PPQs

Question 1

A seller of online educational software would like to evaluate the effectiveness of a recent marketing campaign.

The seller has contacted 80% of schools in a city to provide marketing material about a new software.

Of the schools that were contacted, 40% adopted the new software. Of the schools that were not contacted, 20% adopted the new software.
A school is randomly chosen from this city, and two events are defined as follows:

A = The school was contacted

B = The school adopts the software
Question: Are A and B mutually exclusive? Are they collectively exhaustive? Briefly explain.

Based on the setup in part (a):
What is the probability that a school who adopts the software was contacted?

What is the probability that a school was either contacted or adopted the software? (i.e., find P(A∪B)

Answer 1

Not mutually exclusive: A school can be contacted and adopt the software, so both A and B can happen at the same time.

Not collectively exhaustive: A∪B does not cover all outcomes — e.g., a school could be not contacted and not adopt, so neither A nor
B occur.
✅ Final Answer:

A and B
are not mutually exclusive and not collectively exhaustive.

P(A∪B)=P(A)+P(B)−P(A∩B)

= 0.84

Question 2

(b) Calculate the sample covariance between advertising and sales.

Answer 2

Create a scatter plot with:

X-axis: Advertising Expenditures (in £1,000s)
Y-axis: Total Sales (in £1,000s)
Title: Advertising Expenditures vs Total Sales
🧠 From the plot:

Higher advertising doesn’t clearly increase sales — in fact, there appears to be a negative association.

Question 3

3. Denote the number of accidents in a production facility per month by X. Suppose that random
   variable follows a Poisson distribution with a mean of 3 accidents per month.
   (a) Explain whether the number accidents is a discrete or continuous variable. What is the level
   of measurement for the number of accidents? [2 marks]
   If you have used additional space for working then please tick here:
   (b) Write the probability distribution for X. (c) For a given month, what is the probability there will be less than 2 accidents? [3 marks]
   If you have used additional space for working then please tick here:
   (d) What is the probability that the time between accidents is more than half a month?[3 marks]

Answer 3

X is a discrete variable: it counts whole numbers of accidents (0, 1, 2, …), not fractions.
The level of measurement is ratio:
It has a meaningful zero (0 accidents = none occurred)
Ratios are meaningful (e.g., 4 accidents is twice as many as 2)
✅ Final answer:

Discrete variable; ratio level of measurement.

​

Question 4

Answer 4

Question 5

Answer 5

✅ (c) If
σ
σ increases:

SE increases (since
S
E
=
σ
/
n
SE=σ/
n
​
)
Probability
X
ˉ
<
138
X
ˉ
<138 increases (because the Z-score becomes less negative)
Intuition: More population variability makes extreme sample means (like <138) more likely
✅ (d) If sample size decreases:

SE increases (smaller
n
n makes denominator smaller)
Probability
X
ˉ
<
138
X
ˉ
<138 increases (Z-score shrinks in magnitude)
Intuition: Smaller samples are more variable → more likely to observe unusual sample means