IB Maths AI HL 4.1 Statistics and Probability Sl Content Questions

Practise IB Mathematics AI HL 4.1 by analysing probability models, distributions, regression, inference and technology output with explicit assumptions.

Exam points

Collect, sample and present data, then assess bias, reliability, validity, outliers and the quality of a statistical investigation.
Summarise and compare data with tables, graphs, quartiles, spread, variance, standard deviation and appropriate technology.
Interpret correlation and regression, make predictions and distinguish association from causation while checking extrapolation.
Model probability with sample spaces, conditional/independent events, random variables and expected value.
Select and use binomial, normal and rank-correlation methods, interpreting parameters, probabilities and assumptions.
Form and evaluate hypothesis tests, including chi-square and t-tests, and interpret p-values, decisions and limitations.

Covered syllabus points

SL 4.1—Sampling and data SL 4.2—Data presentation SL 4.3—Summary statistics SL 4.4—Correlation and regression SL 4.5—Probability basics SL 4.6—Combined, conditional and independent events SL 4.7—Discrete random variables SL 4.8—Binomial distribution SL 4.9—Normal distribution SL 4.10—Spearman's rank correlation SL 4.11—Hypothesis testing

[Maximum number: 20]

Juliet is a sociologist who wants to investigate if income affects happiness amongst doctors. This question asks you to review Juliet's methods and conclusions.
Juliet obtained a list of email addresses of doctors who work in her city. She contacted them and asked them to fill in an anonymous questionnaire. Participants were asked to state their annual income and to respond to a set of questions. The responses were used to determine a happiness score out of 100 . Of the 415 doctors on the list, 11 replied.

(a)

Describe one way in which Juliet could improve the reliability of her investigation.

[ 1 ]

(b)

Juliet classifies response K as an outlier and removes it from the data. Suggest one possible justification for her decision to remove it.

[ 1 ]

(c)

For the remaining ten responses in the table, Juliet calculates the mean happiness score to be 52.5.

[ 4 ]

(i)

Calculate the mean annual income for these remaining responses.

[ 2 ]

(ii)

Determine the value of r, Pearson's product-moment correlation coefficient, for these remaining responses.

Juliet decides to carry out a hypothesis test on the correlation coefficient to investigate whether increased annual income is associated with greater happiness.

[ 2 ]

$\quad r=0.558(0.557723 \ldots)$

(d)

State why the hypothesis test should be one-tailed.

[ 1 ]

(e)

State the null and alternative hypotheses for this test.

The critical value for this test, at the 5 % significance level, is 0.549 . Juliet assumes that the population is bivariate normal.

[ 2 ]

$\quad \mathrm{H}_{0}: \rho=0 ; \mathrm{H}_{1}: \rho>0$
Note: Award A1 for $\rho$ seen (do not accept r ), A1 for both correct hypotheses, using their $\rho$ or r. Accept an equivalent statement in words, however reference to "correlation for the population" or "association for the population" must be explicit for the first A1 to be awarded.
Watch out for a null hypothesis in words similar to "Annual income is not associated with greater happiness". This is effectively saying $\rho \leq 0$ and should not be condoned.

(f)

Determine whether there is significant evidence of a positive correlation between annual income and happiness. Justify your answer.

[ 2 ]

METHOD 1 - using critical value of r
$0.558>0.549(0.557723 \ldots>0.549)$
(therefore significant evidence of) a positive correlation
Note: Do not award R0A1.

METHOD 2 - using p-value
$0.0469<0.05(0.0469463 \ldots<0.05)$

Note: Follow through from their r-value from part (c)(ii).
(therefore significant evidence of) a positive correlation
Note: Do not award A0A1.

(g)

Juliet wants to create a model to predict how changing annual income might affect happiness scores. To do this, she assumes that annual income in dollars, X, is the independent variable and the happiness score, Y, is the dependent variable.

She first considers a linear model of the form

Y=a X+b

[ 4 ]

(i)

Use Juliet's data to find the value of a and of b.

[ 1 ]

$a=0.000126(0.000125842 \ldots), b=41.1(41.1490 \ldots)$

(ii)

Interpret, referring to income and happiness, what the value of a represents.

Juliet then considers a quadratic model of the form

Y=c X^{2}+d X+e

[ 1 ]

(iii)

Find the value of c, of d and of e.

[ 1 ]

$\quad c=-2.06 \times 10^{-9}\left(-2.06191 \ldots \times 10^{-9}\right)$ ,
$d=7.05 \times 10^{-4}\left(7.05272 \ldots \times 10^{-4}\right)$ ,
e=12.6 (12.5878...)

(iv)

Comment on the validity of her decision.

After presenting the results of her investigation, a colleague questions whether Juliet's sample is representative of all doctors in the city.

A report states that the mean annual income of doctors in the city is $\$ 80000$ . Juliet decides to carry out a test to determine whether her sample could realistically be taken from a population with a mean of $\$ 80000$ .

[ 1 ]

EITHER
not valid, $R^{2}$ not a useful measure to compare models with different numbers of parameters
OR
not valid, quadratic model will always have a better fit than a linear model
Note: Accept any other sensible critique of the validity of the method. Do not accept any answers which focus on the conclusion rather than the method of model selection.

(h)

State the name of the test which Juliet should use.

[ 1 ]

(i)

State the null and alternative hypotheses for this test.

[ 1 ]

EITHER
$\mathrm{H}_{0}: \mu=80000 ; \mathrm{H}_{1}: \mu \neq 80000$
OR
$\mathrm{H}_{0}$ : (sample is drawn from a population where) the population mean is $\$ 80000$ $\mathrm{H}_{1}$ : the population mean is not $\$ 80000$
Note: Do not allow FT from an incorrect test in part (f)(i) other than a z-test.

(j)

Perform the test, using a 5 % significance level, and state your conclusion in context.

[ 3 ]

$\quad p=0.610(0.610322 \ldots)$
Note: For a z-test follow through from part (f)(i), either 0.578 (from biased estimate of variance) or 0.598 (from unbiased estimate of variance).
0.610>0.05
EITHER
no (significant) evidence that mean differs from $80000
OR
the sample could plausibly have been drawn from the quoted population A1
Note: Allow R1FTA1FT from an incorrect p-value, but the final A1 must still be in the context of the original research question.

[Maximum number: 17]

This question is about applying ideas from logarithms, calculus and probability to an unfamiliar mathematical theory called information theory.
Claude Shannon developed a mathematical theory called information theory to measure the information gained when random events occur. He defined the information, I, that is gained when an event with probability p occurs as

I=-\ln p

where $0<p \leq 1$ . For example, no information is gained ( I=0 ) when an event is certain to $\operatorname{occur}(p=1)$ .

(a)

A computer selects at random an integer x from 1 to 10, inclusive. Each outcome is equally likely.

Alessia is trying to determine the value of x and asks if x is odd.

[ 5 ]

(i)

Write down the probability that x is odd.

[ 1 ]

$\frac{5}{10} \quad\left(=\frac{1}{2}\right)$

(ii)

Alessia is told that x is odd. Find how much information Alessia gains.

The computer then selects at random an integer y from 1 to 10 , inclusive. Each outcome is equally likely.

Daniel is trying to determine the value of y and asks if y is 7 . He is told that it is not 7 .

[ 2 ]

attempt to substitute p= their (b)(i) into $I=-\ln p$ $=0.693 \quad\left(0.693147 \ldots,-\ln \left(\frac{1}{2}\right)\right)$

(iii)

Find how much information Daniel gains.

If a random variable has n possible outcomes with probabilities $p_{1}, p_{2} \ldots p_{n}$ , then the expected information gained, E(I), is defined as

\mathrm{E}(I)=\sum_{r=1}^{n}-p_{r} \ln p_{r}

[ 2 ]

$\frac{9}{10}$ $=0.105 \quad\left(0.105360 \ldots,-\ln \left(\frac{9}{10}\right)\right)$

(b)

For the integer guessing game described in part (b), when Daniel asks if y is 7 , there are two possible outcomes: " y is 7 " or " y is not 7 ".

[ 4 ]

(i)

Show that the expected information gained by Daniel is 0.325 , correct to three significant figures.

[ 2 ]

attempt to substitute into the formula for E(I) and recognise that n=2 (or two terms are needed)
$\mathrm{E}(I)=-\frac{1}{10} \ln \left(\frac{1}{10}\right)-\frac{9}{10} \ln \left(\frac{9}{10}\right)$
0.325 (0.325082...)

(ii)

Alessia asks if x is odd. Show that her expected information gained is greater than Daniel's expected information gained.

Information theory can be applied to a variety of situations.

[ 2 ]

$\quad \mathrm{E}(I)=-\frac{1}{2} \ln \left(\frac{1}{2}\right)-\frac{1}{2} \ln \left(\frac{1}{2}\right)$
0.693 (0.693147..., $\ln (2)$ )
0.693>0.325

(c)

When a coin is flipped, the outcome is either heads or tails. The coin may be biased. Let p be the probability of the outcome being heads.

[ 1 ]

(i)

Find, in terms of p, the expected information gained when the coin is flipped once.

[ 1 ]

$(\mathrm{E}(I)=)-p \ln p-(1-p) \ln (1-p)$

(d)

Angela wants to decide how many balls should be compared to each other in the first weighing. She produces the following table to help plan her strategy.

[ 7 ]

(i)

Find the value of x. Justify your answer.

[ 2 ]

$\quad(x=) \frac{6}{12} \quad\left(=\frac{1}{2}\right)$

EITHER
for the scales to balance, the odd ball must be in the six balls not chosen

OR
for the scales to balance, all the balls chosen must be of equal weight and hence
$\frac{11}{12} \times \frac{10}{11} \times \frac{9}{10} \times \ldots \times \frac{6}{7}$

(ii)

Find the value of y.

[ 2 ]

EITHER
recognition that the sum of the probabilities on the third row of the table equals 1 e.g. x+2 y=1

OR
for one side to be heavier, the odd ball must be one of six balls chosen $\left(\frac{6}{12}\right)$ and half the time this will result in left-side being heavier, therefore $y=\frac{6}{12} \times \frac{1}{2}$ $y=\frac{1}{4}$

(iii)

Find the value of z.

[ 2 ]

$\quad z=-\frac{1}{6} \ln \frac{1}{6}-\frac{1}{6} \ln \frac{1}{6}-\frac{2}{3} \ln \frac{2}{3}$ $=0.868 \quad(0.867563 \ldots)$

(iv)

Use the table to suggest the best choice for Angela's first weighing.

Justify your answer.

[ 1 ]

[Maximum number: 5]

This question compares possible designs for a new computer network between multiple school buildings, and whether they meet specific requirements.
A school's administration team decides to install new fibre-optic internet cables underground. The school has eight buildings that need to be connected by these cables. A map of the school is shown below, with the internet access point of each building labelled A-H.

Jonas is planning where to install the underground cables. He begins by determining the distances, in metres, between the underground access points in each of the buildings.

He finds $\mathrm{AD}=89.2 \mathrm{~m}, \mathrm{DF}=104.9 \mathrm{~m}$ and $\mathrm{A} \hat{\mathrm{DF}}=83^{\circ}$ .

After more research, Jonas decides to install the cables as shown in the diagram below.

Each individual cable is installed such that each end of the cable is connected to a building's access point. The connection between each end of a cable and an access point has a 1.4 % probability of failing after a power surge.

For the network to be successful, each building in the network must be able to communicate with every other building in the network. In other words, there must be a path that connects any two buildings in the network. Jonas would like the network to have less than a 2 % probability of failing to operate after a power surge.

Show that Jonas's network satisfies the requirement of there being less than a 2 % probability of the network failing after a power surge.

METHOD 1 recognition of a binomial distribution
$X \sim \mathrm{~B}(2,0.014)$
finding the probability that a cable fails (at least one of its connections fails)
P(X>0)=0.027804 OR 1-P(X=0)=0.027804
recognition that two cables must fail for the network to go offline
recognition of binomial distribution for network, $Y \sim \mathrm{~B}(8,0.027804)$ $\mathrm{P}(Y \geq 2)=0.0194(0.0193602 \ldots) \quad$ OR $1-\mathrm{P}(Y<2)=0.0194(0.0193602 \ldots)$
therefore, the diagram satisfies the requirement since 1.94 %<2 %

Note: Evidence of binomial distribution may be seen as combinations.

METHOD 2 recognition of a binomial distribution
$X \sim \mathrm{~B}(16,0.014)$
finding the probability that at least two connections fail $\mathrm{P}(X \geq 2)=0.0206473 \ldots$ OR $1-\mathrm{P}(X<2)=0.0206473 \ldots$
recognition that the previous answer is an overestimate
finding probability of two ends of the same cable failing, $F \sim \mathrm{~B}(2,0.014)$ , and the ends of the other 14 cables not failing, $S \sim \mathrm{~B}(14,0.014) \mathrm{P}(F=2) \times \mathrm{P}(S=0)=0.0000160891 \ldots$ $0.0000160891 \ldots \times 8=0.00128713 \ldots$

therefore, the diagram satisfies the requirement since 1.94 %<2 %

METHOD 3 recognition of a binomial distribution
$X \sim \mathrm{~B}(16,0.014)$
finding the probability that the network remains secure if 0 or 1 connections fail or if 2 connections fail provided that the second failed connection occurs at the other end of the cable with the first failure
P( remains secure $)=\mathrm{P}(X \leq 1)+\frac{1}{15} \times \mathrm{P}(X=2)$
=0.9806397625
P( network fails $)=1-0.9806397625=0.0194(0.0193602 \ldots)$
therefore, the diagram satisfies the requirement since 1.94 %<2 %

METHOD 4

P(network failing)
=1-P(0 connections failing )-P(1 connection failing )
- P (2 connections on the same cable failing) M1
$=1-0.986{ }^{16}-{ }^{16} C_{1} \times 0.014 \times 0.986{ }^{15}-{ }^{8} C_{1} \times 0.014^{2} \times 0.986{ }^{14}$
Note: Award A1 for each of $2^{\text {nd }}, 3^{\text {rd }}$ and last terms.
$=0.0194(0.0193602 \ldots)$
therefore, the diagram satisfies the requirement since 1.94 %<2 %
AG

[Maximum number: 1]

Describe one way in which Juliet could improve the reliability of her investigation.

[Maximum number: 1]

Juliet classifies response K as an outlier and removes it from the data. Suggest one possible justification for her decision to remove it.

[Maximum number: 2]

For the remaining ten responses in the table, Juliet calculates the mean happiness score to be 52.5.

Calculate the mean annual income for these remaining responses.

[Maximum number: 3]

The company Fred Express delivers packages. From past experience, the time taken, T, to deliver a package follows a normal distribution with mean 64 hours and standard deviation 12 hours.

For quality control, the manager randomly selects five outgoing packages. These selections are independent.

Find the probability that exactly two of these packages are delivered in less than k hours.

recognizing binomial distribution
$\mathrm{B}(5,0.3) \quad(\mathrm{P}(X=2))$
0.309 (0.3087)

[Maximum number: 3]

This question considers how the assessment of the Air Quality Index (AQI) for a school depends on the method chosen by the person doing the assessing.
Air quality for a district is measured at three monitoring stations. The positions of these stations on a coordinate system with units in kilometres are A(0,5), B(8,9) and C(8,1).
A Voronoi diagram is constructed with the three stations as sites.

Use this model to find the expected number of days per year (correct to the nearest day) on which the AQI value can be classed as "good".

$\quad \mathrm{P}(X<50)=0.00273(0.00272574 \ldots)$
(A1)

\mathrm{P}(X<50) \times 365

=0.994895..
=1

[Maximum number: 8]

A firm wishes to review its recruitment processes. This question considers the validity and reliability of the methods used.
Every year an accountancy firm recruits new employees for a trial period of one year from a large group of applicants.
At the start, all applicants are interviewed and given a rating. Those with a rating of either Excellent, Very good or Good are recruited for the trial period. At the end of this period, some of the new employees will stay with the firm.
It is decided to test how valid the interview rating is as a way of predicting which of the new employees will stay with the firm.
Data is collected and recorded in a contingency table.

(a)

Without calculation, explain why it might not be appropriate to calculate a correlation coefficient for the whole sample of 18 employees.

[ 2 ]

(b)

Find $r_{s}$ for the seven employees working in the international department.

[ 4 ]

L

M

N

O

P

Q

R

Written assessment

rank

1

2

3

4

5

6

Manager score

rank

1

2.5

4.5

2.5

4.5

6
(M1)(A1)
Note: Award (M1) for an attempt to rank the data, and (A1) for correct ranks for both variables. Accept either set of rankings in reverse.

Note: The (M1) is for calculating the PMCC for their ranks.

Note: If a final answer of 0.9107 is seen, from use of $1-\frac{6 \sum d^{2}}{n\left(n^{2}-1\right)}$ , award (M1)(A1)A1.

Accept -0.909 if one set of ranks has been ordered in reverse.

(c)

Hence comment on the validity of the written assessment as a measure of the level of performance of employees in this department. Justify your answer.

The same seven employees are given the written assessment a second time, at the end of the first year, to measure its reliability. Their marks are shown in the table below.

[ 2 ]

(iii) EITHER there is a (strong) association between the written assessment mark and the manager scores.
OR
there is a (strong) agreement in the rank order of the written assessment marks and the rank order of the manager scores.
OR
there is a (strong linear) correlation between the rank order of the written assessment marks and the rank order of the manager scores.
Note: Follow through on a value for their value of $r_{s}$ in c(ii).
THEN
the written assessment is likely to be a valid measure (of the level of employee performance)

(d)

Use an appropriate test, at the 5 % significance level, to determine whether a new employee staying with the firm is independent of their interview rating. State the null and alternative hypotheses, the p-value and the conclusion of the test.

The next year's group of applicants are asked to complete a written assessment which is then analysed. From those recruited as new employees, a random sample of size 18 is selected.

The sample is stratified by department. Of the 91 new employees recruited that year, 55 were placed in the national department and 36 in the international department.

[ 6 ]

Use of $\chi^{2}$ test for independence
$\mathrm{H}_{0}$ : Staying (or leaving) the firm and interview rating are independent.
$\mathrm{H}_{1}$ : Staying (or leaving) the firm and interview rating are not independent

Note: For $\mathrm{H}_{1}$ accept '...are dependent' in place of '...not independent'.
p-value $=0.487(0.487221 \ldots)$

Note: Award A1 for $\chi^{2}=1.438 \ldots$ if p-value is omitted or incorrect.
0.487>0.05
(the result is not significant at the 5 % level)
insufficient evidence to reject the $\mathrm{H}_{0}$ (or "accept $\mathrm{H}_{0}$ ")

Note: Do not award R0A1.
The final R1A1 can follow through from their incorrect p-value

(e)

The written assessment is in five sections, numbered 1 to 5. At the end of the year, the employees are also given a score for each of five professional attributes: V, W, X, Y and Z .

The firm decides to test the hypothesis that there is a correlation between the mark in a section and the score for an attribute.

They compare marks in each of the sections with scores for each of the attributes.

[ 2 ]

(i)

Write down the number of tests they carry out.

[ 1 ]

(ii)

The firm obtains a significant result when comparing section 2 of the written assessment and attribute X. Interpret this result.

[ 1 ]

All question bank results loaded

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question (i)

Question (ii)

Question (d)

Question (e)

Question (f)

Question (g)

Question (i)

Question (ii)

Question (iii)

Question (iv)

Question (h)

Question (i)

Question (j)

Question 2

Question (a)

Question (i)

Question (ii)

Question (iii)

Question (b)

Question (i)

Question (ii)

Question (c)

Question (i)

Question (d)

Question (i)

Question (ii)

Question (iii)

Question (iv)

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question (a)

Question (b)

Question (c)

Question (d)

Question (e)

Question (i)

Question (ii)