IB Maths AI HL 4 Statistics and Probability Questions

Practise IB Maths AI HL statistics and probability through shared-core and HL distributions, inference, regression and technology-supported models in context.

Exam points

Collect, present and summarise data while evaluating validity, bias, outliers, reliability and uncertainty.
Interpret regression and correlation, compare models and make predictions without confusing association with causation.
Model probability with conditional events, random variables, expected value and standard distributions.
Use statistical inference and hypothesis tests to evaluate claims, p-values, confidence/uncertainty and limitations.
Use advanced distributions, sampling theory, confidence intervals and error analysis for HL-only inference questions.
Use transition matrices and Markov chains to calculate future states, steady states and long-run probabilities.

Covered syllabus points

4.1 Statistics and probability - SL content 4.2 Statistics and probability - AHL content

[Maximum number: 24]

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)

Describe one criticism that can be made about the validity of Juliet's investigation.

Juliet's results are summarized in the following table.

[ 1 ]

(c)

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

[ 1 ]

(d)

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 ]

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

(e)

State why the hypothesis test should be one-tailed.

[ 1 ]

(f)

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.

(g)

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.

(h)

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

[ 7 ]

(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)

Find the coefficient of determination for each of the two models she considers.

[ 2 ]

for quadratic model: $R^{2}=0.659(0.659145 \ldots)$
A1
for linear model: $R^{2}=0.311(0.311056 \ldots)$

Note: Follow through from their r value from part (c)(ii).

(v)

Hence compare the two models.

Juliet decides to use the coefficient of determination to choose between these two models.

[ 1 ]

(vi)

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.

(i)

State the name of the test which Juliet should use.

[ 1 ]

(j)

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.

(k)

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: 13]

Taylor is playing a computer game in which they shoot at spaceships and battleships. The number of spaceships they hit per minute can be modelled by a Poisson distribution with mean 4.2. The number of battleships they hit per minute can be modelled by a Poisson distribution with a mean of 2.3. Any single hit occurs independently of all others.

(a)

Find the probability Taylor hits

[ 5 ]

(i)

at most 10 spaceships in 2 minutes.

[ 2 ]

mean =8.4

\mathrm{P}(S \leq 10)=0.774(0.774301 \ldots)

(ii)

a total of more than 10 spaceships and battleships in one minute.

[ 3 ]

attempt to add two means
4.2+2.3=6.5
$\mathrm{P}(S+B>10)=\mathrm{P}(S+B \geq 11)$
0.0668 (0.0668387…)

(b)

Every spaceship that is hit earns Taylor 3 points and every battleship 5 points. Let T be the total points earned in one minute.

Find

[ 3 ]

(i)

E(T).

[ 1 ]

$\mathrm{E}(T)=3 \times 4.2+5 \times 2.3=24.1$

(ii)

$\quad \operatorname{Var}(T)$ .

[ 2 ]

$\quad \operatorname{Var}(T)=3^{2} \times 4.2+5^{2} \times 2.3=95.3$

(c)

State one reason why the distribution of T cannot be Poisson.

[ 1 ]

(d)

Taylor intends to play the game for one hour.

Use the central limit theorem to find the probability that Taylor's mean score per minute is greater than 25 .

[ 4 ]

distribution of mean score is $\mathrm{N}\left(24.1, \frac{95.3}{60}\right)(\mathrm{N}(24.1,1.58833 \ldots))$
(A1)(A1)

Note: Award A1 for normal distribution with mean 24.1, and A1 for variance $\frac{95.3}{60}$ .

\mathrm{P}(\bar{T}>25)=0.238(0.237576 \ldots)

[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: 7]

A random sample of eight packets of Apollo coffee granules are selected from a supermarket shelf.
The weights of the coffee granules present in each packet are as follows:
$222 \mathrm{~g} \quad 226 \mathrm{~g} \quad 221 \mathrm{~g} \quad 228 \mathrm{~g} \quad 227 \mathrm{~g} \quad 225 \mathrm{~g} \quad 222 \mathrm{~g} \quad 223 \mathrm{~g}$

(a)

Find an unbiased estimate for the mean weight of coffee granules in a packet of Apollo coffee.

[ 1 ]

$224 \mathrm{~g}(224.25 \mathrm{~g})$

(b)

Calculate a 95 % confidence interval for the population mean. Give your answer to four significant figures.

[ 3 ]

[222.1, 226.4]

Note: Award A1 for each correct end of the interval. Accept open or closed (weak or strict) interval notation. Inequalities involving $\mu$ would also be accepted, but not involving $\bar{x}$ .

Award A1A0 for correct answers not given correct to 4 sf.

(c)

State one assumption you have made in order for your interval to be valid.

[ 1 ]

(d)

The label of each packet has a description which includes the phrase: "contains 226 g of coffee granules".

Using your answer to part (a)(ii), briefly comment on the claim on the label.

[ 2 ]

All question bank results loaded

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question (d)

Question (i)

Question (ii)

Question (e)

Question (f)

Question (g)

Question (h)

Question (i)

Question (ii)

Question (iii)

Question (iv)

Question (v)

Question (vi)

Question (i)

Question (j)

Question (k)

Question 2

Question (a)

Question (i)

Question (ii)

Question (b)

Question (i)

Question (ii)

Question (c)

Question (d)

Question 3

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 4

Question (a)

Question (b)

Question (c)

Question (d)