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]

Question 2

Paper 1, no calculatorMedium6 marks

The histogram in Figure 1 shows the response times, $x$ seconds, for 32 service calls. The standard deviation of these response times is $6.4$ seconds.

(a)

State the modal class and estimate the mean response time.

[3]

(b)

For each response time $x$ a service index $y$ measured in index units, is defined by

y=2(x-3).

Hence, using your estimate from part (a), determine the estimated mean and the standard deviation of $y$

[3]

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 4

Paper 1, no calculatorMedium6 marks

A grower records the irrigation time, $y$ hours, and the harvested mass, $x$ kg, for five greenhouse beds.

$y$	1	2	3	4	5
$x$	2	3	3	5	6

(a)

Determine, giving your working, the equation of the regression line of $x$ on $y$ in the form $x=a+by$

[3]

(b)

Hence estimate the harvested mass when the irrigation time is 4 hours.

[1]

(c)

The grower wants to estimate the irrigation time from a harvested mass of $4.8$ kg. A trainee rearranges the equation from part (a) and obtains an estimate of 4 hours. Explain whether this is a justified use of a regression line.

[2]

Question 5

Paper 1, no calculatorMedium7 marks

A probability tree models the results of a medical study of a risk factor $R$ and a disease $D$ A person is selected at random from the population. The probabilities on the second set of branches are conditional on the outcome at the first set of branches.

(a)

Find the probability that a randomly selected person has both $R$ and $D$

[1]

(b)

Hence determine the probability that a randomly selected person has $D$

[2]

(c)

Hence determine the probability that a person has $R$ given that they have $D$ Give your answer exactly, and state whether more than half of the people with $D$ have $R$

[4]

Question 6

Paper 1, no calculatorMedium6 marks

The graph in Figure 1 shows the probability density function, $f$ of a continuous random variable $X$

(a)

Write down $f(x)$ as a piecewise function.

[2]

(b)

Determine $E(X)$

[2]

(c)

Hence determine $\operatorname{Var}(X)$

[2]

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