IB Maths AA HL Topic 4: Statistics and probability questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

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Question 1

Paper 2, calculatorMedium6 marks

A transport planner records journey times, $t$ minutes, for shuttle route Cedar. The cumulative frequency is the number of journeys with a time less than the upper boundary of the class.

Class interval	Cumulative frequency at upper boundary
$0\leq t<5$	1
$5\leq t<10$	4
$10\leq t<15$	9
$15\leq t<20$	19
$20\leq t<25$	31
$25\leq t<30$	38
$30\leq t<35$	40

For estimates between class boundaries, assume the cumulative frequency graph consists of straight line segments and starts at $(0,0)$

(a)

State the total number of journeys recorded for route Cedar.

[1]

(b)

Using a cumulative frequency graph, estimate the median and interquartile range of the journey times for route Cedar. Give your answers to the nearest $0.1$ minute.

[3]

(c)

For route Birch, the lower quartile is $12$ minutes, the median is $18$ minutes and the upper quartile is $24$ minutes.

Hence compare the typical journey time and the spread of journey times for the two routes, using the medians and interquartile ranges.

[2]

A teacher records, for seven students, the number $x$ of optional online revision sets each student completed and the student’s examination mark $y$ out of 100. The students chose how many sets to complete.

$x$	2	3	4	5	6	7	8
$y$	52	56	59	65	66	72	74

(a)

Sketch a scatter diagram for the data.

[1]

(b)

Use your GDC to determine Pearson’s product-moment correlation coefficient $r$ and the equation of the regression line of $y$ on $x$ Give all values to three significant figures.

[4]

(c)

Use the regression line to estimate the examination mark for a student who completes 6 revision sets. Interpret the gradient of the regression line in this context.

[3]

(d)

Use the regression line to estimate the examination mark for a student who completes 12 revision sets. Comment on the reliability of this estimate and on the claim that completing more revision sets causes higher examination marks.

[4]

Question 3

Paper 2, calculatorHard6 marks

A running club records the weekly training time, $x$ hours, and 5 km race time, $y$ minutes, for eight recreational runners. The scatter diagram of the results is shown in Figure 1.

(a)

Use a GDC to determine the Pearson product-moment correlation coefficient, $r$ and the equation of the regression line of $y$ on $x$ Give $r$ and the coefficients of the regression equation to three significant figures.

[2]

(b)

Use your regression equation from part (a) to estimate the 5 km time for a runner who trains for 20 hours per week.

[1]

(c)

A coach proposes that the club should:

recommend 20 hours of training per week because the regression predicts a faster 5 km time, claiming that adding training will cause every runner’s time to improve;
estimate the weekly training time needed by a runner targeting a 5 km time of 27.0 minutes by rearranging the regression equation from part (a).

Comment on the statistical validity of these two proposals.

[3]

Question 4

Paper 2, calculatorEasy6 marks

The parking duration $X$ in hours, of a randomly selected vehicle at a car park is modelled by the probability density function

f(x)= \begin{cases} \frac14, & 0\leq x\leq 4,\\ 0, & \text{otherwise}. \end{cases}

(a)

Find the mean parking duration $E(X)$ giving your answer in hours.

[2]

(b)

Determine $\operatorname{Var}(X)$ giving your answer in hours squared.

[2]

(c)

The parking charge $C$ in dollars, is given by $C=2+0.5X$ Hence find $E(C)$ and $\operatorname{Var}(C)$ giving appropriate units.

[2]

Question 5

Paper 2, calculatorEasy4 marks

A psychologist models scores on a test as a normally distributed random variable $X$ measured in points. Figure 1 shows two cumulative probabilities for this distribution.

(a)

Using a GDC, determine the $z$ -value corresponding to $X=42$ Give your answer to four significant figures.

[1]

(b)

Using a GDC, determine the $z$ -value corresponding to $X=58$ Hence use both $z$ -values to determine the mean and standard deviation of $X$ giving each answer to three significant figures.

[3]

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