Faculty of Social Sciences Induction Block: Maths

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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

• 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
• 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 proportion
  is likely to hold in the population.

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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 gt 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

• 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 lt a)
• 4. Compute P and state conclusion.

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P values for one and two tailed tests

• Upper Tail Test
• H1 m gt m0 then P Prob(z gt zi)
• Lower Tail Test
• H1 m lt m0 then P Prob(z lt zi)
• Two Tail Test
• H1 m ? m0 then P 2xProb(z gt zi)

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5. Overview of the Course
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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 peoples 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!

• 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 doesnt 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 elses published research
• E.g. Newspaper examples in MS 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.
• Cant use SPSS to do these examples or to check
  your results

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• (d) Encourages awareness development of Macros
• SPSSs greatest strength
• Customisability/expandability
• Actually dont 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 peoples will greatly increase your
  productivity employability!

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SPSS macros
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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
• MM 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 (MM) 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
• MM 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 IS

• 1. Cost
• Pryce 22 PP (special price of 20 this
  week)
• MM Field 65
• 2. Length
• Pryce 200 pages supplement with further
  reading
• 600 pages (MM) 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 IS

• 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. Im biased in my recommendation!
• But correct, of course.

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