IB Mathematics Analysis and Approaches Topic SL 4.4—pearsons, Scatter Diagrams, Eqn of y on x Questionbank (SL/HL)

IB Mathematics Analysis and Approaches Topic SL 4.4—pearsons, Scatter Diagrams, Eqn of y on x Questionbank (SL/HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic SL 4.4—pearsons, Scatter Diagrams, Eqn of y on x with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 4.4—pearsons, Scatter Diagrams, Eqn of y on x and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

SLPaper 2

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

Using technology, find $r$ , the regression line $y = a x + b$ , and 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 $y = a x + b$ to estimate $x$ when $y = 16$ . Explain the difference.

[4]

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

[3]

Question 2

SLPaper 2

The variables $x$ and $y$ are related by a power law of the form $y = p x^{q}$ , where $p, q \in R^{+}$ . The following table shows values of $ln x$ and $ln y$ .

$\ln x$	0	0.693	1.386	2.079	2.773
$\ln y$	0.693	2.773	4.852	6.931	9.011

The relationship between $ln x$ and $ln y$ can be modelled by $ln y = a ln x + b$ .

Find the value of $a$ and the value of $b$ .

[3]

Hence find the value of $p$ and the value of $q$ .

[3]

Question 3

SLPaper 2

A biologist studies the growth of a colony of bacteria. The table shows the time $t$ (in hours) and the number of bacteria $N$ in the colony at that time.

$t$ (hours)	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$N$	$120$	$180$	$290$	$440$	$680$	$1020$	$1560$

It is proposed that the relationship between $N$ and $t$ can be modelled by $N = a \cdot b^{t}$ , where $a$ and $b$ are positive constants.

Show that $ln N = ln a + t ln b$ .

[2]

Find the equation of the regression line of $ln N$ on $t$ in the form $ln N = m t + c$ .

[3]

Hence find the values of $a$ and $b$ .

[3]

Use the model to estimate the number of bacteria after $9$ hours.

[2]

Find the value of $t$ (to the nearest hour) when the colony first exceeds $10000$ bacteria.

[3]

Comment on the appropriateness of using the model to predict the number of bacteria after $50$ hours.

[2]

Question 4

SLPaper 2

A small coffee shop records the number of customers $x$ and the daily revenue $y$ (in dollars) over ten consecutive days.

$x$	$60$	$72$	$84$	$90$	$98$	$110$	$124$	$135$	$148$	$162$
$y$	$245$	$290$	$336$	$362$	$395$	$445$	$500$	$548$	$602$	$660$

The shop's daily cost $C$ (in dollars) is modelled by $C = 150 + 0.40 y$ , where $y$ is the daily revenue.

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

[2]

Comment on the result.

[1]

Write down the equation of the regression line of $y$ on $x$ in the form $y = a x + b$ .

[2]

Use the regression equation to estimate the revenue when $130$ customers are served on a particular day. Give your answer to the nearest dollar.

[2]

Find an expression for the daily profit in terms of $y$ .

[2]

Find the minimum number of customers required to make a profit on a particular day.

[3]

Comment on the appropriateness of using this regression line to estimate the revenue when the shop serves $400$ customers.

[2]

Question 5

SLPaper 1

A botanist records the number of leaves, $ℓ$ , and the height, $h$ , in centimetres, of ten young plants.

The regression line of $h$ on $ℓ$ is $h = 2.5 ℓ + 30$ . The regression line of $ℓ$ on $h$ is $ℓ = 0.2 h + 4$ .

Estimate the height of a plant with 26 leaves.

[2]

Find the mean number of leaves, $\overset{ˉ}{ℓ}$ .

[3]

IB Mathematics Analysis and Approaches SL 4.4—pearsons, Scatter Diagrams, Eqn of y on x Questionbank

IB Mathematics Analysis and Approaches SL 4.4—pearsons, Scatter Diagrams, Eqn of y on x Questionbank

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IB Mathematics Analysis and Approaches SL 4.4—pearsons, Scatter Diagrams, Eqn of y on x Questionbank

IB Mathematics Analysis and Approaches SL 4.4—pearsons, Scatter Diagrams, Eqn of y on x Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 4.4—pearsons, Scatter Diagrams, Eqn of y on x Questionbank

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