This document provides an outline for a lecture series revising key concepts for Test B, including: - Pythagoras' theorem, trigonometry, sine and cosine rules, and calculating triangle areas. - Probability, probability trees, and examples calculating probabilities of dice rolls. - Descriptive statistics like mode, median, interquartile range, mean, absolute deviation, and standard deviation. - Hypothesis testing using z-tests, t-tests, and chi-squared tests; including setting up hypotheses, finding critical values, calculating test statistics, and making conclusions. The revision is in preparation for Standard Track Test B which will be held the week of April 21st.

1. Lecture 13 - Revision for Test B
C2 Foundation Mathematics (Standard Track)
Dr Linda Stringer Dr Simon Craik
l.stringer@uea.ac.uk s.craik@uea.ac.uk
INTO City/UEA London

2. Outline
Lecture 8
Pythagoras’ Theorem and Trigonometry
Sine and Cosine rules, Areas
Lecture 9
Probability
Lecture 10
Mode, Median and Interquartile range
Mean, Absolute deviation, Standard deviation
Z-test
Lecture 11
T-test
Lecture 12
χ2 -test

3. Pythagoras’ Theorem
Only works in right-angled triangles.
a
b
c
a2
+ b2
= hyp2

4. sin, cos and tan ratios
Only works in right-angled triangles.
opp
adj
hyp A
tan(A) =
opp
adj
cos(A) =
adj
hyp
sin(A) =
opp
hyp

5. Angle rule
Works in any triangle.
a
bc
B C
A
A + B + C = 180

6. Sine rule
Works in any triangle.
a
bc
B C
A
a
sin A
=
b
sin B
=
c
sin C

7. Cosine rule
Works in any triangle.
a
bc
B C
A
a2
= b2
+ c2
− 2bc cos(A)

8. Area
Works in any triangle.
a
bc
B C
A
Area =
1
2
ab sin(C)

9. Probability
The sum of all probabilities is 1.
The probability an event A happens is denoted P(A) and is
a number between 0 and 1.
The probability A does not happen is 1 − P(A).
The probability A and B happens is P(A) × P(B).
The probability A or B happens is P(A) + P(B).

10. Probability trees
You can use a probability tree when there is more than one
event.
Draw different tiers of branches for different events, and
write the probability next to the branch.
Multiply probabilities along the branches (AND)
Add up the probabilities at the end of each branch (OR)
The total of all the probabilities at the end of the branches
should be 1
The probabilities for the second tier may be the same as
for the first tier, or they may be different

11. Probability tree - roll a fair die twice (or roll two fair
dice)
What is the probability that I roll two fours?
What is the probability that I roll no fours?
What is the probability that I roll exactly one four?
What is the probability that I roll at least one four?
First roll Second roll
P(4, 4) = 1
6 × 1
6 = 1
36
P(4, ¬4) = 1
6 × 5
6 = 5
36
P(¬4, 4) = 5
6 × 1
6 = 5
36
P(¬4, ¬4) = 5
6 × 5
6 = 25
36
¬4
¬4
5/6
41/6
5/6
4
¬4
5/6
41/6
1/6

12. Mode
The mode is the most frequent object in your data.
X = 5 8 -3 7 8 2 8
The mode of X is 8.
Y = 0 2 1 6 5 1 2 3
The modes of Y are 1 and 2.

13. Median
The median is the middle data.
X = 5 8 -3 7 8 2 8
Order the data
X = -3 2 5 7 8 8 8
7 × 1/2 = 3.5
Round up to 4.
The median of X is the fourth term 7.

14. Median
Y = 0 2 1 6 5 1 2 3
Order the data
Y = 0 1 1 2 2 3 5 6
8 × 1/2 = 4
Add the fourth term and the fifth term and divide by 2.
The median of Y is 2+2
2 = 2.

15. Interquartile range
First order the data, with values increasing from left to right.
We want to find two values: the first quartile Q1 and the
third quartile Q3.
Let n be the size of the data set (the number of values).
To find Q1 we multiply n by 1
4 . If n
4 is an integer (whole
number) then Q1 is the midpoint of the (n
4 )th value and the
(n
4 + 1)th value
If n
4 is not an integer then round it up to the nearest integer.
Q1 is the corresponding value.
To find Q3 we multiply n by 3
4 . If 3n
4 is an integer then Q3 is
the midpoint of the (3n
4 )th value and the (3n
4 + 1)th value
If 3n
4 is not an integer then round it up to the nearest
integer. Q3 is the corresponding value.
The interquartile range is Q3 − Q1.

16. Interquartile range
The interquartile range is the middle half of the data. Take the
ordered data
X = -3 2 5 7 8 8 8
Find Q1 :
7 × 1/4 = 1.75
Round up to 2.
The first quartile if X is the second term, Q1 = 2.
Find Q3 :
7 × 3/4 = 5.25
Round up to 6.
The third quartile if X is the sixth term, Q3 = 8.
The interquartile range is Q3 − Q1 = 8 − 2 = 6

17. Interquartile range
The interquartile range is the middle half of the data. Take the
ordered data
Y = 0 1 1 2 2 3 5 6
Find Q1 :
8 × 1/4 = 2
Add the second term and the third term and divide by 2.
The first quartile of X is Q1 = 1+1
2 = 1.
Find Q3 :
8 × 3/4 = 6
Add the sixth term and the seventh term and divide by 2.
The third quartile of X is Q3 = 3+5
2 = 4.
The interquartile range is Q3 − Q1 = 4 − 1 = 3

18. Mean
The mean is the average of the data.
¯x =
x
n
5 8 -3 7 8 2 8
¯x =
5 + 8 + (−3) + 7 + 8 + 2 + 8
7
=
35
7
= 5

19. Absolute deviation
The absolute deviation is the average distance of the data from
the mean.
AD =
|x − ¯x|
n
5 8 -3 7 8 2 8
Calculate |x − ¯x| (remember that we calculated ¯x = 5)
0 3 8 2 3 3 3
Add these values together and divide by the number of values
in the list
AD =
0 + 3 + 8 + 2 + 3 + 3 + 3
7
=
22
7

20. Standard deviation
5 8 -3 7 8 2 8
The standard deviation, σ, is a measure of spread.
First find the variance, σ2.
σ2
=
x2
n
− ¯x2
Calculate x2:
25 64 9 49 64 4 64
Add these values together, divide by the size of the data and
subtract the mean squared.
σ2
=
279
7
− 52
=
104
7
The standard deviation is the square root of the variance
σ = 3.85 to 2 d.p.

21. Structure of hypothesis tests
The Z-test, the T-test and the χ2-test all have the same
structure
Hypotheses (H0 and H1)
Critical value (look it up in a table, and for the Z-test and
T-test add a sketch showing rejection regions )
Test statistic (calculate)
Decision (accept/reject H0)
Conclusion (write a sentence)

22. The Z-test
The question will tell you the population mean µ and the
standard deviation of the population σ. It may also give you
information for your alternative hypothesis.
Write your null hypothesis H0 and your alternative
hypothesis H1.
Read your critical value off the table.This will depend on
your significance level and your alternative hypothesis.
You will be given a sample mean ¯x and the size of the
sample data n. Calculate your test statistic.
Z =
¯x − A
σ/
√
n
Compare the test statistic and the critical value. Decide to
accept or reject the null hypothesis. Write a concluding
sentence.

23. Example
The National Association of Florists say the perfect
sunflower should be 90cm tall with a standard deviation of
9.5cm. We grow 100 sunflowers with a mean height of 92
cm. We want to see whether or not our sunflowers are
perfect to a 5% level of significance.
H0 : µ = 90.
H1 : µ = 90.
We are doing a 2-tailed test. Reading off our table we get
critical values of −1.96 and 1.96.
The test statistic is 92−90
9.5/
√
100
= 2.11 to 2 d.p.
The test statistic 2.11 is bigger than the critical value 1.96
so we decide to reject the null hypothesis.
We conclude that our sunflowers are not perfect.

24. The T-test
The question will tell you the population mean µ. It will also
give you a small table of size n with some data. It may also
give you information for your alternative hypothesis.
Write your null hypothesis H0 and your alternative
hypothesis H1.
Read your critical value off the table.This will depend on
your significance level, your alternative hypothesis and the
degree of freedom of your data.
You will calculate the sample mean ¯x and the sample
standard deviation s. Calculate your test statistic.
T =
¯x − A
s/
√
n
Compare the test statistic and the critical value. Decide to
accept or reject the null hypothesis. Write a concluding
sentence.

25. Example
The National Association of Florists say the perfect
sunflower should be 90cm tall. We grow 8 sunflowers.
90 93 87 86 50 92 95 93
We want to see whether our sunflowers are too short at a
5% level of significance.
H0 : µ = 90.
H1 : µ < 90.
We are doing a 1-tailed test, our degrees of freedom is
8 − 1 = 7. Reading off our table we get a critical value of
−1.90.
The sample mean is 86 and the sample standard deviation
is 13.01. It follows the test statistic is −0.87.
The test statistic −0.87 is bigger than the critical value
−1.90 so we decide to accept the null hypothesis.
We conclude that our sunflowers are perfect.

26. The χ2
-test
The question will give you a table.
Write your null hypothesis H0 that the variables are
independent and your alternative hypothesis H1 that the
variables are dependent.
Read your critical value off the table.This will depend on
your significance level and the degrees of freedom of your
data (n − 1)(m − 1).
Calculate the row and column totals of the observed table.
Calculate the expected table. Row total times column total
divide by grand total.
Calculate the residual table. Observed minus expected.
Calculate the χ2 table. Residual value squared divided by
the expected value.
Calculate the test statistic. Add all the values in the χ2
table.
Compare the test statistic and the critical value. Decide to
accept or reject the null hypothesis. Write a concluding
sentence.

27. Example
The Institute for Studies want to know if the weather and
peoples happiness is independent to a 1% level of
significance. They collect the following data.
Happy Indifferent Sad
Rain 4 9 20
Snow 15 2 10
Sun 25 12 3
H0 : The weather and people’s happiness are independent.
H1 : People’s happiness depends on the weather.
Calculate the totals.
Happy Indifferent Sad Row total
Rain 4 9 20 33
Snow 15 2 10 27
Sun 25 12 3 40
Column total 44 23 33 100

28. Example
Calculate the expected table (to 3 d.p.).
14.52 7.59 10.89
11.88 6.21 8.91
17.6 9.2 13.2
Calculate the residual table (to 3 d.p.).
-10.52 1.41 9.11
3.12 -4.21 1.09
7.4 2.8 -10.2

29. Example
Calculate the χ2 table (to 3 d.p.).
7.623 0.262 7.621
0.819 2.854 0.133
3.111 0.8 7.882
The test statistic is 31.10 to 2 d.p.
The degree of freedom is (3 − 1) × (3 − 1) = 4. Our critical
value is 13.28.
Our test statistic is greater than our critical value so we
decide to reject our null hypothesis.
People’s happiness depends on the weather.

30. Standard Track Test B will be held during the
week beginning 21 April