Statistics and Probability — Free IB Maths AI HL Practice Questions

1FoundationSAQ-SRegression analysis5 marksPaper 1~8 min

Data

b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}, \qquad a = \bar{y} - b\bar{x}

The heights (in cm) and weights (in kg) of six adult males are recorded below. Height (

x\,\text{cm}

) — 165 — 170 — 175 — 180 — 185 — 190 Weight (

y\,\text{kg}

) — 62 — 68 — 72 — 78 — 82 — 88

(a)

State the type of regression that would be most appropriate for modelling the relationship between height and weight. [1 mark]

(b)

Find the equation of the regression line of weight on height in the form

y = a + bx

, giving the values of

a

and

b

correct to three significant figures. [4 marks]

Solutions

2MasterySAQ-SRegression analysis5 marksPaper 1~8 min

A researcher measures the heart rate (

y

, in beats per minute) of a person after exercising for

x

minutes. The data for five individuals are:

x

— 2 — 4 — 6 — 8 — 10

y

— 95 — 110 — 125 — 140 — 155 It is given that

\sum x^2 = 220

and

\sum xy = 3900

.

(a)

State the mean point

(\bar{x},\, \bar{y})

and explain why the regression line of

y

on

x

must pass through it. [2 marks]

(b)

Find the equation of the regression line of

y

on

x

in the form

y = a + bx

, giving

a

and

b

correct to three significant figures. [3 marks]

Solutions

3ChallengeSAQ-LRegression analysis8 marksPaper 1~12 min

A biomedical research team investigates the concentration of a drug in a patient's bloodstream. The concentration

y\,\text{mg/L}

is recorded at time

x

hours after administration for six patients.

x

(hours) — 1 — 2 — 3 — 4 — 5 — 6

y

(mg/L) — 8.2 — 6.1 — 4.8 — 3.9 — 3.1 — 2.5 The relationship is modelled by

y = a \times b^x

, where

a

and

b

are constants.

(a)

Show that taking natural logarithms of both sides of

y = a \times b^x

gives a linear model of the form

\ln y = c + dx

, stating

c

and

d

in terms of

a

and

b

. [2 marks]

(b)

Using the data in the table, determine the equation of the regression line of

\ln y

on

x

. Give the coefficients correct to three significant figures. [3 marks]

(c)

Hence, determine the values of

a

and

b

for the exponential model. Give your answers correct to three significant figures. [3 marks]

Solutions

4FoundationSAQ-SRegression analysis5 marksPaper 1~8 min

A researcher investigates the relationship between the number of hours a plant is exposed to light each day,

x

, and the height of the plant after 4 weeks,

y\,\text{cm}

. Data for 8 plants is summarised as follows:

\sum x = 56,\quad \sum y = 120,\quad \sum xy = 912,\quad \sum x^2 = 448,\quad \sum y^2 = 1920

(a)

Calculate the Pearson correlation coefficient,

r

, for this data. [3 marks]

(b)

The researcher concludes that increasing daily light exposure causes plants to grow taller. Evaluate this conclusion with reference to your value of

r

and one limitation of correlation analysis. [2 marks]

Solutions

5FoundationSAQ-SMeasures of central tendency (mean, median, mode)7 marksPaper 1~11 min

A group of 10 students took a mathematics test. Their scores, out of 100, are recorded below:

52,\ 58,\ 63,\ 67,\ 71,\ 71,\ 78,\ 84,\ 96

(a)

State the mode of these scores. [1 mark]

(b)

Find the median of these scores. [2 marks]

(c)

Calculate the mean of these scores, giving your answer correct to one decimal place. [2 marks]

(d)

A student claims that the mean is the most appropriate measure of central tendency to represent this data set. Determine whether this claim is justified. [2 marks]

Solutions

6MasterySAQ-SMeasures of central tendency (mean, median, mode)5 marksPaper 1~8 min

A small business records the number of customer complaints received each day over a 10-day period. The data, in complaints per day, are:

3,\ 5,\ 7,\ 8,\ 8,\ 9,\ 11,\ 12

(a)

Show that the mean number of complaints per day is

7.6

. [2 marks]

(b)

The business manager removes the two extreme values (the smallest and the largest) to reduce the effect of outliers. Calculate the mean of the remaining eight values. [2 marks]

(c)

The manager claims that removing the extreme values gives a more reliable measure of the typical daily complaints. Determine whether the median of the original dataset supports or contradicts this claim, justifying your answer. [1 mark]

Solutions

7ChallengeSAQ-LMeasures of spread (range, variance, standard deviation)8 marksPaper 1~12 min

A factory produces cylindrical metal rods. The diameters (in mm) of a random sample of 8 rods are:

12.4,\ 12.7,\ 12.1,\ 12.9,\ 12.3,\ 12.6,\ 12.2,\ 12.8

(a)

Determine the range and the sample variance of the diameters. [4 marks]

(b)

A second sample of 6 rods from a different production line has diameters with a mean of

12.5\,\text{mm}

and a sample standard deviation of

0.35\,\text{mm}

. Determine the sample standard deviation of the combined sample of all 14 rods, and state whether it is larger or smaller than the standard deviation of the second sample alone. [4 marks]

Solutions

8FoundationSAQ-SMeasures of central tendency (mean, median, mode)5 marksPaper 1~8 min

The heights, in centimetres, of 12 sunflowers are measured:

45,\ 52,\ 48,\ 61,\ 55,\ 48,\ 59,\ 63,\ 48,\ 57,\ 50,\ 54

(a)

State the mode of the heights. [1 mark]

(b)

Find the median height. [2 marks]

(c)

A thirteenth sunflower of height

70\,\text{cm}

is added to the group. Calculate the new mean height, correct to three significant figures. [2 marks]

Solutions

9FoundationSAQ-SBasic probability concepts and rules5 marksPaper 1~8 min

A bag contains 7 red marbles and 3 blue marbles. A marble is selected at random, its colour is noted, and it is not replaced. A second marble is then selected at random.

(a)

Find the probability that both marbles are the same colour. [3 marks]

(b)

Given that both marbles are the same colour, find the probability that both are red. [2 marks]

Solutions

10MasterySAQ-SBasic probability concepts and rules5 marksPaper 1~8 min

In a school,

60\%

of students play football and

45\%

play basketball. It is known that

25\%

of students play both sports.

(a)

Show that the probability that a randomly selected student plays at least one of these two sports is

0.80

. [2 marks]

(b)

Find the probability that a randomly selected student plays exactly one of these two sports. [1 mark]

(c)

Given that a randomly selected student plays at least one of these two sports, find the probability that they play both sports. [2 marks]

Solutions

11ChallengeSAQ-LBasic probability concepts and rules11 marksPaper 1~17 min

A university offers three languages: French (

F

), German (

G

), and Spanish (

S

). From a survey of

200

students, the following data were collected: -

80

students study French -

65

students study German -

70

students study Spanish -

30

students study both French and German -

25

students study both French and Spanish -

20

students study both German and Spanish -

10

students study all three languages

(a)

Determine the number of students who study at least one of the three languages. [3 marks]

(b)

Determine the probability that a randomly selected student from the survey studies exactly one language. [3 marks]

(c)

Given that a student studies French, determine the probability that they study exactly one of German or Spanish (but not both). [2 marks]

(d)

Determine the number of students who study none of the three languages. [1] (e) Determine the probability that a randomly selected student studies at least two languages, given that they study at least one language. [2 marks]

Solutions

12FoundationSAQ-SBasic probability concepts and rules5 marksPaper 1~8 min

A school survey records that

60\%

of students play football,

40\%

play basketball, and

25\%

play both sports.

(a)

State the probability that a randomly selected student plays football but not basketball. [1 mark]

(b)

Find the probability that a randomly selected student plays at least one of the two sports. [2 marks]

(c)

Given that a randomly selected student plays at least one sport, find the probability that they play both sports. [2 marks]

Solutions

13FoundationSAQ-SBinomial distribution and its properties7 marksPaper 1~11 min

A factory produces electronic components. The probability that a randomly selected component is defective is

0.05

. A quality control inspector randomly selects

12

components.

(a)

State two conditions required for the number of defective components,

X

, to be modelled by the distribution

X \sim B(12,\, 0.05)

. [2 marks]

(b)

Find

P(X = 2)

. [2 marks]

(c)

Given that least one component in the sample is defective, find the probability that exactly two are defective. [3 marks]

Solutions

14MasterySAQ-SBinomial distribution and its properties9 marksPaper 1~14 min

A factory produces electronic components. The probability that a randomly selected component is defective is

0.04

. A quality control inspector selects a random sample of

20

components and counts the number of defective components, denoted by

X

.

(a)

State the distribution of

X

, giving two conditions that must be satisfied for this distribution to apply. [2 marks]

(b)

Show that

P(X = 2) = 0.146

, correct to three significant figures. [2 marks]

(c)

Find

P(X \geq 2)

. [2 marks]

(d)

The factory considers a batch of

20

components to be "acceptable" if fewer than

2

defective components are found. A second factory claims its process reduces the defect probability to

0.02

. Determine whether the second factory's process significantly increases the probability that a batch is acceptable. Justify your answer. [3 marks]

Solutions

15ChallengeSAQ-LBinomial distribution and its properties8 marksPaper 1~12 min

A pharmaceutical company is testing a new vaccine. Clinical trials show that the vaccine provides immunity in

85\%

of individuals who receive it. The company administers the vaccine to a random sample of 20 volunteers from a large population. Let

X

be the number of volunteers who develop immunity.

(a)

State the distribution of

X

, including its parameters. [1 mark]

(b)

Calculate the probability that exactly 17 of the 20 volunteers develop immunity. [2 marks]

(c)

The company considers the trial a failure if fewer than 16 volunteers develop immunity. Calculate the probability that the trial is declared a failure. [2 marks]

(d)

State one assumption of the binomial distribution required for the model

X \sim B(20,\, 0.85)

to be valid in this context. [1] (e) Explain how a violation of this assumption could affect the probabilities calculated in parts (b) and (c). [2 marks]

Solutions

16FoundationSAQ-SBinomial distribution and its properties5 marksPaper 1~8 min

A biased coin is tossed

8

times. The probability of obtaining a head on any single toss is

0.3

. Let

X

be the number of heads obtained.

(a)

State the distribution of

X

, including its parameters. [1 mark]

(b)

Find the probability that exactly

3

heads are obtained. [2 marks]

(c)

Given that least

1

head is obtained, find the probability that exactly

3

heads are obtained. [2 marks]

Solutions

17MasterySAQ-SRegression analysis8 marksPaper 2~12 min

A marine biologist studies the relationship between water temperature (

T

, in °C) and diving depth (

D

, in metres) for a species of seal. Data from 10 seals give the following summary statistics:

\sum T = 95.2,\quad \sum D = 246.5,\quad \sum T^2 = 912.04,\quad \sum D^2 = 6090.25,\quad \sum TD = 2345.6,\quad n = 10

The temperature values in the dataset range from

8.6\,°\text{C}

to

10.8\,°\text{C}

.

(a)

Calculate the Pearson product-moment correlation coefficient

r

. [2 marks]

(b)

The regression line of

D

on

T

has the form

D = a + bT

. Calculate the values of

a

and

b

, giving your answers correct to three significant figures. [3 marks]

(c)

Use your regression line to estimate the diving depth when

T = 9.8\,°\text{C}

and when

T = 14.0\,°\text{C}

. Evaluate the reliability of each estimate. [3 marks]

Solutions

18MasterySAQ-SRegression analysis6 marksPaper 2~9 min

A researcher investigates the relationship between years of experience (

x

) and annual salary in thousands of dollars (

y

) for a sample of 8 data analysts. The data are shown in the scatter below. The regression line of

y

on

x

is

y = 33.2 + 2.25x

, calculated from the summary statistics

S_{xx} = 168

,

S_{yy} = 1050

,

S_{xy} = 378

.

(a)

Calculate the correlation coefficient

r

using

r = \dfrac{S_{xy}}{\sqrt{S_{xx}\,S_{yy}}}

. [2 marks]

(b)

Interpret the value of the coefficient of determination

r^2

in the context of this study. [2 marks]

(c)

A data analyst with 18 years of experience earns USD 72 000. Calculate the residual for this data point. [1 mark]

(d)

Evaluate the reliability of using this regression line to predict the salary of a data analyst with 18 years of experience. [1 mark]

Solutions

19MasterySAQ-SRegression analysis6 marksPaper 2~9 min

A biologist is studying the relationship between the temperature (

T

, in °C) and the rate of oxygen consumption (

R

, in arbitrary units) of a species of beetle. The following data was collected for six beetles, each at a different constant temperature.

T

(°C) — 10 — 15 — 20 — 25 — 30 — 35

R

(units) — 2.1 — 3.4 — 5.0 — 6.8 — 9.1 — 12.0 The biologist believes that a linear regression model of the form

R = a + bT

is appropriate for predicting oxygen consumption from temperature.

(a)

Calculate the Pearson correlation coefficient,

r

, for this data. [2 marks]

(b)

Determine the equation of the regression line of

R

on

T

. Give the values of

a

and

b

correct to three significant figures. [2 marks]

(c)

Using your regression line, calculate the predicted oxygen consumption rate when the temperature is

28\,°\text{C}

. Give your answer correct to three significant figures. [2 marks]

Solutions

20MasterySAQ-SRegression analysis6 marksPaper 2~9 min

A market researcher analyses the relationship between advertising spend (

A

, in thousands of dollars) and monthly revenue (

R

, in thousands of dollars) for a chain of six small retail stores.

A

—

2.0

—

3.5

—

5.0

—

6.5

—

8.0

—

9.5

R

—

14.2

—

18.5

—

23.1

—

27.8

—

32.4

—

37.0

(a)

Determine the equation of the regression line of

R

on

A

, giving the slope and intercept correct to three significant figures. State the value of the Pearson correlation coefficient

r

, correct to three significant figures. [3 marks]

(b)

A store spends

7.2

thousand dollars on advertising. Use your regression line to calculate the predicted monthly revenue for this store, giving your answer correct to three significant figures. [1 mark]

(c)

The researcher considers using the regression line to predict the monthly revenue for a store that spends

18.0

thousand dollars on advertising. Evaluate the reliability of this prediction, referring to the value of

r

and the nature of the prediction. [2 marks]

Solutions

21MasterySAQ-SMeasures of central tendency (mean, median, mode)6 marksPaper 2~9 min

A fitness trainer records the number of minutes spent on a treadmill by 12 clients during a session. The data, in minutes, are:

12,\ 15,\ 18,\ 20,\ 22,\ 25,\ 28,\ 30,\ 33,\ 36,\ 40,\ x

where

x

is the time for the 12th client.

(a)

Given that the mean time is

25

minutes, calculate the value of

x

. [2 marks]

(b)

Hence, determine the median time for these 12 clients. [2 marks]

(c)

State whether the mode exists for this dataset and give a reason. [1 mark]

(d)

The trainer wishes to report a single average to best represent the typical session time. Determine which measure of central tendency — mean, median, or mode — is most appropriate, and justify your answer. [1 mark]

Solutions

22MasterySAQ-SMeasures of central tendency (mean, median, mode)6 marksPaper 2~9 min

A small business records the weekly sales (in thousands of dollars) over 10 weeks:

4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10,\ 12,\ y

where

y

is the sales in the 10th week.

(a)

Given that the median weekly sales is

7.5

thousand dollars, calculate the value of

y

. [3 marks]

(b)

Using your value of

y

from part (a), calculate the mean weekly sales. [2 marks]

(c)

The business owner claims that the mean is a better measure of central tendency than the median for this dataset. Determine whether this claim is correct, justifying your answer. [1 mark]

Solutions

23MasterySAQ-SMeasures of central tendency (mean, median, mode)6 marksPaper 2~9 min

A group of 10 students took a mathematics test. Their scores, out of 100, are recorded below.

45,\ 52,\ 58,\ 61,\ 68,\ 72,\ 75,\ 78,\ 83,\ 91

(a)

Calculate the mean score. [2 marks]

(b)

Determine the median score. [2 marks]

(c)

A new student joins the group and takes the same test. After including this student's score, the mean of the 11 scores is 67. Calculate the score of the new student. [2 marks]

Solutions

24MasterySAQ-SMeasures of central tendency (mean, median, mode)6 marksPaper 2~9 min

The annual salaries (in thousands of dollars) of 8 employees at a small company are:

32,\ 35,\ 38,\ 40,\ 42,\ 45,\ 48,\ 52

(a)

Calculate the mean and median salary. [2 marks]

(b)

The company hires a new manager with a salary of USD 120 thousand. Determine the new mean salary for all 9 employees. [2 marks]

(c)

The manager argues that the mean salary should be used to advertise the company's pay to prospective employees, while a current employee argues the median should be used. Evaluate which measure better represents the typical salary of an employee at this company, justifying your answer with reference to your calculated values. [2 marks]

Solutions

25MasterySAQ-SConditional probability and Bayes' theorem6 marksPaper 2~9 min

A factory produces electronic components using two machines, A and B. Machine A produces

60\%

of the components and Machine B produces the remaining

40\%

. Of the components produced by Machine A,

3\%

are defective; of those produced by Machine B,

6\%

are defective.

(a)

Draw a fully labelled tree to represent this information, showing all probabilities on each branch. [2 marks]

(b)

Calculate the probability that a randomly selected component is defective. [2 marks]

(c)

Given that a randomly selected component is defective, calculate the probability that it was produced by Machine A. [2 marks]

Solutions

26MasterySAQ-SBasic probability concepts and rules6 marksPaper 2~9 min

A factory produces electronic components. Each component is tested for two defects: a soldering defect

(S)

and a calibration defect

(C)

. The probability that a component has a soldering defect is

0.12

, the probability that it has a calibration defect is

0.08

, and the probability that it has both defects is

0.03

.

(a)

Calculate

P(S \cup C)

, the probability that a randomly selected component has at least one defect. [2 marks]

(b)

Given that a component has at least one defect, calculate the probability that it has exactly one defect. [2 marks]

(c)

A quality engineer claims that

S

and

C

are independent events. Determine, with justification, whether this claim is correct. [2 marks]

Solutions

27MasterySAQ-SBasic probability concepts and rules8 marksPaper 2~12 min

A bag contains

5

red marbles and

3

blue marbles. Two marbles are drawn from the bag without replacement.

(a)

Calculate the probability that both marbles are red. [2 marks]

(b)

Calculate the probability that the two marbles are of different colours. [2 marks]

(c)

A game is played where a player draws two marbles without replacement. The player wins USD 6 if both marbles are the same colour, and loses USD 4 if the marbles are different colours. Calculate the expected monetary gain for one game, and determine whether a rational player should play. [4 marks]

Solutions

28MasterySAQ-SBasic probability concepts and rules6 marksPaper 2~9 min

A factory produces electronic components. Each component is tested for two distinct faults: a wiring fault

(W)

and a soldering fault

(S)

. The probability that a randomly selected component has a wiring fault is

0.08

, and the probability that it has a soldering fault is

0.05

. The probability that it has neither fault is

0.89

.

(a)

Calculate the probability that a randomly selected component has both faults. [2 marks]

(b)

Determine whether the events

W

and

S

are independent. Justify your answer with a calculation. [2 marks]

(c)

Given that a component has at least one fault, calculate the probability that it has exactly one fault. [2 marks]

Solutions

29MasterySAQ-SBinomial distribution and its properties6 marksPaper 2~9 min

A pharmaceutical company is testing a new vaccine. Clinical trials show that the vaccine is effective in

85\%

of individuals who receive it. A random sample of

12

individuals is selected from a large population. Let

X

be the number of individuals in the sample for whom the vaccine is effective.

(a)

State two conditions required for

X

to be modelled by a binomial distribution, referring to the context of this trial. [2 marks]

(b)

Given that

X \sim B(12,\, 0.85)

, calculate the probability that exactly

10

individuals find the vaccine effective. [2 marks]

(c)

The company considers the trial a concern if fewer than

9

individuals find the vaccine effective. Using your GDC, find the probability that the trial raises a concern, and hence determine whether this outcome is more or less likely than rolling a fair six-sided die and obtaining a

6

. Justify your answer. [2 marks]

Solutions

30MasterySAQ-SBinomial distribution and its properties6 marksPaper 2~9 min

A quality control engineer at a factory inspects batches of

20

electronic components. The probability that any single component is defective is

0.08

, independent of other components.

(a)

State the distribution that models the number of defective components in a batch, including its parameters. [1 mark]

(b)

Calculate the probability that a batch contains exactly

3

defective components. [2 marks]

(c)

The engineer stops the production process if the probability of

3

or more defective components in a batch exceeds

0.25

. Determine, with justification, whether the process should be stopped. [3 marks]

Solutions

31MasterySAQ-SBinomial distribution and its properties6 marksPaper 2~9 min

A factory produces electronic components. The probability that a randomly selected component is defective is

0.04

. A quality control inspector selects a random sample of

20

components.

(a)

State two conditions required for the number of defective components in the sample to follow a binomial distribution. [2 marks]

(b)

Calculate the probability that exactly

2

components in the sample are defective. [2 marks]

(c)

Calculate the expected number of defective components per sample. [1 mark]

(d)

Calculate the standard deviation of the number of defective components per sample. [1 mark]

Solutions

32MasterySAQ-SBinomial distribution and its properties6 marksPaper 2~9 min

A medical research team is testing a new treatment for a disease. The treatment is effective for

85\%

of patients. A random sample of

12

patients is selected to receive the treatment. Let

X

be the number of patients for whom the treatment is effective.

(a)

State the distribution of

X

, including the values of all parameters. [1 mark]

(b)

Calculate

P(X = 10)

. [2 marks]

(c)

Calculate

P(X \geq 11)

. [3 marks]

Solutions

33ChallengeLAQRegression analysis10 marksPaper 3~15 min

A biostatistician investigates the relationship between enzyme concentration,

x

(in

\text{U\,L}^{-1}

), and reaction rate,

y

(in

\mu\text{mol\,min}^{-1}

), from

n = 8

independent experiments.

x

—

2.0

—

3.5

—

5.0

—

6.0

—

7.5

—

9.0

—

10.5

—

12.0

y

—

1.8

—

3.2

—

4.5

—

5.1

—

6.8

—

7.9

—

9.3

—

10.6

A simple linear regression model

y = \alpha + \beta x + \varepsilon

is proposed, where

\varepsilon

represents random error.

(a)

Define the sum of squared residuals

S = \displaystyle\sum_{i=1}^{n}(y_i - \alpha - \beta x_i)^2

. By minimising

S

with respect to

\alpha

and

\beta

using partial differentiation, prove that the least squares estimates satisfy

\hat{\beta} = \frac{S_{xy}}{S_{xx}}, \qquad \hat{\alpha} = \bar{y} - \hat{\beta}\,\bar{x},

where

S_{xy} = \displaystyle\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})

and

S_{xx} = \displaystyle\sum_{i=1}^{n}(x_i - \bar{x})^2

. [5 marks]

(b)

Using the data above, calculate

\hat{\alpha}

and

\hat{\beta}

, giving your answers correct to three significant figures. [2 marks]

(c)

The eight residuals

e_i = y_i - \hat{y}_i

, computed using your regression line from (b), are given below (rounded to two decimal places).

\hat{y}_i

—

1.64

—

3.01

—

4.38

—

5.30

—

6.67

—

8.04

—

9.41

—

10.78

e_i

—

0.16

—

0.19

—

0.12

—

-0.20

—

0.13

—

-0.14

—

-0.11

—

-0.18

By referring to the residuals above, evaluate the validity of the linear regression model with respect to linearity and homoscedasticity. Hence state, with a reason, whether a linear model or a transformed model is more appropriate for these data. [3 marks]

Solutions

34ChallengeLAQRegression analysis10 marksPaper 3~15 min

Data

\sum x_i = 62.8,\quad \sum y_i = 72.3,\quad \sum x_i^2 = 411.34,\quad \sum y_i^2 = 597.77,\quad \sum x_i y_i = 420.69

Define

S_{xx} = \sum x_i^2 - n\bar{x}^2

,

S_{yy} = \sum y_i^2 - n\bar{y}^2

,

S_{xy} = \sum x_i y_i - n\bar{x}\bar{y}

.

An environmental scientist studies the relationship between soil pH (

x

) and the concentration of a heavy metal contaminant (

y

, in ppm) across 10 soil samples. pH,

x

— 4.2 — 4.8 — 5.3 — 5.7 — 6.1 — 6.5 — 6.9 — 7.3 — 7.8 — 8.2 Concentration,

y

(ppm) — 12.5 — 10.8 — 9.2 — 8.1 — 7.0 — 6.2 — 5.5 — 4.9 — 4.3 — 3.8 The scientist fits the linear regression model

\hat{y} = \hat{\alpha} + \hat{\beta}x

using least squares. The following summary statistics are

(a)

Starting from

R^2 = 1 - \dfrac{SS_{\text{res}}}{SS_{\text{tot}}}

, where

SS_{\text{res}} = \sum(y_i - \hat{y}_i)^2

and

SS_{\text{tot}} = \sum(y_i - \bar{y})^2

, prove that

R^2 = \frac{S_{xy}^2}{S_{xx}\,S_{yy}}

In your proof, you must: - substitute

\hat{\alpha} = \bar{y} - \hat{\beta}\bar{x}

to express

SS_{\text{res}}

in terms of

S_{xx}

,

S_{yy}

,

S_{xy}

, and

\hat{\beta}

; - substitute the least-squares estimate

\hat{\beta} = \dfrac{S_{xy}}{S_{xx}}

and simplify; - state the condition

S_{xy}

,

S_{xx}

,

S_{yy}

under which

R^2 = 1

, and give its geometric interpretation. [5 marks]

(b)

Calculate

R^2

using the summary statistics. Give your answer to three significant figures and interpret its value in context. [3 marks]

(c)

The scientist wishes to predict the contaminant concentration at a pH of

5.0

. - Calculate the predicted concentration

\hat{y}

at

x = 5.0

. - Evaluate whether this prediction is reliable, referring to both the value of

R^2

and the range of the data. [2 marks]

Solutions

35ChallengeLAQMeasures of central tendency (mean, median, mode)10 marksPaper 3~15 min

A large-scale agricultural study investigates the relationship between a new organic fertilizer and crop yield. For

100

randomly selected farms, the annual yield (in tonnes per hectare) is recorded. Each farm uses either the organic fertilizer (Group A) or a conventional synthetic fertilizer (Group B). - Group A (

50

farms): mean

= 4.82

, standard deviation

= 0.75

, median

= 4.90

, mode

= 5.10

(all in tonnes per hectare) - Group B (

50

farms): mean

= 4.41

, standard deviation

= 0.82

, median

= 4.35

, mode

= 4.20

(all in tonnes per hectare)

(a)

Show that the combined mean yield for all

100

farms is

4.615

tonnes per hectare. [2 marks]

(b)

Calculate the difference (mean

-

median) for Group A and for Group B. [2 marks]

(c)

A researcher claims: "The median is a more representative measure of central tendency than the mean for comparing these two groups, because the mean is sensitive to extreme values." Using your results from part (b) and the concept of skewness, evaluate this claim. [3 marks]

(d)

A new farm, Farm X, which uses the organic fertilizer and has a yield of

6.50

tonnes per hectare, is added to Group A. (i) Calculate the new mean of Group A (now

51

farms). [1]
(ii) Determine the new median of Group A and hence show that the mean increases by more than the median increases. [2 marks]

Solutions

36ChallengeLAQMeasures of spread (range, variance, standard deviation)12 marksPaper 3~18 min

Data

\mu = \dfrac{\sum x}{n}

,

\quad\sigma^2 = \dfrac{\sum(x-\mu)^2}{n}

A biologist studies wing lengths (in mm) of a butterfly species. She collects two independent random samples from different habitats, Sample A and Sample B, summarised as follows:

\text{Sample A: } n_A = 10,\quad \sum x_A = 240,\quad \sum x_A^2 = 5820

\text{Sample B: } n_B = 12,\quad \sum x_B = 288,\quad \sum x_B^2 = 7008

(a)

Prove that the variance of a data set can be expressed as

\sigma^2 = \dfrac{\sum x^2}{n} - \mu^2

, where

\mu

is the mean. [2 marks]

(b)

Using the result from part (a), calculate the mean and variance for each sample. Give your answers correct to three significant figures. [4 marks]

(c)

The biologist combines both samples into a single data set. Calculate the mean and standard deviation of the combined data set, correct to three significant figures. [3 marks]

(d)

The biologist claims that combining two samples always produces a combined standard deviation smaller than the standard deviation of either individual sample. Using your results and a mathematical argument, evaluate whether this claim is valid in general. [3 marks]

Solutions

37ChallengeLAQBasic probability concepts and rules10 marksPaper 3~15 min

A medical research team investigates a diagnostic test for a rare disease. The disease affects

0.2\%

of the population. - If a person has the disease, the test returns a positive result

95\%

of the time. - If a person does not have the disease, the test returns a negative result

99.8\%

of the time. Let

D

be the event "a randomly selected person has the disease" and

T

be the event "the test returns a positive result".

(a)

Show that

P(T) \approx 0.00390

. [3 marks]

(b)

Calculate

P(D \mid T)

, the probability that a person who tests positive actually has the disease. [2 marks]

(c)

Evaluate whether this test is suitable for widespread population screening. In your answer, refer to your result from (b), the concept of false positives, and the effect of disease prevalence. The following formulae are provided:

P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \qquad P(A') = 1 - P(A)

P(T) = P(D)\,P(T \mid D) + P(D')\,P(T \mid D')

[5 marks]

Solutions

38ChallengeLAQBasic probability concepts and rules10 marksPaper 3~15 min

Three independent assembly lines, A, B, and C, produce identical components at a large manufacturing facility. Line A produces

30\%

of total output, line B produces

45\%

, and line C produces the remaining

25\%

. The probability that a component is defective is

0.02

for line A,

0.03

for line B, and

0.05

for line C.

(a)

Using the law of total probability, calculate

P(D)

, the probability that a randomly selected component is defective. [3 marks]

(b)

Given that a randomly selected component is found to be defective, calculate the probability that it came from each of lines A, B, and C. [4 marks]

(c)

A production manager claims that line B is the most likely source of a defective component because it has the largest share of total output. Evaluate this claim. [3 marks]

Solutions

39ChallengeLAQBinomial distribution and its properties10 marksPaper 3~15 min

A quality control engineer investigates the reliability of a production line for electronic components. The line is considered to be operating correctly if the proportion of defective components is at most

p = 0.05

. A random sample of

n = 20

components is selected. The number of defective components,

X

, is modelled as

X \sim B(20,\, 0.05)

. The binomial PMF is

P(X = x) = \dbinom{n}{x} p^x (1-p)^{n-x}

and the identity

\displaystyle\sum_{x=0}^{n} \binom{n}{x} a^{x} b^{n-x} = (a+b)^n

may be used without proof.

(a)

Using the PMF and the identity above, show that

E(X) = np

for

X \sim B(n, p)

. [4 marks]

(b)

The engineer shuts down the line if

X \geq 3

. Calculate the probability that the line is shut down when it is actually operating correctly. [3 marks]

(c)

In practice, components are produced sequentially on a single machine that gradually wears over a shift. A colleague suggests replacing the binomial model with one in which the probability of a defect increases linearly with the position of the component in the sequence. Evaluate whether the binomial model or the colleague's proposed modification is more appropriate for this context. Justify your answer with reference to the assumptions of the binomial distribution. [3 marks]

Solutions

40ChallengeLAQBinomial distribution and its properties10 marksPaper 3~15 min

A pharmaceutical company is testing a new vaccine. In a clinical trial, a random sample of

n = 20

volunteers each receive the vaccine. The probability that any given volunteer develops immunity is

p

, assumed constant and independent across volunteers. The number of volunteers who develop immunity,

X

, is modelled as

X \sim B(20,\, p)

.

(a)

(i) State the four conditions required for a binomial distribution to be a valid model for

X

. [2]
(ii) Starting from the probability of a single ordered sequence of outcomes, derive the probability mass function

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0, 1, \ldots, n,

and verify that

\displaystyle\sum_{k=0}^{n} P(X = k) = 1

. [3 marks]

(b)

(i) The trial results show that exactly 15 of the 20 volunteers developed immunity. Assuming the company's claim that

p = 0.85

, calculate: -

P(X = 15)

-

P(X \leq 15)

Give your answers correct to three significant figures. [2]
(ii) The company claims

p = 0.85

. Using a one-tailed hypothesis test at the

5\%

significance level, evaluate whether the observed result of

X = 15

provides sufficient evidence to reject this claim. In your answer, state the null and alternative hypotheses, identify the critical region, and justify your conclusion with reference to your calculated probabilities. [3 marks]

Solutions

Statistics and Probability — Free Maths AI HL Practice Questions