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Faculty of Social Sciences
Induction Block:
Maths & Statistics Lecture 6:
Sample size, SPSS and Hypothesis
Testing
Dr Gwilym Pryce
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Plan
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1. Summary of L5
2. Statistical Significance
3. Type 1 and Type II errors
4. Four steps of Hypothesis Testing
5. Overview of the Course
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1. Summary of L5:


Social Research is usually based on samples
We usually want to use our sample to say
something about the population
– I.e. we want to be able to generalise
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How precisely we can estimate the population
mean or proportion depends on our sample
size and the variation within the sample
Using the CLT, statistical inference offers a
systematic way of establishing:
– the range of values in which the population mean
or proportion is likely to lie (‘a confidence interval’).
– Whether a hypothesis about a mean or a
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proportion is likely to hold in the population.
2. Statistical Significance

“Significance” does not refer to “importance”
– but to “real differences in fact” between our
observed sample mean and our assumption about
the population mean

P = significance level = chances of our
observed sample mean occurring given that
our assumption about the population (denoted
by “H0”) is true.
– So if we find that this probability is small, it might
lead us to question our assumption about the
population mean.
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
I.e. if our sample mean is a long way from our
assumed population mean then it is:
– either a freak sample
– or our assumption about the population mean is wrong.

If we draw the conclusion that it is our assumption
that is wrong and reject H0 then we have to bear in
mind that there is a chance that H0 was in fact true.
– I.e. every twenty times we reject H0 when P = 0.05, then on
one of those occasions we would have rejected H0 when it
was in fact true.
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Obviously, as the sample mean moves further away
from our assumption (H0) about the population mean,
we have stronger evidence that H0 is false.
If P is very small, say 0.001, then there is only 1
chance in a thousand of our observed sample mean
occurring if H0 is true.
– This also means that if we reject H0 when P = 0.001, then
there is only one in a thousand chance that we have made a
mistake (I.e. that we have been guilty of a “Type I error”)
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
There is a tradition (initiated by English scientist R.
A. Fisher 1860-1962) of rejecting H0 if the
probability of incorrectly rejecting it is  0.05.
– If P  0.05 then we say that H0 can be rejected at the 5%
significance level.
– If P > 0.05, then, argued Fisher, the chances of incorrectly
rejecting H0 are too high to allow us to do so.

Sig level = P = the probability of a sample mean
at least as extreme as our observed value
occurring, given our assumption about the
population mean.
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3. Type I and Type II errors:

P = significance level = chances of incorrectly
rejecting H0 when it is in fact true.
– Called a “Type I error”

If we accept H0 when in fact the alternative
hypothesis is true
– Called a “Type II error”.

On this course we shall be concerned only
with Type I errors.
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4. The four steps of hypothesis testing

Last lecture we looked at confidence
intervals:
– establish the range of values of the population
mean for a given level of confidence
• e.g. we are 90% confident that population mean age of
HoHs in repossessed dwellings in the Great
Depression lay between 32.17 and 36.83 years (s =
20).
• Based on a sample of 200 with mean = 34.5yrs.
– But what if we want to use our sample to test a
specific hypothesis we may have about the
population mean?
• E.g. does m = 30 years?
– If m does = 30 years, then how likely are we to select a
sample with a mean as extreme as 34.5 years?
» I.e. 4.5 years more or 4.5 years less than the pop
mean?
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One tailed test: P = how likely we are to select a
sample with mean age at least as great as 34.5?
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Finding the value of P:

Because all sampling distributions for
the mean (assuming large n) are
normal, we can convert points on them
to the standard normal curve
– e.g. for 34.5: z = (34.5 - 30)/(20/200)
=4.5/1.4 = 3.2.
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Upper tailed test:
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Two tailed test:
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4 Steps to Hypothesis tests:
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1. Specify null and alternative hypotheses
2. Specify threshold significance level a and
appropriate test statistic formula
3. Specify decision rule (reject H0 if P < a)
4. Compute P and state conclusion.
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P values for one and two tailed
tests:

Upper Tail Test:
H1: m > m0 then P = Prob(z > zi)

Lower Tail Test:
H1: m < m0 then P = Prob(z < zi)

Two Tail Test:
H1: m  m0 then P = 2xProb(z > |zi|)
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Confidence Interval
Find the 90% confidence interval of the
population mean age
1.
Choose the appropriate test
statistic:
Hypothesis Tests
Test the hypothesis that the population mean age = 30
using a significance level of 0.1
1.
Specify null and alternative hypothesis:
2.
Specify the level of significance and the test
statistic
xi  m
* s
zi 
 m  xi  z
s/ n
n
H0: m = 30
H1: m  30
Significance level:
a = likelihood of Type I error that you are
prepared to tolerate
= Prob(Reject H0 when it is true) = 0.1
Test Statistic:
n > 30, therefore we can us z:
zc 
xi  m
s/ n
= zc 
xi  30
s/ n
i.e. we write the zc formula assuming that H0 is correct
2.
Establish the value of z*:
3.
Prob(-z*<z<z*) = 0.9
Area of tails = (0.1)/2 = 0.05
 z* = 1.65
3.
Calculate the confidence interval:
m  34.5  1.65
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200
= 45.5 2.33
Specify the decision rule:
Reject H0 iff P (the calculated level of Type I error)
is no greater than the tolerated level:
i.e. Reject H0 iff P  a
(the smaller is P, the less risk involved in rejecting H0)
4.
Compute P and state your conclusion:
zc = 3.18 ;
PProb(z < 3.18)
Since P < a(i.e. , its safe to reject H0
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Lower Tail Hypothesis Tests
Test the hypothesis that the population mean age
< ? using a significance level of a
Upper Tail Hypothesis Tests
Test the hypothesis that the population mean age
> ? using a significance level of 0.1
1.
1.
Specify null and alternative hypothesis:
H0: m = ?
H1: m < ?
2. Specify the level of significance and the
test statistic
Significance level:
a = likelihood of Type I error that you
are prepared to tolerate
= Prob(Reject H0 when it is true)
Test Statistic:
If n > 30, we can us z:
xi  m
s/ n
H0: m = ?
H1: m > ?
2. Specify the level of significance and the
test statistic
zc 
Specify null and alternative hypothesis:
= zc 
Significance level:
a = likelihood of Type I error that you
are prepared to tolerate
= Prob(Reject H0 when it is true)
Test Statistic:
If n > 30, we can us z:
xi  ?
zc 
s/ n
xi  m
s/ n
= zc 
xi  ?
s/ n
i.e. we write the zc formula assuming that H0 is
correct
i.e. we write the zc formula assuming that H0 is
correct
3.
3.
Specify the decision rule:
Reject H0 iff P (the calculated level of Type I
error) is no greater than the tolerated level:
i.e. Reject H0 iff P  a
4.
Compute P and state your conclusion:
PProb(z < zc)
Note that zc will be negative if x 
m
Specify the decision rule:
Reject H0 iff P (the calculated level of Type I
error) is no greater than the tolerated level:
i.e. Reject H0 iff P  a
4.
Compute P and state your conclusion:
PProb(z > zc)
Note that zc will be positive if m  x
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5. Overview of the Course:
 L1: Density Functions & CLT
 L2: Calculating z-scores
L3: Introduction to Confidence
Intervals
L4: Confidence Intervals for All
Occasions
Quants I
24/09/2005 - v23
L5: Introduction to Hypothesis
Tests
L6: Hypothesis Tests for All
Occasions
L7: Relationships between
Categorical Variables
 L8: Regression
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Nature of the Course:

This is course in applied statistics
– Applied: Not teach theoretical proofs
• prove anything with maths (eg Teletubbies are evil)
• What counts is understanding the concepts
– Statistics: also teach you SPSS,
• But lots of different stats packages out there
– You are likely to use different ones over the course of your
research career
– But statistic concepts remain unchanged


Enable you to critique other people’s work
Also part of a wider research methods
training programme:
– Broader remit is to teach you good practice in
research techniques
• Essential to learn syntax…
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Why learn syntax?
Most texts & courses avoid it!
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A succinct and secure record
Transparency and reproducibility
Efficiency
Paste and Learn
Avoiding obsolescence
– SPSS point-n-click routines change with each new
version of SPSS – changes once a year
– Syntax remained virtually unchanged for 15 years

Accessing Extra Resources & Expanding
SPSS
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Why the macros? 4 reasons:

(a) Get the statistical procedure right, then
choose the program/calculator
– SPSS doesn’t know what sort of data you have
– SPSS canned routine may not be the right one for
your data
– You could compute the procedure by hand, &
indeed it is important to know how to do this.
– but this can be long-winded in repeated
applications & easy to make mistakes
– Macro commands speed the process & are a
useful way to check your calculations.
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
(b) Critiquing/Analysing Published Work
– SPSS routines can only be used if you have the
original data
– Not much use if you want to critique or analyse
someone else’s published research
• E.g. Newspaper examples in M&S tutorial
• E.g.United Nations crime survey
• E.g. MPPI paper by Pryce & Keoghan
– If all you can do is the point-n-click stuff in SPSS
you are going to be severely hampered in what
you can do.
– The Macro commands written specifically for the
course only need summary info (n, xbar, sd, prop.)
• Publicly available via the downloads page of
www.geebeejey.co.uk
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
(c) Working with standard texts
– The exercises and examples in standard
statistical texts (such as Moore and
McCabe) usually only provide summary
information not the original data.
– Can’t use SPSS to do these examples or to
check your results
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
(d) Encourages awareness & development of
Macros
– SPSS’s greatest strength:
• Customisability/expandability
– Actually don’t need to be good at statistics to use
macros
• You can use macros to do anything:
– Manipulate data,
– Automate repetitive tasks
– Formalise and automate complex calculations
– Writing SPSS macros is actually a good way to
acquire basic programming skills
– In real-life applied research, most of your time is
taken up with non-statistical manipulation of data
• Learning how to write your own macros or use other
people’s will greatly increase your productivity &
employability!
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SPSS macros
Confidence Intervals (CI)
Macro
Definition
command
Large sample CI for one mean
CI_L1M
Macro
Command
H_L1M
CI_S1M
Small sample CI for one mean
H_S1M
CI_S2MP
Small independent samples CI for
difference between 2 means
(pooled variance)
Small independent samples CI for
difference between 2 means
(different variances)
Large sample CI for one
proportion (presents output for
both Traditional and Wilson
methods of calculation)
Large sample CI for comparing
two proportions (presents output
for both Traditional and Wilson
methods of calculation)
H_S2MP
CI_S2MD
CI_L1P
CI_L2P
N_L1M
Sample size for desired margin or
error for the mean
H_S2MD
H_L1P
Hypothesis tests
Definition
Large sample significance test on
one mean
Small sample significance test on
one mean
Small
independent
samples
significance test for equality of 2
means (pooled variance)
Small
independent
samples
significance test for equality of 2
means (different variances)
Large sample significance test on
one proportion
H_L2P
Large samples significance test on
two proportions
H_S2VF
Simple small sample F-test on
equality of two variances (see also
Levene’s test in the SPSS help
menu for more sophisticated test
of homogenous variances).
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Guide to Reading:

Essential reading (recommended for
purchase):
– Pryce, G. Inference and Statistics in SPSS
• Lab exercises drawn from this book.

Usually recommended a book on
statistics & a book on SPSS:
– E.g. Moore & McCabe (£40) -- stats
– E.g. Field (£25+)
-- SPSS
– M&M and Field = 2 great books but 4 major
problems:
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2 great books but 4 major problems:
– 1. Cost (to buy both comes to approx £65)
•  many students have tried to make do without one or the
other & struggled.
– 2. Length
• 600 pages (M&M) + 832 pages (Field)
– 3. Content: neither geared to business & soc. sci.
• Field: too shallow/applied:
– Covers huge spectrum of topics (useful for Quants II)
– does not cover some of the basic material we need to do
» tends to cover what can be achieved in SPSS
» Does not use macros
» Does not teach syntax
• M&M: too deep/theoretical
– The Rolls Royce of introductory texts but does not teach SPSS
– But would take 2 semesters to cover material in this depth &
learn SPSS
– 4. Integration
• Leaves you the student with the task of combining the two
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Advantages of Pryce I&S:
– 1. Cost
• Pryce = £22 + P&P (special price of £20 this week)
– M&M + Field = £65
– 2. Length
• Pryce = 200 pages + supplement with further reading
– 600 pages (M&M) + 832 pages (Field)
– 3. Content:
• Pryce:
–
–
–
–
–
–
tries to strike the right balance between theory & application
Based in SPSS
Teaches syntax
Uses the macros
Geared to business and social science
Based on worked examples & exercises
– 4. Integration
• Pryce tries to integrate learning inference with learning SPSS
• But macros will also allow you do do the Moore & McCabe type
of exercise should you want to get more practice
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Disadvantages of Pryce I&S:

1. First edition:
– A few glitches here & there…
– But, rare edition because only a small print run
•
•
•
•

valuable as a collectors item if you keep it for 20 years.
Glitches add value – ask a stamp collector
Even more valuable if I sign it.
Makes a great Xmas gift for friends & family.
2. Wire comb binding
– But actually better for working next to PC

3. I’m biased in my recommendation!
– But correct, of course.
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Feedback forms…
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