , respectively.

[2]

II.

Hence determine the regression line of $v$ on $u$ .

[2]

Determine Pearson's correlation coefficient between $u$ and $v$ , giving a reason for your answer.

[3]

Determine the coefficient of determination for the transformed linear model and explain why it is unchanged by the transformations.

[3]

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