# CHANCE ENCOUNTERS. Making Sense of Hypothesis Tests. Howard Fincher. Learning Development Tutor. Upgrade Study Advice Service

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1 CHANCE ENCOUNTERS Making Sense of Hypothesis Tests Howard Fincher Learning Development Tutor Upgrade Study Advice Service Oxford Brookes University Howard Fincher 2008

2 PREFACE This guide has a restricted aim. This guide will not tell you about the design of experiments or surveys, or about how to use Excel/SPSS. Nor is it a guide about which hypothesis test to choose. And there are no worked examples. This guide is for students following a course with an introductory statistics component, who find themselves directed towards performing a hypothesis test with Excel/SPSS. Such students are often trying to acquire knowledge in three areas simultaneously. First, there is knowledge about the theory of all hypothesis tests. Second, there is the detailed knowledge about one (or more) particular hypothesis test(s). Third, there is the specific knowledge about how to perform such a test with Excel/SPSS software. This guide only covers the first of these branches of knowledge. It aims to give you an overview of the framework of thought that surrounds all hypothesis tests, and seeks to introduce you to the philosophy of chance that lies behind all hypothesis tests. A dictionary will suggest that one of the meanings of chance is the probability of something happening. And the philosophy of chance that lies behind all hypothesis tests is that which has been codified in the mathematics of probability. This guide explains the background that might assist you to write some meaningful and sensible comments about the p-value that Excel/SPSS gives you at the end of a z test, or t test, or F test, or χ 2 test or (almost) any other hypothesis test. ACKNOWLEDGEMENTS I have been inspired by many teachers in the past. I hope that the best of their work has found a place in this guide. The responsibility for shortcomings remains with me. Any comments on this text, or suggestions for improvement of the text, can be made to the author at

3 INTRODUCTION A hypothesis test is not a calculation. A hypothesis test involves many calculations, but the hypothesis test itself is not a calculation. When an Excel/SPSS hypothesis test concludes with a p-value of, say, p = or p = , it gives the misleading impression that a hypothesis test is a calculation. This guide invites you to think of a hypothesis test as a process. This guide would like you to think of a hypothesis test as a carefully nuanced, thoughtfully designed and logically ordered decision-making process whose myriad associated numerical calculations are hidden from view in the Excel/SPSS software. This guide describes what is going on in the background as the Excel/SPSS software performs a hypothesis test. This guide describes eight components that have to come together in a logical order to create the decision making process called a hypothesis test. These eight components are: The Foundation The Null Hypothesis The Alternative Hypothesis The Significance Level of the Test The Key Statistic for the Test The p-value of the Data The Decision The Possibility of an Erroneous Decision This step-by-step guide tells you about the place and the function of these eight components in the process, and how each component provides you with a contribution that will help you to write meaningful and sensible comments about the p-value in the Excel/SPSS output.

4 THE FOUNDATION A hypothesis test is often called a significance test. Let s explore why this is. A researcher who is recording some interesting experimental data will want to be able to distinguish whether the interesting results are due to random chance variation or are due to a special factor (or significant effect) in the research situation. The foundation of a hypothesis test is the researcher s decision to reduce a real research situation to an imaginary competition between two rival hypotheses. The reduction of a real research situation to an imaginary competition between two rival hypotheses is not automatic. (A complex research situation can be reduced to two competing hypotheses in several different ways, and each of these reductions can be the beginning of a fresh hypothesis test.) When such a reduction has been made, a hypothesis test is the name given to a rational way of choosing between these two rival hypotheses. These two rival hypotheses are referred to as the Null Hypothesis (or H 0 or NH) and the Alternative Hypothesis (or H 1 or AH). [This guide will refer to the Null Hypothesis as the NH and the Alternative Hypothesis as the AH.] As you read through this guide, you will notice that a hypothesis test does not give equal weight to each hypothesis. The two rival hypotheses are not on an equal footing. Be aware that there is an asymmetry here. In fact, a hypothesis test places most of its weight on the NH. And that is because it is only the NH that is tested. So the decision at the conclusion of a hypothesis test is always expressed in relation to the NH. This decision will be of the form do not reject the NH or of the form reject the NH in favour of the AH. Since a real research situation is reduced to an imaginary competition between two rival hypotheses, it is important to realise that either of the two rival hypotheses may be true, or neither of them may be true. CONTRIBUTION 1 Either the clause do not reject the null hypothesis or the clause reject the null hypothesis in favour of the alternative hypothesis will contribute to your paragraph of interpretation of the p-value at the conclusion of an Excel/SPSS hypothesis test.

5 THE NULL HYPOTHESIS A hypothesis test gives special (almost exclusive) attention to the NH because this is the hypothesis that is being tested. So which of the two imaginary rival hypotheses do you choose as the NH? Of the two rival hypotheses, you choose as the NH the hypothesis that expresses the idea of no difference, no change, no effect, no correlation, or no association. The NH expresses the idea that there is nothing interesting happening in your research situation, and there are no special factors at work. When you choose your NH like this, your NH is usually the opposite of what you would like to believe is the case. Your NH has to be chosen like this so that your reduced research situation can be analysed from the point of view of random chance variation. Your NH is chosen in this way so that your experimental research data has the opportunity to contradict and to discredit it. Remember that it is only the NH that is being tested. CONTRIBUTION 2 Because a hypothesis test only tests the NH, a sentence concisely describing your NH will contribute to your paragraph of interpretation of the p-value at the conclusion of an Excel/SPSS hypothesis test.

7 THE SIGNIFICANCE LEVEL OF THE TEST You have to choose a significance level for your test. This significance level is also known as the alpha-value. The most common significance level in use is 5%. (You may also be directed to use a significance level of 1% or 0.1%.) The significance level of 5% is equivalent to the alpha-value, α = [This guide assumes that you are familiar with the equivalence of percentages and decimal fractions. Remember, any percentage can be converted to a decimal (fraction) by division by 100, and any decimal (fraction) can be converted to a percentage by multiplication by 100.] On the basis of your NH (which is an imaginary model of the real research situation), the Excel/SPSS software imagines all possible random samples that could be recorded and puts them into two classes. There is the class of samples with the smaller overall differences due to random chance variation, and there is the class of samples with the larger overall differences due to random chance variation. When you choose a significance level for your test, you are choosing the position of the boundary between these two classes. A significance level of 5% means that you are going to put the boundary between a (large) class containing 95% of imaginary samples with the smaller overall differences and a (small) class containing 5% of imaginary samples with the larger overall differences. A little later in the process, you will see that the hypothesis test will want to know to which of these two classes your experimental data belongs. Your choice of a significance level for the test is your way of defining the criterion with which you are going to make a decision about the NH. (Remember that your NH is an imaginary hypothesis which may or may not be true.) CONTRIBUTION 4 The phrase a significance level of 5% (α = 0.05) will contribute to your paragraph of interpretation of the the p-value at the conclusion of an Excel/SPSS hypothesis test.

8 THE KEY STATISTIC FOR THE TEST As hinted in the last section, the Excel/SPSS software imagines all possible random samples that could be recorded based upon your choice of a NH. This work is at the heart of the hypothesis test. The hypothesis test uses a key statistic such as z, or t, or F, or χ 2, etc, to refer to a complex mathematical chance variation model to generate all possible random samples that could be recorded based upon your choice of a NH. (The details of your research situation will suggest which one of these key statistics will provide you with a suitable model.) Numerical values of z, or t, or F, or χ 2, etc, encode the overall difference between random samples and the NH. Probabilities are now associated with each numerical value of the key statistic for your test. The choice of a 5% significance level for the test means that the hypothesis test is tracking (with a numerical value of your key statistic) the position of the boundary between the 95% of samples with the smaller overall differences and the 5% of samples with the larger overall differences in relation to the NH. CONTRIBUTION 5 The identification of the key statistic for your test, such as a z test, or a t test, or an F test, or a χ 2 test will contribute to your paragraph of interpretation of the p-value at the conclusion of an Excel/SPSS hypothesis test.

9 THE p-value OF THE DATA This is the opportunity for your experimental data to contest the imaginary NH. Excel/SPSS reads your experimental data and works out a numerical value of the key statistic. This numerical value of z, or t, or F, or χ 2, etc, is a numerical measure of the overall difference between your experimental data and the NH. Excel/SPSS now assigns a probability to your data value of z, or t, or F, or χ 2, etc. And the probability it assigns is the probability that a numerically larger value of z, or t, or F, or χ 2, etc, than your data value of z, or t, or F, or χ 2, etc, could be expected to occur on the basis that the NH is true. This probability is the p-value. This p-value is the probability of obtaining a key statistic value for your data as different, or more different, from that envisaged by the NH. This p-value of your experimental data is also sometimes known as the significance level of the data. (Remember that the alpha-value is the significance level of the test.) [Excel presents very small values of the p-value in an informal version of scientific notation such as 2.34E-05. This p-value is p = SPSS presents all very small values of the p-value as 0.000] CONTRIBUTION 6 The p-value must be stated at the conclusion of an Excel/SPSS hypothesis test.

12 MAKING SENSE OF HYPOTHESIS TESTS When Excel/SPSS gives you a p-value at the end of a z test, or t test, or F test, or χ 2 test or any other test, you will normally be expected to write a paragraph of interpretation of the p-value. You should consider writing complete sentences with the minimum of numerals and symbols. And you should remember that the focus of this paragraph is the test, and not the real world which produced the experimental data. Here is a model paragraph when the p-value is, say, and you decide to reject the NH in favour of the AH. Make a choice or make an addition at [ ] or ( ). With a 5% significance level [z or t or F or χ 2 ] test (α = 0.05) of the null hypothesis [specify it] against the alternative hypothesis [specify it] (two tail test or right tail test or left tail test), the data significance level was computed by [Excel or SPSS] as p = Since p = < α = 0.05, I reject the null hypothesis in favour of the alternative hypothesis. I conclude that the experimental data is statistically significantly different from the null hypothesis, and may be consistent with the alternative hypothesis. The mathematics of chance suggests that this hypothesis test will reach the correct conclusion on 95% of the occasions when this test is correctly used in this research situation, and when attention is paid to the assumptions on which the test is based. Here is a model paragraph when the p-value is, say, and you decide not to reject the NH. Make a choice or make an addition at [ ] or ( ). With a 5% significance level [z or t or F or χ 2 ] test (α = 0.05) of the null hypothesis [specify it] against the alternative hypothesis [specify it] (two tail test or right tail test or left tail test), the data significance level was computed by [Excel or SPSS] as p = Since p = > α = 0.05, I do not reject the null hypothesis. I conclude that the experimental data is not statistically significantly different from the null hypothesis, and may be consistent with it. The mathematics of chance suggests that this hypothesis test will reach the correct conclusion on 95% of the occasions when this test is correctly used in this research situation, and when attention is paid to the assumptions on which the test is based.

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