Satistics

Greenwich University
Student -Asef Khan
Economics :Statistics for Economics and Finance. Sampling Distributions -Confidence Intervals-
Central Limit Theorem
Tutor: Alexander Guschanski
20.2.2023


GROUP ALLOCATION

 Sent email yesterday - group allocation has ended
 You are all allocated into a group for the empirical project. Click link in email- Excel File and
don’t change any of it
 Those students who had not been allocated are now allocated
 See email addresses of your colleagues. You have been asked to get in touch and establish
contact. At end of week he will release the question to work on, be ready to meet up
 You have till Wed 1st next week to inform him of any problems with the group allocation.
I.e. if someone does not reply to emails/or you cannot reach them. If someone does not
respond he will intervene
 Meeting protocol- worth 10 marks- submit a simple description of activity> we met on the
second of March and x, y were present and we did this etc .Supposed to be an incentive to
work together
 KEY- every group member needs to submit the problem – or fail
 Plagiarism will flag up on Moodle because of everyone submitting same file but ignore
 Exercise will be around portfolio analysis and will be released this week

SAMPLING DISTRIBUTIONS/ CENTRAL LIMIT THEOREM CONFIDENCE INTERVALS/

 Key areas for all your metrics later. Covering sampling distribution and how we analyse
the properties of a statistic to draw conclusions re the population
 See Learning Outcomes
 Not difficult but abstract concept- important to keep structure in mind+ links between topics
 sampling distribution>
 confidence intervals
 central limit theorem>

Here and in other modules lead to inferential stats ; e.g. have a population- can’t observe in total
and can only see the sample we have taken->population parameters are unknown but sample is
known to us. From this we draw inferences

Why sample? Too time consuming to look at whole pop ( e.g., census) and costly

Sample will be generated via simple random sample meaning each individual has the same
probability of being selected

↑↑↑

This represents the ‘ gold standard’ ideal of sampling




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, Greenwich University
Student -Asef Khan
Economics :Statistics for Economics and Finance. Sampling Distributions -Confidence Intervals-
Central Limit Theorem
Tutor: Alexander Guschanski
20.2.2023


SAMPLING DISTRIBUTION

Defined as the probability distribution of all the possible values of a statistic for a given size sample
selected from a population

What does this mean?

o e.g. - a b c d = population of 4
o Say pop =4> random variable is the age/ income/ value of portfolio of specific individual
o Here- we are focusing on age. Value of age= 18, 20, 22 and 24 (a-d)
o If we pick random person re age- can pick a-d. 25% prob of getting one out of a-d.
o This is uniform distribution as all values have the same probability of being chosen and
also a discrete probability distribution; not continuous distribution as that would mean
an infinite number of probabilities while discrete =fixed number of realisations
o There are 4 different realisations of the variable so =discrete

Developing a sampling distribution

if we are picking a sample of two people from the population see slide for possible realisations, e.g.
a- 18, record age and put a back into pool of population then repeat with c-22.

 What kind of samples can we get – can get combinations e.g. aa, ab, ba etc, cb- sees slide for
all possible samples of size Results in> 16 possible samples from population
 Record the mean of each sample e.g.- 18 and 18 (mean 18) 18 and 20 ( 19) etc is average of
2 realisations
 Mean of whole population of 4=21.
 16 samples = 16 sample means

Distribution of sample means

 Certain values appear more often than others. Which? e.g. 21- independent of the sample
chosen you’re most likely to get sample mean of 21 and less likely to get sample mean 20
or 22. See graph re probability- 18 and 24 less likely too
 21 most likely value=discrete probability distribution( definition of sampling distribution-=
probability distribution of all the possible values of a statistic( sample mean) from a given
size sample)
 What is shape of sampling distribution reminiscent of? Is slightly bell shaped and
symmetrical looks like discrete version of normal distribution
 Not just coincidence> see flatness of uniform distribution previously but adding probability
distribution of sample means changes shape to curved> so it looks like normal distribution

Key- this is the basis of the central limit theorem which will be used to generate confidence
intervals




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