Statistics and Probability — Free IB Maths AA SL Practice Questions

1FoundationSAQ-SMeasures of central tendency (mean, median, mode)s5 marksPaper 1~8 min

A group of

8

students recorded the number of books they read during a holiday. The results are:

2,\ 3,\ 5,\ 6,\ 8,\ 10,\ 13

(a)

State the range of the data. [1 mark]

(b)

Find the median number of books read. [2 marks]

(c)

Find the mean number of books read. [2 marks]

Solutions

2MasterySAQ-SMeasures of central tendency (mean, median, mode)s10 marksPaper 1~15 min

The weights (in kg) of eight rugby players are:

73,\ 78,\ 81,\ 85,\ 90,\ 94,\ 97,\ x

The mean weight of the eight players is

84.875\,\text{kg}

.

(a)

Show that

x = 81

. [2 marks]

(b)

Hence find the median weight of the eight players. [3 marks] Correction note (self-verify): Clean integer mean requires clean sum. Let the seven given values be

73, 78, 81, 85, 90, 94, 97

with sum

= 598

. Mean

= \frac{598+x}{8}

. For

x=82

: mean

= 85

. Adopt this clean version below. The weights (in kg) of eight rugby players are:

73,\ 78,\ 81,\ 85,\ 90,\ 94,\ 97,\ x

The mean weight of the eight players is

85\,\text{kg}

. (a) Show that

x = 82

. [2 marks] (b) Hence find the median weight of the eight players. [3 marks]

Solutions

3ChallengeSAQ-LMeasures of spread (range, variance, standard deviation) s8 marksPaper 1~12 min

Data

\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}, \qquad s^2 = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}, \qquad s = \sqrt{s^2}

A biologist studies the growth of a plant species under controlled conditions. She measures the heights (in cm) of a random sample of 10 plants after 4 weeks:

22, \; 25, \; 28, \; 30, \; 32, \; 33, \; 35, \; 38, \; 42, \; 45

(a)

Determine the range of the sample. [1 mark]

(b)

Determine the sample variance of the heights, giving your answer correct to three significant figures. [3 marks]

(c)

The biologist decides to express all heights in millimetres rather than centimetres. Determine the sample variance of the heights measured in millimetres. [2 marks]

(d)

Analyse how the sample standard deviation would change if an 11th plant, with height equal to the mean of the original 10 plants, were added to the sample. Justify your answer without performing a full recalculation. [2 marks]

Solutions

4FoundationSAQ-SMeasures of central tendency (mean, median, mode)s5 marksPaper 1~8 min

The heights (in cm) of 7 plants in a greenhouse are:

30,\; 15,\; 22,\; 12,\; 25,\; 18,\; 20

(a)

State the median height. [1 mark]

(b)

A new plant of height

28\,\text{cm}

is added to the group. Calculate the new median height. [2 marks]

(c)

The mean height of the original 7 plants is

\frac{142}{7}\,\text{cm}

. Determine whether the mean or the median changes by the greater amount when the new plant is added. [2 marks]

Solutions

5MasterySAQ-SMeasures of central tendency (mean, median, mode)s5 marksPaper 1~8 min

The heights (in cm) of five plants are:

12,\, 15,\, 18,\, 21,\, 24

A sixth plant of height

h

cm is added.

(a)

Given that the mean height of the six plants is

19\,\text{cm}

, show that

h = 24

. [2 marks]

(b)

A different sixth plant of height

k

cm is added to the original five plants instead. The median height of these six plants is

17\,\text{cm}

. Determine the possible value(s) of

k

. [3 marks]

Solutions

6FoundationSAQ-SBasic probability rules and conceptss5 marksPaper 1~8 min

In a school survey, 60 students were asked if they play football (

F

) or basketball (

B

). The results show that 35 play football, 22 play basketball, and 12 play both sports.

(a)

Find the probability that a randomly chosen student plays neither sport. [2 marks]

(b)

Given that a student plays basketball, find the probability that they also play football. [2 marks]

(c)

Determine whether playing football and playing basketball are independent events. Justify your answer. [1 mark]

Solutions

7MasterySAQ-SBasic probability rules and conceptss5 marksPaper 1~8 min

In a school,

60\%

of students play basketball and

45\%

play volleyball. It is known that

30\%

of students play both sports.

(a)

Show that the probability that a randomly selected student plays neither sport is

0.25

. [3 marks]

(b)

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

Solutions

8ChallengeSAQ-LBasic probability rules and conceptss10 marksPaper 1~15 min

A bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn at random without replacement. Let

R

be the event that the first marble drawn is red. Let

B

be the event that the second marble drawn is blue.

(a)

Determine

\mathrm{P}(R \cap B)

. [2 marks]

(b)

Determine

\mathrm{P}(B)

. [3 marks]

(c)

Determine

\mathrm{P}(R \cup B)

. [2 marks]

(d)

Determine whether events

R

and

B

are independent. Justify your answer with a calculation, and explain what your result means in the context of this experiment. [3 marks]

Solutions

9FoundationSAQ-SBasic probability rules and conceptss5 marksPaper 1~8 min

Events

X

and

Y

are such that

P(X) = 0.6

,

P(Y) = 0.3

, and

P(X \cap Y) = 0.2

.

(a)

Find

P(X \cup Y)

. [2 marks]

(b)

Find

P(Y \cap X')

. [2 marks]

(c)

Determine whether

X

and

Y

are independent. Justify your answer. [1 mark]

Solutions

10MasterySAQ-SBasic probability rules and conceptss5 marksPaper 1~8 min

Events

A

and

B

are such that

P(A) = 0.4

,

P(B) = 0.5

, and

P(A \cup B) = 0.7

.

(a)

Show that

A

and

B

are independent. [3 marks]

(b)

Hence, find

P(A' \cap B')

. [2 marks]

Solutions

11FoundationSAQ-SHypothesis testing and confidence intervalss5 marksPaper 1~8 min

A factory produces metal rods with a nominal length of

100\,\text{cm}

. The production manager suspects the machine is malfunctioning and producing rods with a mean length different from

100\,\text{cm}

. A random sample of

50

rods gives a sample mean of

\bar{x} = 99.7\,\text{cm}

. The population standard deviation is known to be

\sigma = 0.5\,\text{cm}

. A hypothesis test is conducted at the

5\%

significance level.

(a)

State the null hypothesis

H_0

and the alternative hypothesis

H_1

. [1 mark]

(b)

Calculate the

p

-value for this test. [3 marks]

(c)

State the conclusion of the test, justifying your answer with reference to the

p

-value. [1 mark]

Solutions

12MasterySAQ-SHypothesis testing and confidence intervalss5 marksPaper 1~8 min

A manufacturer claims that the mean lifespan of a certain type of battery is

1200

hours. A consumer group suspects the mean lifespan is less than

1200

hours. They test a random sample of

50

batteries and find a sample mean lifespan of

1175

hours. The population standard deviation is known to be

80

hours. A hypothesis test is carried out at the

5\%

significance level.

(a)

State the null hypothesis

H_0

and the alternative hypothesis

H_1

for this test. [1 mark]

(b)

Calculate the value of the test statistic

z

, where

z = \dfrac{\bar{x} - \mu}{\sigma / \sqrt{n}}

. [2 marks]

(c)

The critical value for this test is

z_c = -1.645

. Calculate the

p

-value of the test and hence determine whether the consumer group's suspicion is supported by the evidence. Justify your answer. [2 marks]

Solutions

13ChallengeSAQ-LZ-scores and t-tests9 marksPaper 1~14 min

The heights of a certain species of plant follow a normal distribution. A botanist claims the mean height of this species is

52.0\,\text{cm}

. To test this claim, the botanist measures the heights (in cm) of a random sample of 12 plants:

49.2,\ 51.8,\ 47.5,\ 53.1,\ 50.4,\ 48.9,\ 52.6,\ 50.7,\ 49.8,\ 51.2,\ 48.3,\ 50.9

(a)

Calculate the sample mean

\bar{x}

and sample standard deviation

s

for these data. [4 marks]

(b)

State the null and alternative hypotheses for a two-tailed

t

-test of the botanist's claim. [1 mark]

(c)

Calculate the value of the

t

-test statistic. [2 marks]

(d)

The critical value for this test at the

5\%

significance level is

2.201

. Determine whether the evidence supports the botanist's claim. [2 marks]

Solutions

14FoundationSAQ-SHypothesis testing and confidence intervalss5 marksPaper 1~8 min

A pharmaceutical company claims that a new drug reduces blood pressure by an average of

12\,\text{mmHg}

. A medical researcher believes the actual mean reduction is less than

12\,\text{mmHg}

. A random sample of 40 patients yields a sample mean reduction of

11.2\,\text{mmHg}

. The population standard deviation is

3.5\,\text{mmHg}

. A hypothesis test is conducted at the

1\%

significance level.

(a)

State the null and alternative hypotheses for this test. [1 mark]

(b)

Calculate the

z

-test statistic. [2 marks]

(c)

State the critical value and hence determine whether the null hypothesis should be rejected. Justify your answer. [2 marks]

Solutions

15MasterySAQ-SHypothesis testing and confidence intervalss5 marksPaper 1~8 min

A school claims that the mean score on a standardised test for its students is

75

. A researcher believes the mean score is actually higher. A random sample of

36

students is taken and the sample mean score is

\bar{x} = 78

. The population standard deviation is known to be

\sigma = 12

. A hypothesis test is conducted at the

1\%

significance level.

z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}

(a)

Write down the null hypothesis

H_0

and the alternative hypothesis

H_1

. [1 mark]

(b)

Calculate the

z

-test statistic. [2 marks]

(c)

Determine the

p

-value for this test and state whether it is less than or greater than

0.01

. [1 mark]

(d)

The researcher concludes that the school's claim should be rejected. Evaluate whether this conclusion is justified, giving a reason based on your results. [1 mark]

Solutions

16MasterySAQ-SMeasures of central tendency (mean, median, mode)s6 marksPaper 2~9 min

A factory produces metal rods. The lengths, in centimetres, of a random sample of 8 rods are:

12.4,\ 13.1,\ 11.8,\ 12.7,\ 13.5,\ 12.2,\ 11.9,\ 13.4

(a)

Calculate the mean length of the rods in this sample. [2 marks]

(b)

Determine the median length of the rods in this sample. [2 marks]

(c)

A ninth rod of length

12.9\,\text{cm}

is added to the sample. Determine the new median and explain why it differs from the median found in part (b). [2 marks]

Solutions

17MasterySAQ-SMeasures of central tendency (mean, median, mode)s6 marksPaper 2~9 min

The heights, in cm, of 10 sunflowers in a garden are:

152,\ 148,\ 165,\ 159,\ 143,\ 171,\ 156,\ 162,\ 149,\ 155

(a)

Calculate the mean height of these sunflowers. [2 marks]

(b)

Determine the median height. [2 marks]

(c)

The gardener plants an eleventh sunflower with height

160\,\text{cm}

. Calculate the new median height of all eleven sunflowers. [2 marks]

Solutions

18MasterySAQ-SMeasures of central tendency (mean, median, mode)s6 marksPaper 2~9 min

The heights, in cm, of 12 sunflowers grown in a greenhouse are recorded below:

112,\ 118,\ 124,\ 126,\ 130,\ 132,\ 134,\ 138,\ 142,\ 146,\ 148,\ x

where

x

is the height of the twelfth sunflower.

(a)

Given that the mean height of the 12 sunflowers is

132\,\text{cm}

, calculate the value of

x

. [3 marks]

(b)

The median height of the 12 sunflowers is

133\,\text{cm}

. A thirteenth sunflower of height

133\,\text{cm}

is added to the greenhouse. Determine whether the mean height of all 13 sunflowers is greater than, equal to, or less than the median height of all 13 sunflowers. Justify your answer. [3 marks]

Solutions

19MasterySAQ-SMeasures of central tendency (mean, median, mode)s6 marksPaper 2~9 min

A fitness instructor records the number of push-ups completed by 10 athletes in one minute:

22,\ 25,\ 28,\ 30,\ 32,\ 35,\ 38,\ 40,\ 42,\ y

where

y

is the number of push-ups completed by the tenth athlete.

(a)

The mean number of push-ups is

33

. Calculate the value of

y

. [3 marks]

(b)

Using your value of

y

from part (a), find the median number of push-ups. [1 mark]

(c)

A different athlete replaces the tenth athlete, completing

y + 7

push-ups instead. Determine whether the median changes, and justify your answer. [2 marks]

Solutions

20MasterySAQ-SMeasures of spread (range, variance, standard deviation) s6 marksPaper 2~9 min

The heights (in cm) of 12 players in a basketball squad are:

185,\ 192,\ 188,\ 196,\ 190,\ 184,\ 198,\ 191,\ 187,\ 194,\ 189,\ 186

(a)

State the range of the heights. [1 mark]

(b)

Calculate the mean and variance of the heights. [3 marks]

(c)

The coach states: *"At least 75% of players have heights within

9\,\text{cm}

of the mean."* Using your results from (b), determine whether the coach's statement is correct. [2 marks]

Solutions

21MasterySAQ-SBasic probability rules and conceptss6 marksPaper 2~9 min

In a school, students can choose to study French, Spanish, or both. The probability that a randomly selected student studies French is

0.6

, the probability that they study Spanish is

0.5

, and the probability that they study both is

0.3

.

(a)

Calculate the probability that a randomly selected student studies at least one of these two languages. [2 marks]

(b)

Show that studying French and studying Spanish are not independent events. [2 marks]

(c)

Given that a student studies at least one of the two languages, calculate the probability that they study both. Hence determine whether, among students who study at least one language, studying both is more or less likely than studying neither French nor Spanish in the whole school. [2 marks]

Solutions

22MasterySAQ-SBasic probability rules and conceptss6 marksPaper 2~9 min

A bag contains

5

red marbles and

3

blue marbles. Two marbles are drawn at random 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)

Determine the probability that least one marble is red. Hence, or otherwise, find the probability that both marbles are red, given that least one marble is red. [2 marks]

Solutions

23MasterySAQ-SBasic probability rules and conceptss6 marksPaper 2~9 min

A bag contains

12

red marbles and

8

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

(a)

Calculate the probability that both marbles are the same colour. [4 marks]

(b)

Given that the two marbles are the same colour, calculate the probability that they are both blue. [2 marks]

Solutions

24MasterySAQ-SBasic probability rules and conceptss7 marksPaper 2~11 min

The Venn below shows the probabilities for events

A

,

B

, and

C

, where

x

is a positive constant.

(a)

Calculate the value of

x

. [2 marks]

(b)

Hence, determine

P\!\left((A \cup B) \cap C'\right)

. [2 marks]

(c)

Given that event

C

occurs, calculate the probability that exactly one of

A

or

B

also occurs. [3 marks]

Solutions

25MasterySAQ-SBasic probability rules and conceptss6 marksPaper 2~9 min

A bag contains 12 marbles: 5 red, 4 blue, and 3 green. Two marbles are drawn without replacement.

(a)

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

(b)

Given that the two marbles are the same colour, calculate the probability that they are both blue. [3 marks]

Solutions

26MasterySAQ-SZ-scores and t-tests6 marksPaper 2~9 min

A biologist measures the wingspan, in mm, of a sample of

12

butterflies from a certain species. The results are as follows:

48.2,\ 51.7,\ 49.5,\ 52.3,\ 50.1,\ 47.8,\ 50.9,\ 49.3,\ 51.6,\ 48.9,\ 51.4,\ 50.8

The biologist claims that the mean wingspan of this species is

50.0\,\text{mm}

.

(a)

Calculate the sample mean and the sample standard deviation of the wingspans. [2 marks]

(b)

Calculate the value of the

t

-test statistic for a one-sample

t

-test of the biologist's claim. [2 marks]

(c)

The test is conducted at the

5\%

significance level. State the critical value and determine whethere is sufficient evidence to reject the biologist's claim. [2 marks]

Solutions

27MasterySAQ-SZ-scores and t-tests6 marksPaper 2~9 min

A manufacturing company produces metal rods with a nominal length of

150.0\,\text{mm}

. The production process is known to produce rods with lengths that are normally distributed with a standard deviation of

2.5\,\text{mm}

. A quality control inspector takes a random sample of 8 rods and measures their lengths (in mm):

149.2,\quad 153.1,\quad 150.8,\quad 154.2,\quad 151.4,\quad 148.9,\quad 152.6,\quad 151.0

(a)

Calculate the sample mean length. [1 mark]

(b)

Calculate the

z

-score for the sample mean relative to the nominal length of

150.0\,\text{mm}

. [2 marks]

(c)

A two-tailed hypothesis test is conducted at the

5\%

significance level to determine whether the mean length of all rods differs from

150.0\,\text{mm}

. State the null and alternative hypotheses, state the critical

z

-values, and determine whethere is sufficient evidence to reject the null hypothesis. Justify your conclusion. [3 marks]

Solutions

28MasterySAQ-SZ-scores and t-tests6 marksPaper 2~9 min

A factory produces metal rods with lengths that are normally distributed. The rods are designed to have a mean length of

150\,\text{mm}

. A quality control inspector measures a random sample of

12

rods and records the following lengths (in mm):

149.2,\ 150.1,\ 148.9,\ 151.0,\ 149.8,\ 150.3,\ 148.7,\ 150.6,\ 149.5,\ 151.2,\ 148.5,\ 150.8

(a)

Calculate the sample mean

\bar{x}

and the unbiased estimate of the population variance

s^2

. [2 marks]

(b)

The inspector suspects the machine is under-filling (producing rods with mean length less than

150\,\text{mm}

). State suitable null and alternative hypotheses for a one-tailed

t

-test. [1 mark]

(c)

Calculate the value of the

t

-test statistic. [2 marks]

(d)

The critical value for this test at the

5\%

significance level is

t = -1.796

. Determine, with a reason, whether the evidence supports the inspector's claim. [1 mark]

Solutions

29MasterySAQ-SZ-scores and t-tests6 marksPaper 2~9 min

The heights of adult females in a certain country are normally distributed with mean

165\,\text{cm}

and standard deviation

6\,\text{cm}

.

(a)

Calculate the probability that a randomly selected female has a height between

160\,\text{cm}

and

170\,\text{cm}

. [2 marks]

(b)

A random sample of

9

females is selected. Calculate the probability that the sample mean height is between

160\,\text{cm}

and

170\,\text{cm}

. [2 marks]

(c)

A female has a height of

178\,\text{cm}

. Calculate her

z

-score. [1 mark]

(d)

Interpret the

z

-score found in part (c) in the context of this population. [1 mark]

Solutions

30MasterySAQ-SZ-scores and t-tests6 marksPaper 2~9 min

A coffee shop claims that the mean caffeine content of its large cappuccino is

150\,\text{mg}

. A consumer group suspects the mean caffeine content is actually higher. They take a random sample of

12

large cappuccinos and measure the caffeine content, in mg:

156,\ 152,\ 148,\ 161,\ 155,\ 149,\ 153,\ 158,\ 151,\ 154,\ 157,\ 150

Assume caffeine content is normally distributed.

(a)

Calculate the sample mean

\bar{x}

and the sample standard deviation

s_{n-1}

. [2 marks]

(b)

State appropriate null and alternative hypotheses for a one-tailed

t

-test, then calculate the value of the test statistic

t

. [3 marks]

(c)

The critical value at the

5\%

significance level is

t_{0.05,\,11} = 1.796

. State the conclusion of the test, justifying your answer and interpreting it in the context of the coffee shop's claim. [1 mark]

Solutions

Statistics and Probability — Free Maths AA SL Practice Questions