LemonmathSelf-study sheet

Grade 12 · Lesson 7.8

Revision: statistics and probability

Name: Date: Time spent: min

Goals

Calculate the mean, median, quartiles and mode of grouped data.
Calculate the range, interquartile range, variance and standard deviation, and use them to compare data sets.
Use the addition rule, the multiplication rule, conditional probability and independent events.
Use tree diagrams, the law of total probability and Bayes' theorem in real-life problems.

Exercises

Exercise 1 Multiple choice

The waiting times (minutes) of $10$ customers are grouped: the class $[0, 4)$ has $3$ people, $[4, 8)$ has $5$ people and $[8, 12)$ has $2$ people. The mean of the data is

A. $5.6$ . B. $6$ . C. $18.67$ . D. $3.6$ .

Exercise 2 Multiple choice

The test marks of $50$ students are given in the table. The median of the data (to 2 decimal places) is

Mark	$[5, 6)$	$[6, 7)$	$[7, 8)$	$[8, 9)$	$[9, 10)$
Number of students	$4$	$8$	$16$	$14$	$8$

A. $7.81$ . B. $7.5$ . C. $6.81$ . D. $7.93$ .

Exercise 3 True/false

The masses (kg) of $20$ students: the class $[45, 50)$ has $5$ students, $[50, 55)$ has $10$ and $[55, 60)$ has $5$ .

a) The mode of the data lies in the class $[55, 60)$ .

b) The lower quartile is $Q_1=50$ .

c) The variance of the data is $12.5$ .

d) The standard deviation of the data is $12.5$ kg.

Exercise 4 Short answer

Find the interquartile range of the following grouped data (to 1 decimal place).

Class	$[0, 10)$	$[10, 20)$	$[20, 30)$	$[30, 40)$
Frequency	$5$	$12$	$15$	$8$

Exercise 5 Multiple choice

$A$ and $B$ are independent events with $P(A)=0.4$ and $P(B)=0.5$ . The probability $P(A\cup B)$ is

A. $0.9$ . B. $0.2$ . C. $0.7$ . D. $0.3$ .

Exercise 6 Short answer

A box holds $10$ cards, $4$ of which are red. Two cards are taken one after the other, without replacement. Find the probability that both cards are red (to 2 decimal places).

Exercise 7 Short answer

In a company's inbox, $30\%$ of the emails are spam. A filter marks $90\%$ of spam emails as "spam" and also $5\%$ of normal emails. An email is marked "spam". Find the probability that it really is spam (to 2 decimal places).

Exercise 8 Written answer

In a city, the probability of rain on a given day is $0.3$ . If it rains, the probability that Minh is late for school is $0.4$ ; if it does not rain, that probability is $0.1$ . Find the probability that Minh is late on a randomly chosen day. Given that Minh is late today, find the probability that it is raining today.

Exercise 9 Short answer

Find the standard deviation of the following grouped data.

Class	$[2, 4)$	$[4, 6)$	$[6, 8)$	$[8, 10)$
Frequency	$4$	$8$	$6$	$2$

Exercise 10 Short answer (challenge)

A shop buys components from three suppliers X, Y, Z in the proportions $50\%$ , $30\%$ , $20\%$ . The defect rates of X, Y, Z are $1\%$ , $2\%$ , $3\%$ respectively. A component chosen at random is found to be faulty. Find the probability that it came from Z (to 2 decimal places).

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. A set of grouped data has the class

[10, 20)

with frequency

2

[20, 30)

with frequency

5

and

[30, 40)

with frequency

3

. The mean of the data is

25

.B.

26

.C.

86.67

.D.

21

Question 2. For the data in the previous question, the class containing the upper quartile

Q_3

[10, 20)

.B.

[20, 30)

.C.

[30, 40)

.D. It cannot be determined.

Question 3. Let

A

B

be events with

P(B)>0

. The probability of

A

given that

B

has happened is

P(A\mid B)=\dfrac{P(A\cap B)}{P(A)}

.B.

P(A\mid B)=P(A)\cdot P(B)

.C.

P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}

.D.

P(A\mid B)=\dfrac{P(A)}{P(B)}

Question 4. Given

P(A)=0.3

and

P(B\mid A)=0.5

, the value of

P(A\cap B)

0.15

.B.

0.8

.C.

0.6

.D.

0.2

Part II. True or false

Question 1. A bag holds

5

red and

3

white marbles. Two marbles are taken one after the other, without replacement.

a) The probability that the first marble is red is $\dfrac58$ .	T / F
b) The probability that the second marble is red, given that the first is red, is $\dfrac47$ .	T / F
c) The probability that both marbles are red is $\dfrac{5}{14}$ .	T / F
d) The probability that the second marble is red is $\dfrac47$ .	T / F

Part III. Short answer

Question 1. A set of grouped data has the class

[0, 2)

with frequency

3

[2, 4)

with frequency

4

and

[4, 6)

with frequency

3

. Find its standard deviation (to 2 decimal places).

Question 2. Given

P(B)=0.4

P(A\mid B)=0.5

and

P(A\mid\overline B)=0.25

, find

P(B\mid A)

(to 2 decimal places).

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: $\bar x=\dfrac{\sum n_ix_i}{n}$ , where $x_i$ are the class midpoints.

Steps:

The midpoints are $2$ , $6$ , $10$ .
Find the sum of $n_ix_i$ and divide by $10$ .

Exercise 2.

What you need: the median class is the first class with cumulative frequency $\ge\dfrac n2$ .

Steps:

Cumulative frequencies: $4$ , $12$ , $28$ , $42$ , $50$ ; $\dfrac n2=25$ .
Substitute into the formula with $C=12$ , $n_m=16$ .

Exercise 3.

What you need: the mode lies in the class with the highest frequency; $s^2=\dfrac{\sum n_i(x_i-\bar x)^2}{n}$ and $s=\sqrt{s^2}$ .

Steps:

The midpoints are $47.5$ , $52.5$ , $57.5$ and the mean is $\bar x=52.5$ .
Cumulative frequencies $5$ , $15$ , $20$ ; $\dfrac n4=5$ .

Exercise 4.

What you need: $\Delta_Q=Q_3-Q_1$ ; the positions of $Q_1$ , $Q_3$ are $\dfrac n4$ , $\dfrac{3n}{4}$ .

Steps:

$n=40$ ; cumulative frequencies $5$ , $17$ , $32$ , $40$ .
Position $10$ is in $[10, 20)$ and position $30$ is in $[20, 30)$ .

Exercise 5.

What you need: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ and $P(A\cap B)=P(A)P(B)$ .

Steps:

Find $P(A\cap B)$ .
Substitute into the addition rule.

Exercise 6.

What you need: $P(A\cap B)=P(A)\cdot P(B\mid A)$ .

Steps:

$P(\text{first red})=\dfrac{4}{10}$ .
After one red card is taken, $3$ of the $9$ remaining cards are red.

Exercise 7.

What you need: the law of total probability and Bayes' theorem.

Steps:

Find $P(\text{marked})=0.3\cdot0.9+0.7\cdot0.05$ .
Divide the branch "spam and marked" by that result.

Exercise 8.

What you need: draw a tree diagram with the branches "rain" and "no rain".

Steps:

Use the law of total probability for the event "late".
Use Bayes' theorem for "rain given late".

Exercise 9.

What you need: $s^2=\dfrac{\sum n_ix_i^2}{n}-\bar x^2$ .

Steps:

The midpoints are $3$ , $5$ , $7$ , $9$ ; find $\bar x$ .
Find the mean of the squares, subtract $\bar x^2$ , then take the square root.

Exercise 10.

What you need: the law of total probability with the three cases X, Y, Z; Bayes' theorem.

Steps:

Find $P(\text{faulty})$ as the sum of three branches.
Divide the branch for Z by $P(\text{faulty})$ .

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