IB Maths AI 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.

Paper

Paper 1 Paper 2

Difficulty

Easy Medium Hard

Question 1

Paper 1Easy6 marks

A transport operator records the journey times, in minutes, for 40 journeys on each of two bus services. The five-number summaries are shown below. No outliers were identified in either sample.

Service	Minimum	Lower quartile	Median	Upper quartile	Maximum
A	12	20	28	36	44
B	8	16	24	32	56

(a)

On a common horizontal scale from 0 to 60 minutes, construct box-and-whisker diagrams for the two services.

[5]

(b)

State which service’s journey-time distribution may be normally distributed, giving a reason based on the symmetry of its box and whiskers.

[1]

Question 2

Paper 1Easy6 marks

A student records the distance downstream from a spring and the number of aquatic invertebrate taxa found at each of six fieldwork sites. Figure 1 shows the results.

(a)

Using your GDC, find Spearman’s rank correlation coefficient, $r_s$ for these data. Give your answer to three significant figures.

[2]

(b)

Using your GDC, find Pearson’s product moment correlation coefficient, $r$ for these data. Give your answer to three significant figures.

[2]

(c)

The student wants to describe a monotonic association, and considers the point $(10,5)$ to be unusual compared with the other observations. State which coefficient is more appropriate for this purpose and give a reason.

[2]

Question 3

Paper 1Easy6 marks

A running coach records the weekly training time, $x$ hours, and 5 km race time, $y$ minutes, for six runners.

$x$	2	4	5	7	9	10
$y$	31.8	29.4	28.7	26.3	25.5	23.9

(a)

Use your graphic display calculator to find Pearson’s product-moment correlation coefficient, $r$ Give your answer to three significant figures.

[1]

(b)

Describe the direction and strength of the correlation.

[1]

(c)

Use your graphic display calculator to find the equation of the regression line of $y$ on $x$ in the form $y=ax+b$ Give the values of $a$ and $b$ to three significant figures.

[2]

(d)

Use the regression line to estimate the 5 km race time for a runner who trains for 6.5 hours per week. Give your answer to three significant figures.

[1]

(e)

Interpret the value of $a$ in this context.

[1]

Question 4

Paper 1Medium6 marks

A parcel-delivery company defines its daily forecast error as

\text{actual number of parcels}-\text{forecast number of parcels}.

A random sample of 16 independent days has a mean forecast error of 18.4 parcels and a sample standard deviation of 12.0 parcels. The daily forecast errors may be modelled by a normal population, whose standard deviation is unknown.

(a)

State which distribution should be used to construct a confidence interval for the population mean forecast error, $\mu$

[1]

(b)

Calculate a 95% confidence interval for $\mu$ Give both endpoints to one decimal place.

[3]

(c)

The planning director claims that the forecasts are unbiased, so that $\mu=0$ Use your interval to comment on this claim and state the direction of any bias indicated.

[2]

Question 5

Paper 1Easy6 marks

A campus club records which of two cafés its members visit each Friday. The transition diagram in Figure 1 gives the probabilities that a member visits each café on the next Friday, based on the café visited this Friday.

Let $s_n$ be the column state matrix after $n$ weekly transitions, using the state order North, South. On the Friday of the survey, the state matrix is

s_0=\begin{pmatrix}0.75\\0.25\end{pmatrix}.

(a)

Write down the transition matrix $T$ where $T_{ij}$ is the probability of moving from state $j$ to state $i$

[2]

(b)

Calculate $s_1=Ts_0$

[2]

(c)

Hence, using your result from part (b), calculate $s_2$

[2]

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