IB Maths AA HL 4.1 Statistics and Probability Sl Content Questions

Practise IB Mathematics AA HL 4.1 by analysing distributions, inference, regression and probability models with precise assumptions.

Exam points

HL syllabus concepts and methods
HL extended reasoning

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—Regression x on y SL 4.11—Formal conditional probability SL 4.12—Standardizing normal variables

[Maximum number: 6]

A supermarket analyses the shopping habits of its customers.
The number of times, X, each customer visits the supermarket in a week is given by the following probability distribution.

(a)

Find the value of a.

[ 2 ]

(b)

Write down the mode of X.

[ 1 ]

(c)

Find the mean of X.

The manager wants to know why customers come to their supermarket. They survey the first 50 customers to arrive at the supermarket on a particular day.

[ 2 ]

(d)

Identify which one of the following best describes the manager's sampling method. Circle your answer.

Simple random / Systematic / Convenience / Quota / Stratified

[ 1 ]

[Maximum number: 4]

The continuous random variable X has a probability density function given by

f(x)=\left\{\begin{array}{cc} k \sin \left(\frac{\pi x}{6}\right), & 0 \leq x \leq 6 \\ 0, & \text { otherwise } \end{array}\right.

(a)

Hence state the interquartile range of X.

[ 2 ]

from (c)(i) $Q_{1}=2$
as the graph is symmetrical about the middle value $x=3 \Rightarrow Q_{3}=4$
so interquartile range is
4-2
=2

(b)

Calculate $P(X \leq 4 \mid X \geq 3)$ .

[ 2 ]

$\quad P(X \leq 4 \mid X \geq 3)=\frac{P(3 \leq X \leq 4)}{P(X \geq 3)}$ $=\frac{\frac{1}{4}}{\frac{1}{2}}$ $=\frac{1}{2}$

[Maximum number: 4]

There are 75 players in a golf club who take part in a golf tournament. The scores obtained on the 18th hole are as shown in the following table.

(a)

One of the players is chosen at random. Find the probability that this player's score was 5 or more.

[ 2 ]

$\quad \mathrm{P}(5$ or more $)=\frac{29}{75}(=0.387)$
Marking guidance:

[2 marks]

(b)

Calculate the mean score.

[ 2 ]

mean score $=\frac{2 \times 3+3 \times 15+4 \times 28+5 \times 17+6 \times 9+7 \times 3}{75}$
(M1)
$=\frac{323}{75}(=4.31)$

[Maximum number: 1]

A supermarket analyses the shopping habits of its customers.
The number of times, X, each customer visits the supermarket in a week is given by the following probability distribution.

Identify which one of the following best describes the manager's sampling method. Circle your answer.

Simple random / Systematic / Convenience / Quota / Stratified

[Maximum number: 2]

The continuous random variable X has a probability density function given by

f(x)=\left\{\begin{array}{cc} k \sin \left(\frac{\pi x}{6}\right), & 0 \leq x \leq 6 \\ 0, & \text { otherwise } \end{array}\right.

Hence state the interquartile range of X.

from (c)(i) $Q_{1}=2$
as the graph is symmetrical about the middle value $x=3 \Rightarrow Q_{3}=4$
so interquartile range is
4-2
=2

[Maximum number: 2]

There are 75 players in a golf club who take part in a golf tournament. The scores obtained on the 18th hole are as shown in the following table.

Calculate the mean score.

mean score $=\frac{2 \times 3+3 \times 15+4 \times 28+5 \times 17+6 \times 9+7 \times 3}{75}$
(M1)
$=\frac{323}{75}(=4.31)$

[Maximum number: 3]

A class is given two tests, Test A and Test B. Each test is scored out of a total of 100 marks. The scores of the students are shown in the following table.

Let x be the score on Test A and y be the score on Test B.
The teacher finds that the equation of the regression line of y on x for these scores is y=0.822 x+18.4.

(a)

Find the value of Pearson's product-moment correlation coefficient, r.

Giovanni was absent for Test A and Paulo was absent for Test B.

The teacher uses the regression line of y on x to estimate the missing scores.
Paulo scored 10 on Test A.

The teacher estimated his score on Test B to be 27 to the nearest integer using the following calculation:

y=0.822(10)+18.4 \approx 27

[ 2 ]

$r=0.901017 \ldots$
r=0.901

(b)

Give a reason why this method is not appropriate for Paulo.

Giovanni scored 90 on Test B.

The teacher estimated his score on Test A to be 87 to the nearest integer using the following calculation:

90=0.822 x+18.4, \text { so } x=\frac{90-18.4}{0.822} \approx 87

[ 1 ]

(c)

Give a reason why this method is not appropriate for Giovanni.

[ 1 ]

(d)

Use an appropriate method to show that the estimated Test A score for Giovanni is 86 to the nearest integer.

[ 3 ]

attempt to find the equation of the regression line of x on y
$(x=) 0.987124 \ldots y-3.21970 \ldots((x=) 0.987 y-3.22)$ $(x=) 0.987124 \ldots(90)-3.21970 \ldots(=85.6214 \ldots)$
=86 to nearest integer.

Note: Condone notation for x and y switched if values are correct.

[Maximum number: 2]

There are 75 players in a golf club who take part in a golf tournament. The scores obtained on the 18th hole are as shown in the following table.

One of the players is chosen at random. Find the probability that this player's score was 5 or more.

$\quad \mathrm{P}(5$ or more $)=\frac{29}{75}(=0.387)$
Marking guidance:

[2 marks]

[Maximum number: 4]

Amanda enters data from surveys into a database. It can be assumed that the accuracy of any survey entered is independent of all other surveys entered.
From previous records, it is known that Amanda enters 8 % of the surveys inaccurately.

Find the value of x.

METHOD 1

\begin{aligned} & \mathrm{P}(I \mid C)=\frac{x}{100} \text { (seen anywhere) } \\ & \mathrm{P}(I)=0.55 \times 0.08+0.25 \times 0.06+0.2 \times \frac{x}{100} \end{aligned}

equating their expression for P(I) to $\frac{x}{100}$

\begin{aligned} & \frac{x}{100}=0.55 \times 0.08+0.25 \times 0.06+0.2 \times \frac{x}{100} \\ & x=7.375 \\ & x=7.38 \text { (accept } 7.38 \% \text { ) } \end{aligned}

METHOD 2

\mathrm{P}(I)=0.55 \times 0.08+0.25 \times 0.06+0.2 \times y

equating their expression for P(I) to y
$y=0.55 \times 0.08+0.25 \times 0.06+0.2 \times y$
y=0.07375
recognizing to multiply their y value by 100
x=7.38 (accept 7.38 % )

[Maximum number: 5]

(a)

On a particular day Amanda enters data from 50 surveys.

[ 2 ]

(i)

Find the probability that Amanda entered at most six surveys inaccurately.

[ 2 ]

recognizing binomial distribution
$E \sim \mathrm{~B}(50,0.08)$ , where E is the number of inaccurate surveys
$\mathrm{P}(E \leq 6)=0.898128 \ldots$ $(\mathrm{P}(E \leq 6)=) 0.898$ (accept 89.8 % )

(b)

Find the value of n.

Bryce and Carmen also enter data from surveys into the same database. It is known that surveys entered by Bryce and Carmen are inaccurate 6 % and 11 % of the time respectively. It can again be assumed that the accuracy of any survey entered is independent of all other surveys entered.

From the surveys assigned to the three of them, Amanda enters 55 %, Bryce 25 % and Carmen 20\%.

[ 3 ]

use of tables

Note: Award (M1) for at least one correct value of $\mathrm{P}(E \leq 6)$ for a value of n, where $n \neq 50$ .

(n=94 \Rightarrow) \mathrm{P}(E \leq 6)=0.366746 \ldots

n=94

[Maximum number: 3]

The random variable X has probability distribution $\operatorname{Po}(8)$ .

A random sample of 40 observations is taken from the distribution for X.

Find $\mathrm{P}(7.1<\bar{X}<8.5)$ .

EITHER

Note: M1 for normality, A1 for parameters.

OR
The expression is equivalent to $P\left(283 \leq \sum X \leq 339\right)$ where $\sum X$ is $\operatorname{Po}(320)$
= 0.840
Note: Accept 284, 340 instead of 283, 339

Accept any answer that rounds correctly to 0.84 or 0.85 .

[Maximum number: 3]

The random variable X has probability distribution $\operatorname{Po}(8)$ .

Find $\mathrm{P}(X=6 \mid 5 \leq X \leq 8)$ .

$\mathrm{P}(X=6 \mid 5 \leq X \leq 8)=\frac{\mathrm{P}(X=6)}{\mathrm{P}(5 \leq X \leq 8)}=\frac{0.122 \ldots}{0.592 \ldots-0.0996 \ldots} =0.248$

[Maximum number: 3]

The random variable X has probability distribution $\operatorname{Po}(8)$ .

A random sample of 40 observations is taken from the distribution for X.

Given that $\mathrm{P}(|\bar{X}-8| \leq k)=0.95$ , find the value of k.

EITHER
$k=1.96 \frac{\sigma}{\sqrt{n}}$ or $1.96 \operatorname{std}(\bar{X})$
k=0.877 or $1.96 \sqrt{0.2}$

OR
The expression is equivalent to
$P\left(320-40 k \leq \sum X \leq 320+40 k\right)=0.95$
k=0.875

Note: Accept any answer that rounds to 0.87 or 0.88 .
Award M1A0 if modulus sign ignored and answer obtained rounds to 0.74 or 0.75
Total

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Covered syllabus points

Question 1

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Question (b)

Question (c)

Question (d)

Question 2

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Question (b)

Question 3

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Question 4

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Question 6

Question 7

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Question (b)

Question (c)

Question (d)

Question 8

Question 9

Question 10

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Question (i)

Question (b)

Question 11

Question 12

Question 13