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RES 342 RESEARCH AND EVALUATION - II

RES 342 RESEARCH AND EVALUATION - II. WORKSHOP 1 By Dr. Serhat Eren University OF PHOENIX . CHAPTER X HYPOTHESIS TESTING 10.1 CHAPTER OBJECTIVES. What Is a Hypothesis Test? Overview of Hypotheses to Be Tested The Pieces of a Hypothesis Test Two-Tail Tests of the Mean: Large Sample

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RES 342 RESEARCH AND EVALUATION - II

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  1. RES 342RESEARCH AND EVALUATION - II WORKSHOP 1 By Dr. Serhat Eren University OF PHOENIX Dr. Serhat Eren

  2. CHAPTER XHYPOTHESIS TESTING 10.1 CHAPTER OBJECTIVES • What Is a Hypothesis Test? • Overview of Hypotheses to Be Tested • The Pieces of a Hypothesis Test • Two-Tail Tests of the Mean: Large Sample • Which Theory Should Go Into the Null Hypothesis? • One-Tail Tests of the Mean: Large Sample • What Error Could You Be Making? Dr. Serhat Eren

  3. CHAPTER XHYPOTHESIS TESTING 10.2 WHAT IS A HYPOTHESIS TEST? • The word hypothesis has the same meaning in statistics as it does in everyday use. What does this word mean to you? Some possibilities are • an idea • an assumption • a guess • a theory • In statistics, a hypothesisis an idea, and assumption, or a theory about the behavior of one or more variables in one or more populations. • A hypothesis testis a statistical procedure that involves formulating a hypothesis and using sample data to decide on the validity of the hypothesis. Dr. Serhat Eren

  4. CHAPTER XHYPOTHESIS TESTING 10.3 DESIGNING HYPOTHESES TO BE TESTEDAN OVERVIEW 10.3.1 Hypotheses About quantitative Variables • If we are constructing a theory or hypothesis about a quantitative variable it might be a statement about: • The shape of the distribution of the variable in one population • The mean value, , of the variable in one population • How the mean value of the variable in one population, 1 compares to the mean value of the variable in a second population, 2 • The equality of the mean values of the variable in more than two populations Dr. Serhat Eren

  5. CHAPTER XHYPOTHESIS TESTING 10.3 DESIGNING HYPOTHESES TO BE TESTEDAN OVERVIEW 10.3.1 Hypotheses About quantitative Variables • The amount of variability, ², of the variable in one population • How the amount of variability of the variable in one population, ²1, compares to the amount of variability of a variable in a second population, ²2. • All but the first bulleted item are generally referred to as “tests on means” and “tests on variances”. The first item requires a "goodness of fit" test. Dr. Serhat Eren

  6. Dr. Serhat Eren

  7. CHAPTER XHYPOTHESIS TESTING 10.3 DESIGNING HYPOTHESES TO BE TESTEDAN OVERVIEW 10.3.2 Hypotheses About Nominal Variables • In addition to using nominal variables as a way to divide the data into two groups, we are often interested in what percentage or proportion of a population has a particular characteristic. • For example, we might be interested in what proportion of the product we are manufacturing is defective. • In this case, the nominal variable is the quality status of the product, non-defective or defective, and we are interested in the percentage or proportion of the population that has the quality status "defective." Dr. Serhat Eren

  8. CHAPTER XHYPOTHESIS TESTING 10.3 DESIGNING HYPOTHESES TO BE TESTEDAN OVERVIEW 10.3.2 Hypotheses About Nominal Variables • If the data we are analyzing are nominal data, the hypothesis might be a statement about: • The value of the proportion, , of population members that have a certain characteristic (one of the categories of the nominal variable). • How the proportion who have a certain characteristic in one population, 1, compares with the corresponding proportion in a second population, 2. Dr. Serhat Eren

  9. Dr. Serhat Eren

  10. CHAPTER XHYPOTHESIS TESTING 10.3 DESIGNING HYPOTHESES TO BE TESTEDAN OVERVIEW 10.3.3 Hypotheses About Ordinal Variables • When numbers are used to name ordered categories the data are called ordinal. If we are analyzing ordinal data, the hypothesis might be a statement about • The mean value, , of the variable in one population • How the mean value of the variable in one population, 1, compares to the mean value of the variable in a second population, 2 • The equality of the mean values of the variable in more than two populations Dr. Serhat Eren

  11. Dr. Serhat Eren

  12. CHAPTER XHYPOTHESIS TESTING 10.3 DESIGNING HYPOTHESES TO BE TESTEDAN OVERVIEW 10.3.4 Hypotheses About Two Variables • In this case we could treat the number of hours per day that a person exercises as a quantitative variable and use the effectiveness rating to divide the data into 5 populations. • We could then construct the hypothesis that the average number of hours exercised is the same for students with an effectiveness rating of 1, 2, 3, 4, and 5. • However, if the data are collected in such a way that both variables are qualitative variables, then we need to construct the hypothesis that says that the variable exercise rating and study effectiveness rating are independent. This is typically called a test for independence. Dr. Serhat Eren

  13. CHAPTER XHYPOTHESIS TESTING 10.3 DESIGNING HYPOTHESES TO BE TESTEDAN OVERVIEW 10.3.5 Summary of Kinds of Hypotheses • Table 10.1 summarizes the various kinds of hypotheses you are likely to need to analyze sample data and make informed business decisions. Dr. Serhat Eren

  14. Dr. Serhat Eren

  15. CHAPTER XHYPOTHESIS TESTING 10.4 THE PIECES OF A HYPOTHESIS TEST 10.4.1 The Null and Alternative Hypotheses • The first step is to take your idea or hypothesis and construct two opposing views. Dr. Serhat Eren

  16. CHAPTER XHYPOTHESIS TESTING 10.4 THE PIECES OF A HYPOTHESIS TEST 10.4.1 The Null and Alternative Hypotheses • One of these is called the null hypothesis and the other is called the alternative hypothesis. • Using the notation of hypothesis testing, you would rewrite this as H0:  = 1000 lb/ream HA:   1000 lb/ream • The null hypothesisis a statement about the population(s). It is referred to as H0. Dr. Serhat Eren

  17. CHAPTER XHYPOTHESIS TESTING 10.4 THE PIECES OF A HYPOTHESIS TEST 10.4.1 The Null and Alternative Hypotheses • The alternative hypothesisis a statement about the population(s) that is opposite to the null hypothesis. It is referred to as HA. • You should notice that no value of is part of both the null and the alternative hypotheses. The null and alternative hypotheses cannot overlap. That is, the two hypotheses are mutually exclusive. • The null and the alternative hypotheses must cover all the possibilities. • Finally, note that the "=" sign is in the null hypothesis. This will always be the case. Dr. Serhat Eren

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  20. CHAPTER XHYPOTHESIS TESTING 10.4 THE PIECES OF A HYPOTHESIS TEST 10.4.2 The Rejection Region • The next step is to determine how you will decide between the null and the alternative hypotheses. • Your decision is always phrased with regard to the null hypothesis. If you choose not to reject the null hypothesis this means that the sample data are consistent with the null hypothesis. • To make this decision you must set up what is known as the rejection region. • The rejection regionis the range of values of the test statistic that will lead you to reject the null hypothesis. Dr. Serhat Eren

  21. CHAPTER XHYPOTHESIS TESTING 10.4 THE PIECES OF A HYPOTHESIS TEST 10.4.2 The Rejection Region • For example, the rejection region might be all values of Z larger than 1.96. This is shown as the shaded region in Figure 10.1. Dr. Serhat Eren

  22. CHAPTER XHYPOTHESIS TESTING 10.4 THE PIECES OF A HYPOTHESIS TEST 10.4.2 The Rejection Region • If this is the rejection region, we would reject the null hypothesis if the value of Z calculated from the data is greater than 1.96. • If the value of Z calculated is less than or equal to1.96, we would not reject the null hypothesis. Dr. Serhat Eren

  23. CHAPTER XHYPOTHESIS TESTING 10.4 THE PIECES OF A HYPOTHESIS TEST 10.4.3 The Test Statistic • A test statistic is a number that captures the information in the sample. • It will be used to choose between the null and alternative hypotheses. Dr. Serhat Eren

  24. CHAPTER XHYPOTHESIS TESTING 10.6 WHICH THEORY SHOULD GO INTO THE NULL HYPOTHESIS? 10.6.1 Two-Tail Tests and One-Tail Tests • A two-tail test of the population mean has the following null and alternative hypotheses: H0:  = [a specific number] HA:   [a specific number] • The null hypothesis of a two-tail test claims that the mean (or whatever parameter you are testing) is actually equal to the particular number stated. Dr. Serhat Eren

  25. CHAPTER XHYPOTHESIS TESTING 10.6 WHICH THEORY SHOULD GO INTO THE NULL HYPOTHESIS? 10.6.1 Two-Tail Tests and One-Tail Tests • The opposing view is clearly that the mean is not equalto that particular value, and this is the alternative hypothesis. • When you use a two-tail test you are interested in seeing if the true mean is differentfrom the number specified. • You wish to know if the true mean is higher thanthe number or lower thanthe number. In other words, you want to test for deviations from the number in either direction-on the high side or the low side. Dr. Serhat Eren

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  28. CHAPTER XHYPOTHESIS TESTING 10.6 WHICH THEORY SHOULD GO INTO THE NULL HYPOTHESIS? 10.6.1 Two-Tail Tests and One-Tail Tests • Sometimes you really wish to see only if the population mean (or whatever parameter you are testing) is lower than the stated value. • In this case, you are interested only in testing if the true mean, proportion, or variance is less thansome number. • Then you should use what is called a lower-tail test. A lower-tail test is one of two types of one-tail tests. Dr. Serhat Eren

  29. CHAPTER XHYPOTHESIS TESTING 10.6 WHICH THEORY SHOULD GO INTO THE NULL HYPOTHESIS? 10.6.1 Two-Tail Tests and One-Tail Tests • A lower-tail test of a population mean has the following null and alternative hypotheses: H0:  [a specific number] HA:  < [a specific number] Dr. Serhat Eren

  30. Dr. Serhat Eren

  31. CHAPTER XHYPOTHESIS TESTING 10.6 WHICH THEORY SHOULD GO INTO THE NULL HYPOTHESIS? 10.6.1 Two-Tail Tests and One-Tail Tests • You may also wish to see if the true mean is greater than the stated value. • In this case, you are interested only in testing if the true mean is greater thansome number. This is also called an upper-tail test. • An upper-tail test is the other kind of one-tail test. In general, such a test would look like this: H0:  [a specific number] HA:  > [a specific number] Dr. Serhat Eren

  32. Dr. Serhat Eren

  33. CHAPTER XHYPOTHESIS TESTING 10.6 WHICH THEORY SHOULD GO INTO THE NULL HYPOTHESIS? 10.6.1 Two-Tail Tests and One-Tail Tests 10.6.1.1 Summary • A Two-Tail Test • Is used to test if the parameter has shifted away from a certain number in either direction, increased or decreased. • Must always be set up so the "=" theory is the null hypothesis. • Is used when the problem statement has the key words changedor differentin the problem statement. Dr. Serhat Eren

  34. CHAPTER XHYPOTHESIS TESTING 10.6 WHICH THEORY SHOULD GO INTO THE NULL HYPOTHESIS? 10.6.1 Two-Tail Tests and One-Tail Tests 10.6.1.1 Summary • One-Tail Tests • A Lower-Tail Test • Is used to test if the parameter has shifted to a number less than a certain number. • Must always be set up with the "=" as part of the null hypothesis. • Is used when the problem statement has the key wordsdecreased, reduced, or less than. • The theory that you wish to "prove" is placed into the alternative hypothesis. Dr. Serhat Eren

  35. CHAPTER XHYPOTHESIS TESTING 10.6 WHICH THEORY SHOULD GO INTO THE NULL HYPOTHESIS? 10.6.1 Two-Tail Tests and One-Tail Tests 10.6.1.1 Summary • One-Tail Tests • An Upper-Tail Test • Is used to test if the parameter has shifted to a number more than a certain number. • Must always be set up with the "=" as part of the null hypothesis. • Is used when the problem statement has the key words increased, greater than. • The theory that you wish to "prove" is placed into the alternative hypothesis. Dr. Serhat Eren

  36. CHAPTER XHYPOTHESIS TESTING 10.6 WHICH THEORY SHOULD GO INTO THE NULL HYPOTHESIS? 10.6.2 What View Requires No Action? • What view requires that I take no action? • Typically, this is the view that the population is behaving as it should be or as someone claims it should be. • This view becomes the null hypothesis. Sometimes people call this the status quo. Dr. Serhat Eren

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  38. CHAPTER XHYPOTHESIS TESTING 10.8 WHAT ERROR COULD YOU BE MAKING? • You may choose to believe the null hypothesis when, in fact, it is not true. Alternatively, you may choose to believe the alternative hypothesis (rejecting the null hypothesis) when in fact it is not true. • These are the two ways that you could be wrong when you perform a hypothesis test. • In hypothesis testing, we use probabilities to measure the chance of being wrong. If we can state that there is a 5% chance that we have made an error, then we have a sense of how often we will be wrong. Dr. Serhat Eren

  39. CHAPTER XHYPOTHESIS TESTING 10.8 WHAT ERROR COULD YOU BE MAKING? 10.8.1 Two Types of Errors • Four outcomes could result from the decisions you make in each hypothesis test situation. • Look at Table 10.2 and think carefully about how each of these outcomes could occur. Dr. Serhat Eren

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  41. CHAPTER XHYPOTHESIS TESTING 10.8 WHAT ERROR COULD YOU BE MAKING? 10.8.1 Two Types of Errors • Let’s look at these four possible outcomes from the perspective of the tissue manufacturer, in Table 10.3. Dr. Serhat Eren

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  43. CHAPTER XHYPOTHESIS TESTING 10.8 WHAT ERROR COULD YOU BE MAKING? 10.8.1 Two Types of Errors • To be clear which of the two possible errors we are talking about, we need to give them names, in Table 10.4. Dr. Serhat Eren

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  45. CHAPTER XHYPOTHESIS TESTING 10.8 WHAT ERROR COULD YOU BE MAKING? 10.8.1 Two Types of Errors • A Type I error is made when you reject the null hypothesis and the null hypothesis is actually true. In other words, you incorrectly reject a true null hypothesis. • A Type II error is made when you fail to reject the null hypothesis and the null hypothesis is actually false. In other words, you continue to believe a false null hypothesis. Dr. Serhat Eren

  46. Dr. Serhat Eren

  47. CHAPTER XHYPOTHESIS TESTING 10.8 WHAT ERROR COULD YOU BE MAKING? 10.8.2 Probability of Making an Error • The probability of making a Type I error is called (alpha) and the probability of making a Type II error is called (beta). • Clearly,  and must be numbers between 0 and 1 since they are probabilities. • As the investigator, you will get to decide the value of . This means that you can specify the chances of making a Type I error to be anything you wish. Dr. Serhat Eren

  48. CHAPTER XHYPOTHESIS TESTING 10.8 WHAT ERROR COULD YOU BE MAKING? 10.8.2 Probability of Making an Error • Once you set , the value of is completely determined. You can force the chance of making a Type I error to be really small but then you have to live with a greater chance of making a Type II error. • This trade-off is pictured in Figure 10.3. Dr. Serhat Eren

  49. Dr. Serhat Eren

  50. CHAPTER XHYPOTHESIS TESTING 10.8 WHAT ERROR COULD YOU BE MAKING? 10.8.2 Probability of Making an Error • If we look at Figure 10.4, we can see that the value of  has been reduced. • This means we will be less likely to rejectH0, when we should not reject it. Dr. Serhat Eren

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