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 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]

Question 3

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 4

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 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 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 7

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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