Statistics & Probability Exam Questions

The following are exam style questions for the sub-topics in the functions unit. There are questions with video solutions here for here for Statistical concepts, cumulative frequency and box plots, central tendency and dispersion, bivariate data and linear correlation, probability, discrete random variables, binomial distribution, normal distribution, spearman's rank and chi squared tests!

Statistical Concepts - Exam Style Questions

The following questions are based on IB exam style questions from past exams. You should print these off (from the document at the top) and try to do these questions under exam conditions. Then you can check your work with the video solution.

Question 1

A headteacher wants to find out in how many subjects students have Internal Assessments (IA). He conducts a survey:

Number of subjects with IAs	0	1	2	3	4	5	6
Number of students	15	20	20	32	30	17	16

State whether the data is discrete or continuous (1)
It was not possible to ask the whole school, so he conducts a survey. The school is 55% boys and 45% girls, so he makes sure his sample includes 55% boys and 45% girls. What type of sampling does he use? (1)
He then wants to investigate further, so asks all the students that are studying Geography. What type of sampling is this? (1)
The coursework marks of the students studying Geography are listed below.
1. Identify the outlier
2. By removing the outlier, calculate the new mean (3)

45%

39%

46%

29%

39%

43%

42%

48%

39%

41%

(6 marks)

Video Solution

Question 2

A reservoir contains 3 types of fish. There are estimated to be 700 carp, 500 bass and 300 pike. A survey of the fish is taken, and a sample of 50 is used.

a)Explain why a simple random sample would not be appropriate (1)

b)Name a potential sampling method that could be used (1)

c) Give one advantage and one disadvantage of this method (2)

d) Explain briefly how you could conduct this survey (3)

(7 marks)

Video Solution

Question 3

A survey of hours taken revising for a test is conducted by a teacher. She asks all the students in her Grade 12 class: “Revising hard will make you do better in this exam. How many hours did you revise?”

a)What type of sampling did the teacher use? (1)

b) Her question might have made her results unreliable. Suggest what change could be made, and why. (2)

The table shows the answers that her students gave.

c)Showing your working, identify the outlier and explain whether or not you think the value should be included in the teacher’s analysis. (5)

(8 marks)

Video Solution

Cumulative frequency and Box plots - Exam Style Questions

The following are exam style questions on this topic that you should try on your own before checking against the video solution.

Question 1

See the solution in the hidden box

Question 2

See the solution in the hidden box

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Question 3

See the solution in the hidden box

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Central tendency and dispersion - Exam Style Questions

Question 1

The table below shows the number of languages a class of 25 children speak

No of languages	1	2	3	4	5
Frequency	7	p	6	3	1

Find the value of p

(1)

Use your graphical display calculator to find
1. The median number of languages spoken
2. The mean number of languages spoken
3. The standard deviation of the number of languages spoken

(3)

Showing your working, identify any outliers in this data.

(2)

(6 marks)

Video solution

Question 2

The mean of the ten numbers listed below is 11.

12, 9, 14, 10, 19, 8, 13, 11, a, b

(a) Write down an equation in terms of a and b.

(2)

The mode of these ten numbers is nine and a is less than b.

(b) Write down the value of

(i) a;

(ii) b.

(2)

(Total 6 marks)

Video solution

Question 3

10 dogs were weighed and their results written down in kilograms. The following box and whisker plot was used to represent their weights.

a) Write down

(i) the median weight;

(ii) the 75th percentile weight;

(iii) the range of marks.

(4)

b) How many dogs weighed more than 39 kg?

(2)

The mean weight of the dogs was found to be 40.5kg, and the standard deviation 13.8. After being put on a strict exercise routine, all the dogs had lost 2kg the next time they were weighed.

c) What is the new mean and standard deviation of the dogs?

(2)

(Total 8 marks)

Video solution

Bivariate data & linear correlation - Exam Style Questions

Question 1

Video solution

Question 2

Video solution

Question 3

Video solution

Question 4

Mikel believes there is a linear relationship between time spent studying and performance in an IB exam.To test this, he collected data on ten of his classmates. He recorded the time spent studying, x, alongside the percentage they received in their most recent exam, y.

The following table shows the data he found.

Hours studying, x.	4	5	4	3	7	2	7	9	10	11
Percentage Score, y	58	60	43	40	65	35	70	75	75	80

a. Why does Pearson’s Product Moment Correlation Coefficient appear to be valid for this data?

b. Find the value of Pearson’s Product Moment Correlation Coefficient, r.

c. Comment on the strength of the correlation

Mikel decides to study for 8 hours for the test.

d. By using the equation of the regression line, what score could he expect to get?

Video Solution

Probability - Exam Style Questions

Question 1

Video solution

Question 2

Video solution

Question 3

Video solution

Question 4

Video solution

Question 5

Video solution

Question 6

Ross and Dylan are playing padel. The probability that Dylan wins the first game is 0.6. If Ross wins the game the probability that he wins the next game is 0.9. If Dylan wins the game, the probability that he wins the next game is 0.5. Let R be the probability that Ross wins a game, and D be the probability that Dylan wins the game.

a. Complete the tree diagram to represent the padel game.

b. Find the probability that Dylan wins both games

c. Find the probability that Ross wins the second game, given that he wins the first game.

Video Solution

Discrete Random Variables - Exam Style Questions

Question 1

Jenni plays a game using a spinner. The values on the spinner are labelled -5,0,3 and 10. The score for the game, X, is the value the spinner lands on. The following table shows the probability distribution for X.

Score X	-5	0	3	10
P(X=x)	0.29	a	0.39	0.19

What is the value of a?

(1)

What is Jenni’s expected score if she plays the game once?

(2)

Jenni plays this game at a fair, and wins the amount in $ that the spinner lands on.

What would be a fair price for Jenni to pay?

(1)

Jenni plays the game twice and adds up her scores. What is the probability that she gets a total score of 3?

(3)

(7 marks)

Video solution

Question 2

The following table shows the probability distribution of a discrete random variable A.

a	2	3	4	5
P(A=a)	0.3	x	y	0.2

Given that E(A)=3.4, find the value of x and y.

(4 marks)

Binomial Distribution - Exam Style Questions

Question 1

Ms Roberts teaches a Maths class with 17 students. 6 of them are working from school, and 11 of them are working remotely.

Each day, Ms Roberts chooses one student at random to make a presentation.

a. Find the probability that on a given day, Ms Roberts chooses someone who is working from home (1)

In September, Ms Roberts will teach 15 lessons with her class.

b. Find the probability that she chooses a student working remotely 8 times. (2)

c. Find the probability that she chooses a student working in school at least 9 times. (3)

(6 marks)

Video Solution

Question 2

The probability that my train is late on any given day is 0.3. What is the probability that it will be late at least three times between Monday and Friday? (4)

Video Solution

Question 3

An infectious virus is spreading through a school. The probability of a randomly selected student having the virus next week is 0.2. Mr Santos has a class of 24 students.

a. Calculate the probability that 3 or more students in Mr Santos’ class will have the virus next week. (3)

b. If more than 25% of the students have the virus next week, the exam will be cancelled. What is the probability that the exam is cancelled? (4)

(6 marks)

Question 4

A wind turbine has 15 different working components, and works as long as at least of them is working. The probability that an individual component fails in a given year is 0.9. Given that the probability of each individual component failing is independent of the others:

Find the probability that the wind turbine will fail in the first year. (2)