Quantitative Methods - Mock Question one (6 Marks) Three coins are tossed together. a) List all possible combinations of heads (H) and tails (T). (2 marks) b) What is the probability of having exactly one head? (2 marks) c) What is the probability of at least 2 heads? (2 marks) Question two (8 marks): In a game of selecting a 3-digit number, you pay 50p to play a game. (Each selected digit can be any number from 0 to 9 and may be selected more than once.) If you select the correct number, you win £250 a) How many different selections are possible? (2 marks) b) What is the probability of winning? (2 marks) c) If you win, what is your profit? (1 marks) a) d) Find the expected value of winning and say whether you would play this game. (3 marks) Question three (6 marks) The table below summarises the job applications made by some graduates during their last year: Was the application successful? Yes No Male graduates 263 148 Female graduates 178 122 If you were to randomly select a graduate: a) What is the probability that an application was not successful? (2 marks) b) What is the probability the application was made by a male graduate or was successful? (4 marks) Question four (20 marks): X is a random variable which represents the weight in kg of crates of apples in the local market. Assume it has a normal distribution with a mean of 50 and variance of 49. a) What proportion of the crates weigh over 80kg? (4 marks) b) 68% of the crates will weigh between [?, ?]. (2 marks) c) What proportion of the crates will weigh between 40kg and 53kg? Interpret it. (6 marks) d) A sample of 25 crates was randomly drawn. What is the probability that the sample mean is between 47kg and 54kg? (8 marks) Question five (15 marks): For a sample of 201 households from Norwich it was found that the average monthly expenditure on food was £500 with sample variance £2,500. a) Using the given formula construct a 95% confidence interval for the average monthly expenditure on food. (4 marks) b) You were informed that the monthly expenditure on food is £510; you believe that this is too high. Conduct a test in order to prove that you are right. (6 marks) c) You think that household monthly expenditure on food would be £505, test your guess. (5 marks) Question six (20 marks): The ‘golden ratio’ 1 to 0.618. It is claimed that rectangles whose sides are in this ratio are pleasing to the eye (i.e. they look nice). An experiment was conducted to see if this claim is true of the shape of paintings. That is, the ratio of the length to the width of a picture frame. Twenty people were asked to look at different sized picture frames and choose the most pleasing shape. The data below are the width ratios they chose as part of the study: 0.632 0.690 0.606 0.570 0.749 0.670 0.672 0.628 0.844 0.654 0.615 0. ...

1. Quantitative Methods - Mock
Question one (6 Marks)
Three coins are tossed together.
a) List all possible combinations of heads (H) and tails (T).
(2 marks)
b) What is the probability of having exactly one head?
(2 marks)
c) What is the probability of at least 2 heads?
(2 marks)
Question two (8 marks):
In a game of selecting a 3-digit number, you pay 50p to play a
game.
(Each selected digit can be any number from 0 to 9 and may be
selected more than once.)
If you select the correct number, you win £250
a) How many different selections are possible?
(2 marks)

2. b) What is the probability of winning?
(2 marks)
c) If you win, what is your profit?
(1 marks)
a) d) Find the expected value of winning and say whether you
would play this game.
(3 marks)
Question three (6 marks)
The table below summarises the job applications made by some
graduates during their last year:
Was the application successful?
Yes No
Male graduates 263 148
Female graduates 178 122
If you were to randomly select a graduate:
a) What is the probability that an application was not
successful? (2 marks)
b) What is the probability the application was made by a male
graduate or was successful?
(4 marks)

3. Question four (20 marks):
X is a random variable which represents the weight in kg
of crates of apples in the local market. Assume it has a normal
distribution with a mean of 50 and variance of 49.
a) What proportion of the crates weigh over 80kg?
(4 marks)
b) 68% of the crates will weigh between [?, ?].
(2 marks)
c) What proportion of the crates will weigh between 40kg and
53kg? Interpret it.
(6 marks)
d) A sample of 25 crates was randomly drawn. What is the
probability that the sample mean is between 47kg and 54kg?
(8 marks)
Question five (15 marks):

4. For a sample of 201 households from Norwich it was found that
the average monthly expenditure on food was £500 with sample
variance £2,500.
a) Using the given formula construct a 95% confidence interval
for the average monthly expenditure on food. (4 marks)
b) You were informed that the monthly expenditure on food is
£510; you believe that this is too high. Conduct a test in order
to prove that you are right.
(6 marks)
c) You think that household monthly expenditure on food would
be £505, test your guess.
(5 marks)
Question six (20 marks):
The ‘golden ratio’ 1 to 0.618. It is claimed that rectangles
whose sides are in this ratio are pleasing to the eye (i.e. they
look nice).
An experiment was conducted to see if this claim is true of the
shape of paintings. That is, the ratio of the length to the width
of a picture frame. Twenty people were asked to look at
different sized picture frames and choose the most pleasing
shape. The data below are the width ratios they chose as part of
the study:
0.632
0.690
0.606

5. 0.570
0.749
0.670
0.672
0.628
0.844
0.654
0.615
0.606
0.601
0.576
0.614
0.553
0.668
0.693
0.609
0.901
The significance level is 10%.
a) State formally and clearly the hypotheses. (2 marks)
b) Is this a one or two-tail test? Why? (2
marks)
c) Calculate the t-statistic (6 marks)
d) State the critical value (4 marks)
e) Is the claim true? Briefly explain your answer. (6
marks)

6. Statistical Formulae
Variance:
s 2 = Σx2 - (Σx)2/n
n– 1
Z score:
Z = X – mean
S
Coefficient of variation:
CV = S x 100
Mean
Probability (equally likely):
P(Event) = Number of outcomes in that the Event occurs
Total number of possible outcomes
Probability (not equally likely):
P(Event) = Number of times the Event of interestoccurs
Total number of times
Complementary events:
P(A) + P(Not A) = 1

7. or
P(Not A) = 1 – P(A)
Combining events:
P(A or B) = P(A) + P(B) – P(A AND B)
or
P(A AND B) = P(A) + P(B) – P(A OR B)
Conditional probability:
P(A|B) =P(A AND B) or P(B|A) =P(A AND B)
P(B) P(A)
Can be written as …
P(A AND B) = P(A|B) P(B) or P(A AND B) = P(B|A)
P(A)
An independent event:
P(A AND B) = P(A) P(B)
Expected value:

8. μ = E(x) = ΣxP(x)
Expected value of x2:
E(x2) = Σx2P(x)
Binomial combinations:
nCx = n ! .
x!(n – x)!
Binomial probability:
P(x) = nCx p x(1 – p) n-x
Where:
nCx = n ! .
x !(n – x)!
Expected value and variance of a continuous distribution:
The formula for mean and the variance of a uniformly
distributed random variable:
f(x) = 1 .
b – a
a <x<b

9. The mean is:
a + b
μ = 2
The variance is:
σ2 = (b – a)2
12
Standardising a normal random variable:
Z = X – μ
σ
The distribution of sample mean:
95% confidence interval:
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To test H0: =0:

10. .
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To test H0: 1=2:
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Revision Questions:
Question 1
In a game you have to select a 3-digit number, you pay 50p to
play a game.
(Each selected digit can be any number from 0 to 9 and may be
selected more than once.)
If you select the correct number, you win £250
a) How many different selections are possible? (2
marks)
b) What is the probability of winning? (2 marks)
c) If you win, what is your profit? (1
mark)
d) Find the expected value of winning and say whether you
would play this game.
(3 marks)

14. (Total 8 marks)
Question 3
The table below summarises the job applications made by some
graduates during their last year:
Was the application successful?
Yes No
Male graduates 263 148
Female graduates 178 122
If you were to randomly select a graduate:
a) What is the probability that an application was not
successful? (2 marks)
b) What is the probability the application was made by a male
graduate or was successful? (4
marks)
(Total 6 marks)
Question 4
X is a random variable which represents the weight in kg of
crates of apples in the local market. Assume it has a normal
distribution with a mean of 50 and variance of 49.
a) What proportion of the crates weigh over 80kg?
(4 marks)
b) 68% of the crates will weigh between [what?, what?].
(2 marks)

15. c) What proportion of the crates will weigh between 40kg and
53kg?
(6 marks)
d) A sample of 25 crates was randomly drawn. What is the
probability that the sample mean is between 47kg and 54kg?
(8 marks)
(Total 20 marks)
Question 5
For a sample of 201 households from Norwich it was found that
the average monthly expenditure on food was £500 with sample
variance £2,500.
a) Using the given formula construct a 95% confidence interval
for the average monthly expenditure on food. (4
marks)
b) You were informed that, for the population as a whole, the
monthly expenditure on food is £510; you believe that this is
too high. Conduct a test in order to prove that you are right.
(6 marks)
(Total 10 marks)
Question 6

16. An employer claims that bonuses paid to workers in a factory
average £1000. A random sample of 22 workers gives an
average bonus of only £975 with a standard deviation of £100.
Conduct a one-tail t test of the employer’s claim against an
alternative hypothesis of H1: µ<1000, Use a 5% significance
level.
(Total 25 marks)