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.

Paper

Paper 1, no calculator Paper 2, calculator

Difficulty

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Question 1

Paper 1, no calculatorEasy6 marks

The table shows the five-number summaries for daily journey times at two depots.

Depot	Minimum (min)	Lower quartile (min)	Median (min)	Upper quartile (min)	Maximum (min)
A	12	18	24	30	36
B	10	16	22	29	40

(a)

Write down the median journey time for depot A.

[1]

(b)

Calculate the interquartile range for each depot.

[2]

(c)

State which depot has less variation among the middle 50% of journey times.

[1]

(d)

State which depot’s distribution appears more nearly symmetric, giving a feature of its five-number summary to support your answer.

[2]

Question 2

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 3

Paper 1, no calculatorEasy5 marks

A clinic records whether each of 100 patients has a particular exposure, $E$ and whether they have a condition, $D$ Figure 1 shows the results.

(a)

Write down the formula for $P(D\mid E)$ assuming $P(E)>0$

[1]

(b)

Use Figure 1 for both parts.

(i)

Calculate $P(D\mid E)$

[2]

(ii)

Determine whether $D$ and $E$ are independent. Give a reason.

[2]

Question 4

Paper 1, no calculatorHard6 marks

A health service records whether each of 300 participants has a risk factor $F$ and whether each is diagnosed with a disease $D$ Figure 1 shows the results.

A report notes that more participants diagnosed with $D$ had $F$ than did not have $F$ and concludes that diagnosis is more likely among participants with $F$

(a)

State the definition of $P(A\mid B)$ in terms of $P(A\cap B)$ and $P(B)$

[1]

(b)

Use Figure 1 to calculate $P(F\mid D)$

[1]

(c)

In light of your answer to (b), test whether $F$ and $D$ are independent and assess the report’s conclusion.

[4]

Question 5

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 6

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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