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

Easy Medium Hard

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, 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 3

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 5

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 6

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 7

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 8

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 9

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 10

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 11

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]

Question 12

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]

The best questions are still waiting

Unlock every question in this topic, then type, photograph or draw your answer and have it marked against the markscheme. Find out where you are dropping marks.

Unlock all questions Compare Student and Student Plus