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1 Faculty of Social Sciences Induction Block: Maths & Statistics Lecture 6: Sample size, SPSS and Hypothesis Testing Dr Gwilym Pryce

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2 Plan n 1. Summary of L5 n 2. Statistical Significance n 3. Type 1 and Type II errors n 4. Four steps of Hypothesis Testing n 5. Overview of the Course

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3 1. Summary of L5: n Social Research is usually based on samples n We usually want to use our sample to say something about the population –I.e. we want to be able to generalise n How precisely we can estimate the population mean or proportion depends on our sample size and the variation within the sample n 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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4 2. Statistical Significance n “Significance” does not refer to “importance” –but to “real differences in fact” between our observed sample mean and our assumption about the population mean n P = significance level = chances of our observed sample mean occurring given that our assumption about the population (denoted by “H 0 ”) 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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5 n 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. n If we draw the conclusion that it is our assumption that is wrong and reject H 0 then we have to bear in mind that there is a chance that H 0 was in fact true. –I.e. every twenty times we reject H 0 when P = 0.05, then on one of those occasions we would have rejected H 0 when it was in fact true.

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6 n Obviously, as the sample mean moves further away from our assumption (H 0 ) about the population mean, we have stronger evidence that H 0 is false. n If P is very small, say 0.001, then there is only 1 chance in a thousand of our observed sample mean occurring if H 0 is true. –This also means that if we reject H 0 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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7 n There is a tradition (initiated by English scientist R. A. Fisher ) of rejecting H 0 if the probability of incorrectly rejecting it is –If P 0.05 then we say that H 0 can be rejected at the 5% significance level. –If P > 0.05, then, argued Fisher, the chances of incorrectly rejecting H 0 are too high to allow us to do so. n 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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8 3. Type I and Type II errors: n P = significance level = chances of incorrectly rejecting H 0 when it is in fact true. –Called a “Type I error” n If we accept H 0 when in fact the alternative hypothesis is true –Called a “Type II error”. n On this course we shall be concerned only with Type I errors.

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9 4. The four steps of hypothesis testing n 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 and 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 = 30 years? –If 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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11 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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12 Finding the value of P: n 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 = ( )/(20/ 200) =4.5/1.4 = 3.2.

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15 Upper tailed test:

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16 Two tailed test:

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17 4 Steps to Hypothesis tests: n 1. Specify null and alternative hypotheses 2. Specify threshold significance level and appropriate test statistic formula 3. Specify decision rule (reject H 0 if P < ) n 4. Compute P and state conclusion.

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18 P values for one and two tailed tests: n Upper Tail Test: H 1 : > 0 then P = Prob( z > z i ) n Lower Tail Test: H 1 : < 0 then P = Prob( z < zi ) n Two Tail Test: H 1 : 0 then P = 2xProb( z > |zi|)

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21 5. Overview of the Course:

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22 Nature of the Course: n 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 n Enable you to critique other people’s work n 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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23 Why learn syntax? Most texts & courses avoid it! n A succinct and secure record n Transparency and reproducibility n Efficiency n Paste and Learn n 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 n Accessing Extra Resources & Expanding SPSS

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24 Why the macros? 4 reasons: n (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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25 n (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

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26 n (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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27 n (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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28 SPSS macros

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29 Guide to Reading: n Essential reading (recommended for purchase): –Pryce, G. Inference and Statistics in SPSS Lab exercises drawn from this book. n 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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30 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) 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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31 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) 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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32 Disadvantages of Pryce I&S: n 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. n 2. Wire comb binding –But actually better for working next to PC n 3. I’m biased in my recommendation! –But correct, of course.

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