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Date	May 2022	Marks available	1	Reference code	22M.2.AHL.TZ2.4
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Find	Question number	4	Adapted from	N/A

Question

A student investigating the relationship between chemical reactions and temperature finds the Arrhenius equation on the internet.

$k = A e^{- \frac{c}{T}}$

This equation links a variable $k$ with the temperature $T$ , where $A$ and $c$ are positive constants and $T > 0$ .

The Arrhenius equation predicts that the graph of $\ln k$ against $\frac{1}{T}$ is a straight line.

Write down

The following data are found for a particular reaction, where $T$ is measured in Kelvin and $k$ is measured in ${cm}^{3} {mol}^{- 1} s^{- 1}$ :

Find an estimate of

Show that $\frac{d k}{d T}$ is always positive.

[3]

Given that $\lim_{T \to \infty} k = A$ and $\lim_{T \to 0} k = 0$ , sketch the graph of $k$ against $T$ .

[3]

(i) the gradient of this line in terms of $c$ ;

(ii) the $y$ -intercept of this line in terms of $A$ .

[4]

Find the equation of the regression line for $\ln k$ on $\frac{1}{T}$ .

[2]

$c$ .

It is not required to state units for this value.

[1]

e.i.

$A$ .

It is not required to state units for this value.

[2]

e.ii.

Markscheme

attempt to use chain rule, including the differentiation of $\frac{1}{T}$ (M1)

$\frac{d k}{d T} = A \times \frac{c}{T^{2}} \times e^{- \frac{c}{T}}$ A1

this is the product of positive quantities so must be positive R1

Note: The R1 may be awarded for correct argument from their derivative. R1 is not possible if their derivative is not always positive.

[3 marks]

A1A1A1

Note: Award A1 for an increasing graph, entirely in first quadrant, becoming concave down for larger values of $T$ , A1 for tending towards the origin and A1 for asymptote labelled at $k = A$ .

[3 marks]

taking $\ln$ of both sides OR substituting $y = \ln x$ and $x = \frac{1}{T}$ (M1)

$\ln k = \ln A - \frac{c}{T}$ OR $y = - c x + \ln A$ (A1)

(i) so gradient is $- c$ A1

(ii) $y$ -intercept is $\ln A$ A1

Note: The implied (M1) and (A1) can only be awarded if both correct answers are seen. Award zero if only one value is correct and no working is seen.

[4 marks]

an attempt to convert data to $\frac{1}{T}$ and $\ln k$ (M1)

e.g. at least one correct row in the following table

line is $\ln k = - 13400 \times \frac{1}{T} + 15.0 (= - 13383.1 \dots \times \frac{1}{T} + 15.0107 \dots)$ A1

[2 marks]

$c = 13400 (13383.1 \dots)$ A1

[1 mark]

e.i.

attempt to rearrange or solve graphically $\ln A = 15.0107 \dots$ (M1)

$A = 3 300 000 (3 304 258 \dots)$ A1

Note: Accept an $A$ value of $3269017$ … from use of $3 sf$ value.

[2 marks]

e.ii.

Examiners report

This question caused significant difficulties for many candidates and many did not even attempt the question. Very few candidates were able to differentiate the expression in part (a) resulting in difficulties for part (b). Responses to parts (c) to (e) illustrated a lack of understanding of linearizing a set of data. Those candidates that were able to do part (d) frequently lost a mark as their answer was given in x and y.

[N/A]

e.i.

[N/A]

e.ii.

Syllabus sections

Topic 4—Statistics and probability » SL 4.4—Pearsons, scatter diagrams, eqn of y on x

Show 146 related questions

18N.2.SL.TZ0.T_1a.iii:
Use your graphic display calculator to write down r , Pearson’s product–moment correlation coefficient.
18N.2.SL.TZ0.T_1c.i:
Use the regression line y on x to estimate Jerome’s examination score.
19M.2.SL.TZ1.T_1d.ii:
Hence describe the correlation between temperature difference from 37 °C and heartbeat.
22M.3.AHL.TZ1.1a.ii:
Write down the value of $r$ , Pearson’s product-moment correlation coefficient.
22M.2.SL.TZ1.3c:
State whether it would be appropriate for Chester to use the equation of a regression line for $y$ on $x$ to predict a country’s Eurovision score. Justify your answer.
17M.2.SL.TZ1.T_1b.ii:
For the data from these seven species describe the correlation between the average body weight and the average weight of the brain.
18M.1.SL.TZ2.T_1c:
Using your line of best fit, estimate the physics test score for a student with a score of 20 in their mathematics test.
19N.1.SL.TZ0.T_6c:
Explain why it is valid to use the regression equation to estimate the airfare between Hong Kong and Tokyo.
18N.2.SL.TZ0.T_1b.i:
Find the exact value of m and of c for these data.
19M.2.SL.TZ1.T_1d.i:
Use your graphic display calculator to find the Pearson’s product–moment correlation coefficient, $r$ .
22M.3.AHL.TZ1.1f.i:
Use linear regression to estimate the value of $k$ and of $L$ .
22M.2.AHL.TZ2.4e.ii:
$A$ .

It is not required to state units for this value.
18M.2.SL.TZ1.S_8a:
Find the value of a and of b.
21M.3.AHL.TZ2.1c.ii:
Determine the value of $r$ , Pearson’s product-moment correlation coefficient, for these remaining responses.
17M.2.SL.TZ1.T_1d:
Use your regression line to estimate the average weight of the brain of grey wolves.
21N.1.SL.TZ0.1a:
For this data, find the value of the Pearson’s product-moment correlation coefficient, $r$ .
22M.3.AHL.TZ1.1a.i:
Find the equation of the regression line of $Q (t)$ on $t$ .
18N.2.SL.TZ0.T_1a.ii:
Use your graphic display calculator to write down ${\bar y}$ , the mean examination score.
18N.3.AHL.TZ0.Hsp_3b.i:
State the distribution of your test statistic, including the parameter.
18M.2.SL.TZ2.S_1a.ii:
Write down the correlation coefficient.
17M.1.SL.TZ1.S_4a.i:
Write down the independent variable.
17M.1.SL.TZ1.S_4a.ii:
Write down the boiling temperature of the liquid.
16N.2.SL.TZ0.T_1h:
Write down a reason why this estimate is not reliable.
18M.1.SL.TZ1.T_4b:
Using the equation of the regression line, estimate the concentration of dissolved oxygen in the river when the temperature is 18 °C.
17N.1.SL.TZ0.T_5a:
Find the mean number of hours that the people surveyed watch television per week.
17N.1.SL.TZ0.T_5c:
By placing a tick (✔) in the correct box, determine which of the following statements is true:
18M.1.SL.TZ2.T_1a:
Plot and label the point M $\left( {\bar m,\,\,\bar p} \right)$ on the scatter diagram.
19M.2.SL.TZ2.S_1a.ii:
Find the correlation coefficient.
18N.3.AHL.TZ0.Hsp_3c.iii:
State the p-value and interpret it in this context.
18M.1.SL.TZ1.T_4a.ii:
For these data, find the equation of the regression line y on x.
SPM.2.SL.TZ0.3f:
The Commissioner believes Minsun’s score for competitor G is too high and so decreases the score from 9.5 to 9.1.

Explain why the value of the Spearman’s rank correlation coefficient ${r_s}$ does not change.
18M.2.SL.TZ2.S_1b:
Use your equation to estimate the mean weight of a child that is 1.95 years old.
17M.2.SL.TZ2.S_2b:
Use the regression equation to estimate the number of visitors on a day when the maximum temperature is 15 °C.
17N.2.SL.TZ0.S_8c:
Write down the number of low production hives.
17M.2.SL.TZ2.T_3b.i:
Write down, for this set of data the mean number of folders produced, $\bar x$ ;
19M.2.SL.TZ1.T_1b.ii:
Write down, for this set of data the mean number of heartbeats per minute, $\bar y$ .
19M.2.SL.TZ1.T_1f:
Draw the regression line $y$ on $x$ on the scatter diagram.
17M.2.SL.TZ1.T_1a:
Find the range of the average body weights for these seven species of mammal.
EXM.1.AHL.TZ0.15a:
Show that ${\text{ln}}\left( {T - 25} \right) = bt + {\text{ln}}\,a$ .
17N.1.SL.TZ0.T_5d:
Diogo is 18 years old. Give a reason why the regression line should not be used to estimate the number of hours Diogo watches television per week.
18M.1.SL.TZ1.T_4a.i:
For these data, find Pearson’s product-moment correlation coefficient, r.
18M.2.SL.TZ2.S_1a.i:
Find the value of a and of b.
16N.2.SL.TZ0.T_1f:
Draw the regression line, from part (e), on your scatter diagram.
SPM.2.SL.TZ0.3a.ii:
Using the value of $r$ , interpret the relationship between Stan’s score and Minsun’s score.
SPM.2.SL.TZ0.3d:
Copy and complete the information in the following table.
SPM.2.SL.TZ0.3c.ii:
State whether this estimate is reliable. Justify your answer.
17M.2.SL.TZ1.T_1b.i:
For the data from these seven species calculate $r$ , the Pearson’s product–moment correlation coefficient;
SPM.2.SL.TZ0.3e.ii:
Comment on the result obtained for ${r_s}$ .
18M.2.SL.TZ1.S_8b:
Use the regression equation to estimate the value of y when x = 3.57.
SPM.2.SL.TZ0.3c.i:
Use your regression equation from part (b) to estimate Minsun’s score when Stan awards a perfect 10.
SPM.2.SL.TZ0.3e.i:
Find the value of the Spearman’s rank correlation coefficient, ${r_s}$ .
17M.2.SL.TZ2.S_2a.ii:
Write down the value of $r$ .
17N.2.SL.TZ0.S_8e:
Adam decides to increase the number of bees in each low production hive. Research suggests that there is a probability of 0.75 that a low production hive becomes a regular production hive. Calculate the probability that 30 low production hives become regular production hives.
19M.2.SL.TZ1.T_1c:
Plot and label the point M( $\bar x$ , $\bar y$ ) on the scatter diagram.
18N.3.AHL.TZ0.Hsp_3c.i:
State suitable hypotheses for the test.
17M.2.SL.TZ2.S_2a.i:
Find the value of $a$ and of $b$ .
17M.1.SL.TZ1.S_4b:
Jim describes the correlation as very strong. Circle the value below which best represents the correlation coefficient.

$0.992\quad \quad \quad 0.251\quad \quad \quad 0\quad \quad \quad - 0.251\quad \quad \quad - 0.992$
17M.1.SL.TZ1.S_4c:
Jim’s model is $d = - 2.24t + 105$ , for $0 \leqslant t \leqslant 20$ . Use his model to predict the decrease in temperature for any 2 minute interval.
19M.2.SL.TZ2.S_1b:
Use the regression equation to estimate the BMI of an adult man whose waist size is 95 cm.
18N.2.SL.TZ0.S_2b:
Another athlete on this sports team has a hand length of 21.5 cm. Use the regression equation to estimate the height of this athlete.
17N.2.SL.TZ0.S_8a:
Write down the value of $a$ and of $b$ .
SPM.2.SL.TZ0.3b:
Write down the equation of the regression line $y$ on $x$ .
EXM.1.AHL.TZ0.15c.ii:
predict the temperature of the metal rod after 3 minutes.
18M.1.SL.TZ2.T_1b:
Draw the line of best fit, by eye, on the scatter diagram.
17N.2.SL.TZ0.S_8d.i:
Find the value of $k$ ;
17N.2.SL.TZ0.S_8b:
Use this regression line to estimate the monthly honey production from a hive that has 270 bees.
19N.3.AHL.TZ0.Hsp_1a.ii:
Carry out a suitable test at the 5 % significance level. With reference to the $p$ -value, state your conclusion in the context of Peter’s claim.
SPM.2.SL.TZ0.3a.i:
Write down the value of the Pearson’s product–moment correlation coefficient, $r$ .
18N.3.AHL.TZ0.Hsp_3d:
Use an appropriate regression line to estimate the weight of a fish with length 360 mm.
16N.2.SL.TZ0.T_1a:
On graph paper, draw a scatter diagram for these data. Use a scale of 2 cm to represent 5 hours on the $x$ -axis and 2 cm to represent 10 points on the $y$ -axis.
17M.2.SL.TZ2.T_3e:
State a reason why the regression line $C$ on $x$ is appropriate to model the relationship between these variables.
21M.3.AHL.TZ2.1c.i:
Calculate the mean annual income for these remaining responses.
17M.2.SL.TZ2.T_3g:
Draw the regression line $C$ on $x$ on the scatter diagram.
18N.3.AHL.TZ0.Hsp_3a.i:
Find the probability that a fish from this lake will have a weight of more than 560 grams.
17N.2.SL.TZ0.S_8d.ii:
Find the number of hives that have a high production.
19N.3.AHL.TZ0.Hsp_1a.i:
State suitable hypotheses ${H_0}$ and ${H_1}$ to test Peter’s claim, using a two-tailed test.
18N.3.AHL.TZ0.Hsp_3b.iii:
State the conclusion of the test, justifying your answer.
19M.2.SL.TZ1.T_1b.i:
Write down, for this set of data the mean temperature difference from 37 °C, $\bar x$ .
20N.2.SL.TZ0.S_2a.ii:
Write down the correlation coefficient.
20N.2.SL.TZ0.S_2b:
Using the regression equation, estimate the number of hot chocolates that Lucy will sell on a day when the maximum temperature is $12 ° C$ .
20N.2.SL.TZ0.T_1a:
On graph paper, draw a scatter diagram to show the results of Don’s investigation. Use a scale of $1 cm$ to represent $2$ units on the $x$ -axis, and $1 cm$ to represent $5$ units on the $y$ -axis.
20N.2.SL.TZ0.T_1f:
Don concluded from his investigation: “There is no causation between wind speed and the time to fully charge the robot”.

In the context of the question, briefly explain the meaning of “no causation”.
18N.2.SL.TZ0.T_1c.ii:
Justify whether it is valid to use the regression line y on x to estimate Jerome’s examination score.
17N.1.SL.TZ0.T_5b:
Draw the regression line on the scatter diagram.
20N.2.SL.TZ0.S_2a.i:
Find the value of $a$ and of $b$ .
20N.2.SL.TZ0.T_1b.ii:
Calculate $y$ , the mean time to fully charge the robot.
16N.2.SL.TZ0.T_1b:
(i) ${\bar x}$ , the mean number of hours spent on social media;

(ii) ${\bar y}$ , the mean number of IB Diploma points.
18N.3.AHL.TZ0.Hsp_3c.ii:
Find the product-moment correlation coefficient $r$ .
18M.2.SL.TZ1.S_8c:
The relationship between x and y can be modelled using the formula y = kxⁿ, where k ≠ 0 , n ≠ 0 , n ≠ 1.

By expressing ln y in terms of ln x, find the value of n and of k.
17M.2.SL.TZ2.T_3b.ii:
Write down, for this set of data the mean production cost, $\bar C$ .
EXN.1.SL.TZ0.8c.i:
Explain why your line should not be used to estimate the value of $m$ at which the temperature is $10.0$ $° C$ .
EXN.1.SL.TZ0.8d:
State a more appropriate model for the water temperature in the lake over an extended period of time. You are not expected to calculate any parameters.
EXN.1.SL.TZ0.8b:
Use your line to find an estimate for the water temperature on the first day of May.
EXN.1.SL.TZ0.8c.ii:
Explain in context why your line should not be used to predict the value for December (month $12$ ).
EXN.1.AHL.TZ0.12a:
Explain why this graph indicates that $P$ is inversely proportional to $d^{n}$ .
EXN.1.AHL.TZ0.12c.i:
Use your answer to part (b) to write down the value of $n$ to the nearest integer.
17M.2.SL.TZ2.T_3c:
Label the point ${\text{M}}(\bar x,{\text{ }}\bar C)$ on the scatter diagram.
EXN.2.SL.TZ0.1c:
Which of the correlation coefficients would you recommend is used to assess whether or not there is an association between total number of minutes late and distance from school? Fully justify your answer.
EXM.1.AHL.TZ0.15b:
Find the equation of the regression line of ${\text{ln}}\left( {T - 25} \right)$ on $t$ .
18N.3.AHL.TZ0.Hsp_3a.ii:
The maximum weight a hand net can hold is 6 kg. Find the probability that a catch of 11 fish can be carried in the hand net.
18N.3.AHL.TZ0.Hsp_3b.ii:
Find the p-value for the test.
16N.2.SL.TZ0.T_1g:
Use the given equation of the regression line to estimate the number of IB Diploma points that this girl obtained.
17M.2.SL.TZ2.T_3h:
Use the equation of the regression line to estimate the least number of folders that the factory needs to sell in a month to exceed its production cost for that month.
20N.2.SL.TZ0.T_1b.i:
Calculate $x$ , the mean wind speed.
20N.2.SL.TZ0.T_1d.i:
Calculate $r$ , Pearson’s product–moment correlation coefficient.
20N.2.SL.TZ0.T_1d.ii:
Describe the correlation between the wind speed and the time to fully charge the robot.
20N.2.SL.TZ0.T_1e.i:
Write down the equation of the regression line $y$ on $x$ , in the form $y = m x + c$ .
19M.2.SL.TZ2.S_1a.i:
Write down the value of $a$ and of $b$ .
16N.2.SL.TZ0.T_1c:
Plot the point $(\bar x,{\text{ }}\bar y)$ on your scatter diagram and label this point M.
20N.2.SL.TZ0.T_1e.ii:
Draw this regression line on your scatter diagram.
19N.3.AHL.TZ0.Hsp_1b:
Peter uses the regression line of $y$ on $x$ as $y = 0.248x + 83.0$ and calculates that a student with a Mathematics test score of 73 will have a running time of 101 seconds. Comment on the validity of his calculation.
17M.2.SL.TZ1.T_1e:
Find the percentage error in your estimate in part (d).
20N.2.SL.TZ0.T_1e.iii:
Hence or otherwise estimate the charging time when the wind speed is $27 {km h}^{- 1}$ .
17M.2.SL.TZ1.T_1c:
Write down the equation of the regression line $y$ on $x$ , in the form $y = mx + c$ .
17M.2.SL.TZ2.T_3a:
Draw a scatter diagram for this data. Use a scale of 2 cm for 5000 folders on the horizontal axis and 2 cm for 10 000 Euros on the vertical axis.
21M.3.AHL.TZ2.1e.i:
Use Juliet’s data to find the value of $a$ and of $b$ .
21M.2.SL.TZ1.1f:
Find the percentage score calculated by Jason.
21M.2.SL.TZ1.1g:
State whether it is valid to use the regression line $y$ on $x$ for Jason’s estimate. Give a reason for your answer.
21M.2.SL.TZ1.1e:
Describe the correlation.
19N.1.SL.TZ0.T_6a:
Use your graphic display calculator to find the equation of the regression line $y$ on $x$ .
16N.2.SL.TZ0.T_1e:
Write down the equation of the regression line $y$ on $x$ for these eight male students.
19N.1.SL.TZ0.T_6b:
Use your regression equation to estimate the cost of a flight from Hong Kong to Tokyo with Galois Airways.
21M.3.AHL.TZ2.1e.ii:
Interpret, referring to income and happiness, what the value of $a$ represents.
17M.2.SL.TZ2.T_3f:
Use your graphic display calculator to find the equation of the regression line $C$ on $x$ .
20N.2.SL.TZ0.T_1c:
Plot and label the point $M$ on your scatter diagram.
EXN.1.SL.TZ0.8a:
Assuming the data follows a linear model for this period, find the regression line of $T$ on $m$ for the remaining data.
EXM.1.AHL.TZ0.15c.i:
find the value of $a$ and of $b$ .
EXN.1.AHL.TZ0.12b:
Find the equation of the least squares regression line of $\log_{10} P$ against $\log_{10} d$ .
EXN.1.AHL.TZ0.12c.ii:
Find an expression for $P$ in terms of $d$ .
21N.1.SL.TZ0.1b:
Comment on your answer to part (a), using the information that Eduardo found.
21N.1.AHL.TZ0.12a:
Use the data in the second table to find the value of $m$ and the value of $b$ for the regression line, $\ln x = m (\ln d) + b$ .
21N.1.AHL.TZ0.12b:
Assuming that the model found in part (a) remains valid, estimate the percentage of trees in stock when $d = 25$ .
21N.1.SL.TZ0.1c:
Write down the equation of the regression line of $t$ on $x$ , in the form $t = a x + b$ .
21N.1.SL.TZ0.1d:
A 57-year-old male also ran in the $5000 m$ race.

Use the equation of the regression line to estimate the time he took to complete the $5000 m$ race.
21N.3.AHL.TZ0.2a.i:
Find the equation of the regression line of $h$ on $t$ .
21N.3.AHL.TZ0.2a.ii:
Interpret the meaning of parameter $a$ in the context of the model.
21N.3.AHL.TZ0.2a.iii:
Suggest why Eva’s use of the linear regression equation in this way could be unreliable.
21N.3.AHL.TZ0.2b.i:
Find the equation of the least squares quadratic regression curve.
21N.3.AHL.TZ0.2b.ii:
Find the value of $k$ .
21N.3.AHL.TZ0.2b.iii:
Hence, write down a suitable domain for Eva’s function $h (t) = p t^{2} + q t + r$ .
21N.3.AHL.TZ0.2c:
Show that $\frac{d h}{d t} = - R^{2} \sqrt{70 560 h}$ .
21N.3.AHL.TZ0.2d:
By solving the differential equation $\frac{d h}{d t} = - R^{2} \sqrt{70 560 h}$ , show that the general solution is given by $h = 17 640 {(c - R^{2} t)}^{2}$ , where $c \in ℝ$ .
21N.3.AHL.TZ0.2e:
Use the general solution from part (d) and the initial condition $h (0) = 3.2$ to predict the value of $T$ .
21N.3.AHL.TZ0.2f:
Find this new height.
21N.3.AHL.TZ0.2g.i:
Show that $\frac{d H}{d t} \approx 0.2514 - 0.009873 t - 0.1405 \sqrt{H}$ , where $0 \leq t \leq T$ .
21N.3.AHL.TZ0.2g.ii:
Use Euler’s method with a step length of $0.5$ minutes to estimate the maximum value of $H$ .

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Topic 4—Statistics and probability

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