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Date	May 2021	Marks available	2	Reference code	21M.3.AHL.TZ2.1
Level	Additional Higher Level	Paper	Paper 3	Time zone	Time zone 2
Command term	Calculate	Question number	1	Adapted from	N/A

Question

Juliet is a sociologist who wants to investigate if income affects happiness amongst doctors. This question asks you to review Juliet’s methods and conclusions.

Juliet obtained a list of email addresses of doctors who work in her city. She contacted them and asked them to fill in an anonymous questionnaire. Participants were asked to state their annual income and to respond to a set of questions. The responses were used to determine a happiness score out of $100$ . Of the $415$ doctors on the list, $11$ replied.

Juliet’s results are summarized in the following table.

For the remaining ten responses in the table, Juliet calculates the mean happiness score to be $52.5$ .

Juliet decides to carry out a hypothesis test on the correlation coefficient to investigate whether increased annual income is associated with greater happiness.

Juliet wants to create a model to predict how changing annual income might affect happiness scores. To do this, she assumes that annual income in dollars, $X$ , is the independent variable and the happiness score, $Y$ , is the dependent variable.

She first considers a linear model of the form

$Y = a X + b$ .

Juliet then considers a quadratic model of the form

$Y = c X^{2} + d X + e$ .

After presenting the results of her investigation, a colleague questions whether Juliet’s sample is representative of all doctors in the city.

A report states that the mean annual income of doctors in the city is $$ 80 000$ . Juliet decides to carry out a test to determine whether her sample could realistically be taken from a population with a mean of $$ 80 000$ .

Describe one way in which Juliet could improve the reliability of her investigation.

[1]

a.i.

Describe one criticism that can be made about the validity of Juliet’s investigation.

[1]

a.ii.

Juliet classifies response $K$ as an outlier and removes it from the data. Suggest one possible justification for her decision to remove it.

[1]

Calculate the mean annual income for these remaining responses.

[2]

c.i.

Determine the value of $r$ , Pearson’s product-moment correlation coefficient, for these remaining responses.

[2]

c.ii.

State why the hypothesis test should be one-tailed.

[1]

d.i.

State the null and alternative hypotheses for this test.

[2]

d.ii.

The critical value for this test, at the $5 %$ significance level, is $0.549$ . Juliet assumes that the population is bivariate normal.

Determine whether there is significant evidence of a positive correlation between annual income and happiness. Justify your answer.

[2]

d.iii.

Use Juliet’s data to find the value of $a$ and of $b$ .

[1]

e.i.

Interpret, referring to income and happiness, what the value of $a$ represents.

[1]

e.ii.

Find the value of $c$ , of $d$ and of $e$ .

[1]

e.iii.

Find the coefficient of determination for each of the two models she considers.

[2]

e.iv.

Hence compare the two models.

[1]

e.v.

Juliet decides to use the coefficient of determination to choose between these two models.

Comment on the validity of her decision.

[1]

e.vi.

State the name of the test which Juliet should use.

[1]

f.i.

State the null and alternative hypotheses for this test.

[1]

f.ii.

Perform the test, using a $5 %$ significance level, and state your conclusion in context.

[3]

f.iii.

Markscheme

Any one from: R1

increase sample size / increase response rate / repeat process
check whether sample is representative
test-retest participants or do a parallel test
use a stratified sample
use a random sample

Note: Do not condone:
Ask different types of doctor
Ask for proof of income
Ask for proof of being a doctor
Remove anonymity
Remove response $K$ .

[1 mark]

a.i.

Any one from: R1

non-random sampling means a subset of population might be responding
self-reported happiness is not the same as happiness
happiness is not a constant / cannot be quantified / is difficult to measure
income might include external sources
Juliet is only sampling doctors in her city
correlation does not imply causation
sample might be biased

Note: Do not condone the following common but vague responses unless they make a clear link to validity:
Sample size is too small
Result is not generalizable
There may be other variables Juliet is ignoring
Sample might not be representative

[1 mark]

a.ii.

because the income is very different / implausible / clearly contrived R1

Note: Answers must explicitly reference "income" to get credit.

[1 mark]

$($) 90 200$ (M1)A1

[2 marks]

c.i.

$r = 0.558 (0.557723 \dots)$ A2

[2 marks]

c.ii.

EITHER
only looking for change in one direction R1

OR
only looking for greater happiness with greater income R1

OR
only looking for evidence of positive correlation R1

[1 mark]

d.i.

$H_{0} : ρ = 0; H_{1} : ρ > 0$ A1A1

Note: Award A1 for $ρ$ seen (do not accept $r$ ), A1 for both correct hypotheses, using their $ρ$ or $r$ . Accept an equivalent statement in words, however reference to “correlation for the population” or “association for the population” must be explicit for the first A1 to be awarded.

Watch out for a null hypothesis in words similar to “Annual income is not associated with greater happiness”. This is effectively saying $ρ \leq 0$ and should not be condoned.

[2 marks]

d.ii.

METHOD 1 – using critical value of $r$

$0.558 > 0.549 (0.557723 \dots > 0.549)$ R1

(therefore significant evidence of) a positive correlation A1

Note: Do not award R0A1.

METHOD 2 – using $p$ -value

$0.0469 < 0.05 (0.0469463 \dots < 0.05)$ A1

Note: Follow through from their $r$ -value from part (c)(ii).

(therefore significant evidence of) a positive correlation A1

Note: Do not award A0A1.

[2 marks]

d.iii.

$a = 0.000126 (0.000125842 \dots), b = 41.1 (41.1490 \dots)$ A1

[1 mark]

e.i.

EITHER
the amount the happiness score increases for every $$ 1$ increase in (annual) income A1

OR
rate of change of happiness with respect to (annual) income A1

Note: Accept equivalent responses e.g. an increase of $1.26$ in happiness for every $$ 10000$ increase in salary.

[1 mark]

e.ii.

$c = - 2.06 \times 10^{- 9} (- 2.06191 \dots \times 10^{- 9})$ ,

$d = 7.05 \times 10^{- 4} (7.05272 \dots \times 10^{- 4})$ ,

$e = 12.6 (12.5878 \dots)$ A1

[1 mark]

e.iii.

for quadratic model: $R^{2} = 0.659 (0.659145 \dots)$ A1

for linear model: $R^{2} = 0.311 (0.311056 \dots)$ A1

Note: Follow through from their $r$ value from part (c)(ii).

[2 marks]

e.iv.

EITHER
quadratic model is a better fit to the data / more accurate A1

OR
quadratic model explains a higher proportion of the variance A1

[1 mark]

e.v.

EITHER
not valid, $R^{2}$ not a useful measure to compare models with different numbers of parameters A1

OR
not valid, quadratic model will always have a better fit than a linear model A1

Note: Accept any other sensible critique of the validity of the method. Do not accept any answers which focus on the conclusion rather than the method of model selection.

[1 mark]

e.vi.

(single sample) $t$ -test A1

[1 mark]

f.i.

EITHER

$H_{0} : μ = 80 000; H_{1} : μ \neq 80 000$ A1

$H_{0} :$ (sample is drawn from a population where) the population mean is $$ 80 000$
$H_{1} :$ the population mean is not $$ 80 000$ A1

Note: Do not allow FT from an incorrect test in part (f)(i) other than a $z$ -test.

[1 mark]

f.ii.

$p = 0.610 (0.610322 \dots)$ A1

Note: For a $z$ -test follow through from part (f)(i), either $0.578$ (from biased estimate of variance) or $0.598$ (from unbiased estimate of variance).

$0.610 > 0.05$ R1

EITHER

no (significant) evidence that mean differs from $$ 80 000$ A1

the sample could plausibly have been drawn from the quoted population A1

Note: Allow R1FTA1FT from an incorrect $p$ -value, but the final A1 must still be in the context of the original research question.

[3 marks]

f.iii.

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

c.i.

[N/A]

c.ii.

[N/A]

d.i.

[N/A]

d.ii.

[N/A]

d.iii.

[N/A]

e.i.

[N/A]

e.ii.

[N/A]

e.iii.

[N/A]

e.iv.

[N/A]

e.v.

[N/A]

e.vi.

[N/A]

f.i.

[N/A]

f.ii.

[N/A]

f.iii.

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.
22M.2.AHL.TZ2.4e.i:
$c$ .

It is not required to state units for this value.
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.
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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