Test

Question 1

What is an outcome or an elementary event?

Answer 1

• one of the possible things that can happen.

For example, suppose that we are interested in the (UK) shoe size of the next customer to come into a shoe shop. Possible outcomes: “eight” “twelve” “nine and a half”

In any experiment, one, and only one, outcome occurs.

Question 1

What is the sample space

Answer 1

• the set of all possible outcomes. For example, it could be the set of all shoe sizes

Question 2

What does mutually exclusive mean?

Answer 2

Two events are said to be mutually exclusive if both cannot
occur simultaneously: The outcomes success and failure are mutually exclusive.

Question 3

What does it mean if 2 events are independent?

Answer 3

Two events are said to be independent if the occurrence of
one does not affect the probability of the second occurring.

For example, if you toss a coin and look out of the window, it
would be reasonable to suppose that the events “get heads”
and “it is raining” would be independent.

Question 4

There are three main ways in which we can measure
probability:

Answer 4

1. The classical interpretation
2. The frequentist interpretation
3. The subjective or Bayesian interpretation

Question 5

The Classical Interpretation

When do we use it?
Example?
What is the underlying idea behind the view?
What probability do the outcomes have?
What is the general equation?

Answer 5

If all possible outcomes are “equally likely” then we can adopt the classical approach to measuring probability.
For example, if we tossed a fair coin, there are only two
possible outcomes – a head or a tail – both of which are equally likely

The underlying idea behind this view of probability is
symmetry.
In this example, there is no reason to think that the outcome
Head and the outcome Tail have different probabilities and so they should have the same probability.
Since there are two outcomes and one of them must occur,
both outcomes must have probability 1/2.

Generally,

P (Event) = Total number of outcomes in which event occurs/Total number of possible outcomes

Question 6

The frequentist approach

When do we use it?
What do we do?
Example?
What 2 factors do we use?
What is the general equation?

Answer 6

When the outcomes of an experiment are not equally likely, we can conduct experiments to give us some idea of how likely the different outcomes are.

For example, suppose we were interested in measuring the
probability of producing a defective item in a manufacturing
process.

• Monitor the process over a long period
• Calculate the proportion of defective items

However what constitutes a “long period of time”?

Generally,

P(Event) = Number of times an event occurs/Total number of times experiment done

Question 7

The Subjective/Bayesian interpretation

Examples?
Difference between this and the frequentist interpretation?

Answer 7

When we board a plane, we judge the probability of it
crashing to be sufficiently small that we are happy to
undertake the journey.

Similarly, the odds given by bookmakers on a horse race
reflect people’s beliefs about which horse will win.
This probability does not fit within the frequentist definition as the same race cannot be run a large number of times.

Question 8

Laws of Probability: Multiplication law

The probability of two independent events E1 and E2 both
occurring can be written as…

And it is known as?

Answer 8

The probability of two independent events E1 and E2 both
occurring can be written as…
P(E1 and E2) = P(E1) × P(E2),
and this is known as the multiplication law of probability.

Question 9

Laws of probability: Addition Law

What does it describe?

The addition law for two events E1 and E2 is

What is the more basic version and when can we use it?

Why can we use that?

Answer 9

The addition law describes the probability of any of two or
more events occurring.

The addition law for two events E1 and E2 is

P(E1 or E2) = P(E1) + P(E2) − P(E1 and E2)

A more basic version of the rule works where events are
mutually exclusive.

If events E1 and E2 are mutually exclusive then

P(E1 or E2) = P(E1) + P(E2)

This simplification occurs because when two events are
mutually exclusive they cannot happen together and so
P(E1 and E2) = 0.

Question 10

The binomial distribution

What is the probability of rolling a dice three times

What about the probability that we get two sixes — i.e. P (X = 2)?

Answer 10

Each roll of the dice is an experiment or trial which gives a “six” (success, or s) or “not a six” (failure, or f ). The probability of a success is p = P (six) = 1/6. We have n = 3 independent experiments or trials (rolls of the dice). Let X be the number of sixes obtained.

We can now obtain the full probability distribution of X; a probability distribution is a list of all the possible outcomes for X with along with their associated probabilities.

For example, suppose we want to work out the probability of obtaining three sixes (three “successes” — i.e. sss — or P (X = 3)). Since the rolls of the die can be considered independent, we get

P (sss) = P (s) × P (s) × P (s) = 1/6 × 1/6 × 1/6 = (1/6)^3
That one’s easy! What about the probability that we get two sixes — i.e. P (X = 2)?
This one’s a bit more tricky, because that means we need two s’s and one f — i.e. two sixes and one “not six” — but the “not six” could appear on the first roll, or the second roll, or the third! Thinking about it, there are actually eight possible outcomes for the three rolls of the die

Question 11

Experiment

Answer 11

An experiment is an activity where we do not know for certain what will happen, but we will observe what happens.

Question 12

Event

Answer 12

An event is a set of outcomes. For example “the shoe size of the next customer is less than 9” is an event. It is made up of all of the outcomes where the shoe size is less than 9. Of course an event might contain just one outcome.

Question 13

Random variables (definition)

Answer 13

The quantities measured in a study

Question 14

Data (definition)

Answer 14

A collection of such observations

Question 15

Observation (definition)

Answer 15

A particular outcome

Question 16

Population (definition)

Answer 16

The collection of all possible outcomes

Question 17

Example

Study on the height of students of A&F course at Newcastle

What would be the random variable?
What would a value of Joe Blogs measured height be called?

What would be our data?

This would be a _______ from the ____________ which consists of all students registered on A&F degrees

Answer 17

• Our random variable is “the height of students on A&F courses at Newcastle”.
• If Joe Bloggs is an A&F student, and we measured his height, then that value would be a single observation.
• If we measured the height of every first year A&F student, we would have a collection of such observations which would be our data.
• This would be a sample from the population which consists of all students registered on A&F degrees

Question 18

Ideally, to get a true idea of what is going on, we’d like to observe the whole population (take a _______). However, this can be difficult:

Why would it be difficult?

Answer 18

A census

• If the population is huge, then this would take ages!
• And it would be very costly!
• In reality, we usually observe a subset of the population… but how do we choose who to observe?

Question 19

Quantitative variables (2 types and explanation)

Answer 19

Discrete random variables

• can only take a sequence of distinct values (usually integers);
• are usually countable - e.g. the number of people attending a tutorial group;
• can be ordinal - where the outcomes are ordered.

Continuous random variables

• can take any value over some continuous scale - e.g. height or weight.
• can be measured to a very high degree of accuracy (provided we have the equipment to do so) (often decimals)
• however, we can never say precisely how much someone weighs, for example,
  might be measured to the nearest whole number - and so could “look” discrete - be careful!

Question 20

Sampling

What is a sample?
What is the difficulty?
What is a biased sample?

Answer 20

• Subset of the whole population
• Obtaining a representative sample
• Unrepresentative and unfair

Question 21

What are the general forms of sampling techniques?

Answer 21

1. Random sampling - where the members of the sample are chosen by some random (i.e. unpredictable) mechanism.
2. Quasi-random sampling - where the mechanism for choosing the sample is only partly random.
3. Non-random sampling - where the sample is specifically selected rather than randomly selected.

Question 22

Simple Random Sampling disadvantages

Answer 22

• We don’t have a complete list of the population
• Not all elements, of the population are equally accessible
• By chance, you could pick an unrepresentative sample

Question 23

Stratified sampling

What is it?
What is its main idea?

Answer 23

• Form of random sample where clearly defined groups or strata exist within the population
• If we know the overall proportion of the population that falls into each of these groups, we can take a simple random sample from each f the groups and then adjust the results according to the known proportions

Question 24

Systematic sampling

What is it a form of?
Example?
Disadvantage?

Answer 24

• Form of quasi-random sampling
• For example picking every 10th item to come off the production line
• Not entirely random and can be biased

Question 25

Multi–stage Sampling

What is it a form of?
When is it common?
How does it work?
Example?
Advantage?
Disadvantage?

Answer 25

• This is another form of quasi–random sampling.
• These types of sampling schemes are common where the population is spread over a wide geographic area which might be
  difficult or expensive to sample from.
• Multi–stage sampling works, for example, by dividing the area into geographically distinct smaller areas, randomly selecting one (or more) of these areas and then sampling, whether by random, stratified or systematic sampling schemes within these areas.
• For example, if we were interested in sampling school children, we might take a random (or stratified) sample of education authorities, then, within each selected authority, a random (or stratified) sample of schools, then, within each selected school, a random (or stratified) sample of pupils.
• This is likely to save time and cost less than sampling from the whole population.
• The sample can be biased if the stages are not carefully selected. Indeed, the whole scheme needs to be carefully thought through and designed to be truly representative.

Question 26

Cluster Sampling

What is it?
What does it differ from?
Advantage?
Disadvantage?
Example?

Answer 26

• This is a method of non–random sampling. For example, a geographic area is
  sub–divided into clusters and all the members of a particular cluster are then surveyed.

This differs from multi–stage sampling covered in Section 3.2.4 where the members of the cluster were sampled randomly. Here, no random sampling occurs.

• The advantage of this method is that,
  because the sampling takes place in a concentrated area, it is relatively inexpensive to perform.
• The very fact that small clusters are picked to allow an entire cluster to be surveyed introduces the strong possibility of bias within the sample. If you were interested in the take up of organic foods and were sampling via the cluster method you could easily get biased results;
• if, for example, you picked an economically deprived area, the proportion of those surveyed that ate organically might be very low, while if you picked a middle class suburb the proportion is likely to be higher than the overall population

Question 27

Judgemental sampling

What is it?
Advantage?
Example?
Disadvantage?

Answer 27

• Here, the person interested in obtaining the data decides whom they are going to ask.
• This can provide a coherent and focused sample by choosing people with experience
  and relevant knowledge to provide their opinions.
• For example, the head of a service
  department might suggest particular clients to survey based on his judgement. They
  might be people he believes will be honest or have strong opinions.
• This methodology is non–random and relies on the judgement of the person making the choice. Hence, it cannot be guaranteed to be representative. It is prone to bias

Question 28

Accessibility sampling

What is it?
Disadvantage?
Example?

Answer 28

• Here, only the most easily accessible individuals are sampled.
• This is clearly prone to bias and only has convenience and cheapness in its favour.
• For example, a sample of grain taken from the top of a silo might be quite unrepresentative of the silo as a whole
  in terms of moisture content.

Question 29

Quota Sampling

How is it similar/different?
What do we do?
Example?
Advantages?
Disadvantages?

Answer 29

• This method is similar to stratified sampling but uses judgemental (or some other)
  sampling rather than random sampling within groups.
• We would classify the population by any set of criteria we choose to sample individuals and stop when we have reached our quota.
• For example, if we were interested in the purchasing habits of 18–23 year old male students, we would stop likely candidates in the street; if they matched the requirements we would ask our questions until we had reached our quota of 50 such students.
• This type of sampling can lead to very accurate results as it is specifically targeted, which saves time and expense.
• The accurate identification of the appropriate quotas can be problematic. This method is highly reliant on the individual interviewer selecting people to fill the quota. If this is done poorly bias can be introduced into the sample.

Question 30

Frequency tables for categorical data

Answer 30

This gives us a much clearer picture of the methods of transport used. Also of interest
might be the relative frequency of each of the modes of transport. The relative
frequency is simply the frequency expressed as a proportion of the total number of
students surveyed. If this is given as a percentage, as here, this is known as the
percentage relative frequency

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Question 31

Frequency tables for continuous data

What are some things to think about:

Answer 31

With discrete data, and especially with small data sets, it is easy to count the
quantities in the defined categories. With continuous data this is not possible. Strictly
speaking, no two observations are precisely the same. With such observations we group
the data together

Some things to think about:

• Often for simplicity we would write the class intervals up to the number of
  decimal places in the data and avoid using the inequalities; for example, 20 up to
  29.999 if we were working to 3 decimal places.
• We need to include the full range of data in our table and so we need to identify
  the minimum and maximum points (sometimes our last class might be “greater
  than such and such”).
• The class interval width should be a convenient number – for example 5, 10, or
  100, depending on the data. Obviously we do not want so many classes that each
  one has only one or two observations in it.
• The appropriate number of classes will vary from data set to data set; however,
  with simple examples that you would work through by hand, it is unlikely that
  you would have more than ten to fifteen classes

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Question 32

Stem and Leaf plots

Answer 32

Stem and leaf plots are a quick and easy way of representing data graphically. They
can be used with both discrete and continuous data

Extra digits are cut and not rounded

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Question 33

hy use percentage relative frequency?

Answer 33

• It puts both samples on the same scale

Question 34

how do you use polygons ?

Answer 34

• Join the midpoints with straight lines in the histogram

Question 35

How do you do cumulative relative polygons ?

Answer 35

• Add data on top oh each other
• take the endpoints instead of the midpoints
• Start with 0

Question 36

What to do with grouped data for means

Answer 36

Multiply the midpoint by the quantities, add it all together and divide by the frequency

Question 37

What to note about quartiles?

Answer 37

If there are 20 observations, you will pick the 5 1/4th smallest observation because that is 20+1=21/4

To find the upper quartile it would be the 21/4 *3 = 15 3/4th smallest observation

Question 38

How to calculate variance

Answer 38

1. Determine the mean of your data.
2. Find the difference of each value from the mean.
3. Square each difference.
4. Calculate the squared values.
5. Divide this sum of squares by n – 1 (sample) or N (population).