SL 4.4—Pearsons, scatter diagrams, eqn of y on x Questionbank

Practice SL 4.4—Pearsons, scatter diagrams, eqn of y on x with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SL & HLPaper 1

A librarian records the number of books borrowed, $B$ , and the number of library visits, $V$ , by eight members over a month. The data are shown below.

Books borrowed $(B)$	Library visits $(V)$
2	1
4	2
6	3
8	4
10	5
12	6
14	7
16	8

Find Pearson's product-moment correlation coefficient, $r$ , and interpret its value in context.

[3]

Find the equation of the regression line $V$ on $B$ .

[2]

Estimate the number of library visits for a member who borrows 9 books.

[2]

Draw a scatter diagram of the data with the regression line.

[3]

Question 2

SL & HLPaper 1

A researcher studies the relationship between the number of hours, $H$ , spent studying per week and the average test score, $S$ , out of 100, for eight randomly selected students. The data are shown in the following table.

Hours studying $(H)$	Test score $(S)$
2	55
4	60
6	65
8	70
10	75
12	80
14	85
16	90

The relationship is modeled by the regression equation $S = aH + b$ .

Write down the value of $a$ and of $b$ .

[3]

Use the regression equation to estimate the test score for a student who studies for 9 hours per week.

[2]

Draw a scatter diagram of the data, including the regression line.

[3]

Question 3

SLPaper 2

A dataset records study-hours $x$ and test scores $y$ for eight students: $\begin{array}{c|cccccccc} x & 2&4&5&7&9&12&15&18\\ \hline y & 30&28&25&21&20&17&14&12 \end{array}$

(a) Using technology, find (i) $r$ ; (ii) the regression line $y=ax+b$ ; (iii) the regression line $x=cy+d$ . [5]

Using technology, find (i) $r$ ; (ii) the regression line $y=ax+b$ ; (iii) the regression line $x=cy+d$ .

[5]

Predict $y$ at $x=10$ and find the residual for $(7,21)$ .

[3]

Using $x$ on $y$ , estimate $x$ for $y=16$ ; then invert $y=ax+b$ to estimate $x$ when $y=16$ . Explain the difference.

[4]

Compute $r^2$ and interpret; comment on extrapolating to $x=25$ .

[3]

Question 4

SL & HLPaper 2

A biologist studies the relationship between the length of a fish, $L$ (in cm), and its weight, $W$ (in grams), for a sample of 10 fish. The data are shown below.

Length $(L)$	Weight $(W)$
20	150
25	200
30	270
35	350
40	450
45	570
50	700
55	850
60	1000
65	1200

It is assumed that $(L, W)$ follow a bivariate normal distribution with product moment correlation coefficient $\rho$ . The relationship can be modeled by the regression line $W=$ $a L+b$ .

(i) State suitable hypotheses $H_{0}$ and $H_{1}$ to test for a positive linear association between length and weight, using a one-tailed test.

[1]

(ii) Calculate Pearson's product-moment correlation coefficient, $r$ , and the corresponding $p$ -value at the $5 \%$ significance level. Interpret the result in context.

[4]

Find the equation of the regression line $W$ on $L$ .

[3]

The regression line of $L$ on $W$ is given by $L=0.045 W+14.5$ . Find the coordinates of the point where the two regression lines intersect, and interpret this point in context.

[3]

Estimate the weight of a fish with a length of 70 cm , and explain why this estimate may be unreliable.

[3]

Draw a scatter diagram of the data with both regression lines and their intersection point.

[4]

Question 5

SL & HLPaper 2

A gardener records the amount of water, $W$ , in liters, given to a plant each week and the height of the plant, $P$ , in centimeters, after 10 weeks. The data for seven plants are shown below.

Water $(W)$	Height $(P)$
1	20
2	25
3	28
4	30
5	35
6	38
7	40

It is assumed that ( $W, P$ ) follow a bivariate normal distribution with product moment correlation coefficient $\rho$ .

(i) State suitable hypotheses $H_{0}$ and $H_{1}$ to test whether there is a correlation between the amount of water and plant height, using a two-tailed test.

[1]

(ii) Calculate Pearson's product-moment correlation coefficient, $r$ , for these data.

[2]

(iii) Using a $5 \%$ significance level, state your conclusion in the context of the gardener's study.

[2]

(b) Comment on whether the regression line of $P$ on $W$ should be used to predict the height of a plant receiving 8 liters of water.

[2]

Question 6

SLPaper 2

A nutrition researcher investigates the relationship between the amount of protein (in grams) in a breakfast meal and the time (in minutes) after which a person begins to feel hungry again. The following data were obtained from eight participants.

Protein (g) $x$	10	14	18	21	25	28	32	36
Hunger time (min) $y$	60	75	82	90	110	115	124	130

Using technology, find: (i) the mean and standard deviation of $x$ and $y$ ; (ii) the value of the Pearson product–moment correlation coefficient $r$ .

[3]

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

[2]

A new breakfast bar contains 30 g of protein. Estimate, using your regression model, how long it will take for an average person to feel hungry again.

[1]

The researcher claims that hunger time increases by about 2.5 minutes for each additional gram of protein. Test this claim against your model and comment on whether it is supported.

[2]

Calculate the coefficient of determination, and interpret its meaning in context.

[2]

Question 7

SL & HLPaper 2

A teacher records the number of pages read, $P$ , and the time taken, $T$ , in minutes, for six students completing a reading task. The data are shown below.

Pages read $(P)$	Time taken $(T)$
10	15
15	22
20	28
25	34
30	40
35	45

Calculate Pearson's product-moment correlation coefficient, $r$ , and interpret its value in context.

[3]

Find the equation of the regression line $T$ on $P$ .

[2]

Estimate the time taken to read 18 pages.

[2]

Interpret the slope of the regression line in context.

[1]

Question 8

SL & HLPaper 2

A farmer records the number of seeds planted, $S$ (in thousands), and the crop yield, $Y$ (in kg), for 10 fields. The data for $\ln S$ and $\ln Y$ are shown below.

$\ln S$	$\ln Y$
2.30	4.61
2.71	5.01
3.00	5.30
3.22	5.52
3.40	5.70
3.50	5.80
3.69	5.99
3.91	6.21
4.09	6.39
4.20	6.50

The relationship between $\ln Y$ and $\ln S$ can be modeled by the regression equation $\ln Y = a \ln S + b$ . The relationship between $S$ and $Y$ can be modeled as $Y=k S^{n}$ .

Find the equation of the regression line $\ln Y$ on $\ln S$ .

[4]

Use the regression equation to estimate the crop yield when 15,000 seeds are planted.

[4]

Find the values of $k$ and $n$ in the model $Y=k S^{n}$ .

[4]

Calculate Pearson's product-moment correlation coefficient for $\ln S$ and $\ln Y$ , and interpret its value in context.

[2]

If the farmer increases the number of seeds by $10 \%$ in a field with 20,000 seeds, estimate the expected percentage increase in crop yield.

[3]

Question 9

SL & HLPaper 2

A store manager records the daily advertising budget, $A$ , in dollars, and the number of customers, $C$ , visiting the store over seven days. The data are shown below.

Advertising budget $(A)$	Customers $(C)$
50	20
100	25
150	30
200	35
250	40
300	45
350	50

Find the equation of the regression line $C$ on $A$ .

[2]

Write down the mean values $\bar{A}$ and $\bar{C}$ .

[1]

Draw a scatter diagram of the data, including the regression line and the point $(\bar{A}, \bar{C})$ .

[3]

Estimate the number of customers if the advertising budget is 400 dollars, and explain why this estimate may not be reliable.

[3]

Question 10

SL & HLPaper 2

A scientist studies the effect of temperature, $T$ , in degrees Celsius, on the reaction time, $R$ , in seconds, of a chemical process. The data for six experiments are shown below.

Temperature $(T)$	Reaction time $(R)$
10	8.0
15	7.5
20	6.8
25	6.2
30	5.5
35	5.0

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

[2]

Find the equation of the regression line $R$ on $T$ .

[2]

Estimate the reaction time at $22^{\circ} \mathrm{C}$ .

[2]

State one reason why the regression line may not be suitable for predicting the reaction time at $50^{\circ} \mathrm{C}$ .

[1]