IB Maths AI SL 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.

Paper

Paper 1 Paper 2

Difficulty

Easy Medium Hard

Question 1

Paper 2Easy6 marks

A city council wants to describe the repair times for 240 potholes repaired during April. It selects 12 repair records at random. For these 12 records, the lower quartile is 4 hours and the upper quartile is 8 hours. One recorded repair time is 18 hours.

(a)

State the population and the sample.

[2]

(b)

Calculate the interquartile range and determine whether the 18-hour repair time is an outlier.

[2]

(c)

The maintenance log confirms that this repair took 18 hours because a lane closure delayed the crew. State whether the value should be removed before describing the sample, giving a reason.

[2]

Question 2

Paper 2Easy5 marks

A school records the commute times of students travelling by two bus routes. Figure 1 shows box-and-whisker diagrams of the times, in minutes.

(a)

Write down the median commute time for Route A.

[1]

(b)

Calculate the interquartile range of the commute times for Route B. Give your answer in minutes.

[2]

(c)

State which route’s commute times may be normally distributed. Give one feature of its box-and-whisker diagram to support your answer.

[2]

Question 3

Paper 2Easy5 marks

A delivery service compares the delivery times for two routes. The values below summarize the box-and-whisker plots for each route. All values are in minutes.

Route	Lower whisker end	Lower quartile	Median	Upper quartile	Upper whisker end
A	12	18	24	30	36
B	10	15	20	28	40

(a)

Calculate the interquartile range for each route.

[2]

(b)

State which route has the higher median delivery time.

[1]

(c)

State which route’s delivery times may be normally distributed. Give a reason based on the symmetry of its box and whiskers.

[2]

Question 4

Paper 2Easy6 marks

A technician records the volume of water in a container as it is filled. Let $t$ be the time, in minutes, after filling begins, and let $V$ be the volume of water, in litres.

$t$	0	2	4	6	8	10	12	14
$V$	10	17	27	30	44	46	60	64

(a)

Use a GDC to calculate Pearson’s product-moment correlation coefficient, $r$ to three significant figures. Describe the direction and strength of the linear correlation.

[2]

(b)

Use a GDC to find the equation of the regression line of $V$ on $t$ giving its coefficients to four significant figures.

[1]

(c)

Use your regression line to estimate the volume of water after 9 minutes. Give your answer in litres to three significant figures.

[1]

(d)

Interpret the gradient and the vertical intercept of the regression line in this context.

[2]

Question 5

Paper 1Easy7 marks

A fieldworker records the distance of seven sample plots from a road and the number of native plant species found in each plot.

Distance from road (m)	5	10	20	35	50	75	100
Number of native plant species	2	6	8	11	12	14	13

(a)

Use a graphic display calculator to find Spearman’s rank correlation coefficient, $r_s$ for these data. Give your answer to three significant figures.

[2]

(b)

Interpret your value of $r_s$ in the context of the fieldwork.

[1]

(c)

The scatter plot suggests an overall increasing, curved relationship between distance from the road and the number of native plant species. State which coefficient, Pearson’s product moment correlation coefficient or Spearman’s rank correlation coefficient, is more appropriate for describing this relationship. Give a reason.

[2]

(d)

If one additional observation is an extreme value, explain how Pearson’s product moment correlation coefficient and Spearman’s rank correlation coefficient are likely to differ in their sensitivity to it.

[2]

Question 6

Paper 1Easy5 marks

A fieldwork study records the daily hours of sunlight and the height of a plant in six plots.

Plot	A	B	C	D	E	F
Daily sunlight (hours)	2	4	6	8	10	12
Plant height (cm)	4	9	13	18	22	7

(a)

Use a graphic display calculator to find Spearman’s rank correlation coefficient, $r_s$ for these data. Give your answer to three significant figures.

[2]

(b)

State the direction of the association indicated by $r_s$

[1]

(c)

If the researchers want to assess a linear relationship, rather than a monotonic relationship in general, name the more appropriate coefficient: Pearson’s product-moment correlation coefficient or Spearman’s rank correlation coefficient.

[1]

(d)

Plot F appears to be an outlier. State which of these two coefficients is generally less sensitive to an outlier.

[1]

Question 7

Paper 2Easy6 marks

A filling machine produces packets of dried fruit. The mass $X$ in grams, of a randomly selected packet is modelled by a normal distribution with mean $500$ g and standard deviation $4$ g.

(a)

Sketch the normal curve for $X$ Label the horizontal axis with $\mu-\sigma$ $\mu$ $\mu+\sigma$ and their corresponding masses.

[2]

(b)

Use your GDC to calculate the probability that a packet has a mass between $496$ g and $504$ g. Give your answer to four significant figures.

[2]

(c)

Use your GDC to find the mass $x$ such that $90\%$ of packets have a mass less than $x$ Give your answer to the nearest $0.1$ g.

[2]

Question 8

Paper 2Easy5 marks

A game pays the player different amounts on each play. Let $X$ be the player’s net gain, in USD, on one play. The probability distribution of $X$ is shown in Figure 1.

(a)

Write down $P(X=1)$

[1]

(b)

Show that $E(X)=0$

[3]

(c)

Hence state whether the game is fair.

[1]

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