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Chapter Ten

Chapter Ten. Introduction to Hypothesis Testing. New Statistical Notation. The symbol for greater than is > The symbol for less than is < The symbol for greater than or equal to is ≥ The symbol for less than or equal to is ≤ The symbol for not equal to is ≠. Chapter 10 - 2.

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Chapter Ten

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  1. Chapter Ten Introduction to Hypothesis Testing

  2. New Statistical Notation The symbol for greater than is > The symbol for less than is < The symbol for greater than or equal tois ≥ The symbol for less than or equal tois ≤ The symbol for not equal to is ≠ Chapter 10 - 2

  3. The Role of Inferential Statistics in Research Chapter 10 - 3

  4. Sampling Error Remember: Sampling error results when random chance produces a sample statistic that does not equal the population parameter it represents. Chapter 10 - 4

  5. Inferential Statistics Inferential statistics are used to decide if sample data represent a particular relationship in the population. Chapter 10 - 5

  6. Parametric Statistics Parametric statistics are procedures that require certain assumptions about the characteristics of the populations Two assumptions are common to all parametric procedures: The population of dependent scores forms a normal distribution and The scores are interval or ratio Chapter 10 - 6

  7. Nonparametric Procedures Nonparametric statistics are inferential procedures that do not require stringent assumptions about the populations Nonparametric statistics are typically used with nominal or ordinal scores, or with skewed interval or ratio distributions Chapter 10 - 7

  8. Setting up Inferential Procedures Chapter 10 - 8

  9. Experimental Hypotheses Experimental hypotheses describe the predicted relationship we may or may not find. Chapter 10 - 9

  10. Predicting a Relationship A two-tailed test is used when we predict a relationship, but do not predict the direction in which scores will change A one-tailed test is used when we predict the direction in which scores will change Chapter 10 - 10

  11. Designing a One-Sample Experiment To perform a one-sample experiment, we must already know the population mean under some other condition of the independent variable. Chapter 10 - 11

  12. Statistical Hypotheses Statistical hypotheses describe the population parameters the sample data represent if the predicted relationship does or does not exist. Chapter 10 - 12

  13. Statistical Hypotheses The alternative hypothesis describes the population parameters the sample data represent if the predicted relationship exists. Chapter 10 - 13

  14. Statistical Hypotheses The null hypothesis describes the population parameters the sample data represent if the predicted relationship does not exist. Chapter 10 - 14

  15. A Graph Showing the Existence of a Relationship Chapter 10 - 15

  16. A Graph Showing a Relationship Does Not Exist Chapter 10 - 16

  17. Performing the z-Test Chapter 10 - 17

  18. The z-Test The z-test is the procedure for computing a z-score for a sample mean on the sampling distribution of means. Chapter 10 - 18

  19. We have randomly selected one sample The dependent variable is at least approximately normally distributed in the population and involves an interval or ratio scale We know the mean of the population of raw scores under some other condition of the independent variable We know the true standard deviation of the population described by the null hypothesis Assumptions of the z-Test Chapter 10 - 19

  20. Setting up for a Two-Tailed Test Choose alpha. Common values are 0.05 and 0.01. Locate the region of rejection. For a two-tailed test, this will involve defining an area in both tails of the sampling distribution. Determine the critical value. Using the chosen alpha, find the zcrit value that gives the appropriate region of rejection. Chapter 10 - 20

  21. A Sampling Distribution for H0 Showing the Region of Rejection for a = 0.05 in a Two-tailed Test Chapter 10 - 21

  22. In a two-tailed test, the null hypothesis states the population mean equals a given value. For example, H0: m = 100. In a two-tailed test, the alternative hypothesis states the population mean does not equal the same given value as in the null hypothesis. For example, Ha: m 100. Two-Tailed Hypotheses Chapter 10 - 22

  23. Computing z The z-score is computed using the same formula as before where Chapter 10 - 23

  24. Rejecting H0 When the zobt falls beyond the critical value, the statistic lies in the region of rejection, so we reject H0 and accept Ha. When we reject H0 and accept Ha we say the results are significant. Significant indicates the results are too unlikely to occur if the predicted relationship does not exist in the population. Chapter 10 - 24

  25. Interpreting Significant Results When we reject H0 and accept Ha, we do not prove that H0 is false While it is unlikely for a mean that lies within the rejection region to occur, the sampling distribution shows such means do occur once in a while Chapter 10 - 25

  26. Failing to Reject H0 When the zobt does not fall beyond the critical value, the statistic does not lie within the region of rejection, so we “fail to reject H0.” When we fail to reject H0 we say the results are nonsignificant. Nonsignificant indicates the results are likely to occur if the predicted relationship does not exist in the population. Chapter 10 - 26

  27. Interpreting Nonsignificant Results When we fail to reject H0, we do not prove that H0 is true Nonsignificant results provide no convincing evidence—one way or the other—as to whether a relationship exists in nature Chapter 10 - 27

  28. Determine the experimental hypotheses and create the statistical hypothesis Compute compute and compute zobt Set up the sampling distribution Compare zobt to zcrit Summary of the z-Test Chapter 10 - 28

  29. The One-Tailed Test Chapter 10 - 29

  30. One-Tailed Hypotheses In a one-tailed test, if it is hypothesized the independent variable causes an increase in scores, the null hypothesis states the population mean is less than or equal to a given value and the alternative hypothesis states the population mean is greater than the same value. For example: H0: m ≤ 50 Ha: m > 50 Chapter 10 - 30

  31. A Sampling Distribution Showing the Region of Rejection for a One-tailed Test of Whether Scores Increase Chapter 10 - 31

  32. One-Tailed Hypotheses In a one-tailed test, if it is hypothesized the independent variable causes a decrease in scores, the null hypothesis states the population mean is greater than or equal to a given value and the alternative hypothesis states the population mean is less than the same value. For example: H0: m ≥ 50 Ha: m < 50 Chapter 10 - 32

  33. A Sampling Distribution Showing the Region of Rejection for a One-tailed Test of Whether Scores Decrease Chapter 10 - 33

  34. Choosing One-Tailed Versus Two-Tailed Tests Use a one-tailed test only when confident of the direction in which the dependent variable scores will change. When in doubt, use a two-tailed test. Chapter 10 - 34

  35. Errors in Statistical Decision Making Chapter 10 - 35

  36. Type I Errors A Type I error is defined as rejecting H0 when H0 is true In a Type I error, there is so much sampling error we are fooled into concluding the predicted relationship exists when it really does not The theoretical probability of a Type I error equals a Chapter 10 - 36

  37. Type II Errors A Type II error is defined as retaining H0 when H0 is false (and Ha is true) In a Type II error, the sample mean is so close to the m described by H0 that we are fooled into concluding the predicted relationship does not exist when it really does The probability of a Type II error is b Chapter 10 - 37

  38. Power The goal of research is to reject H0 when H0 is false The probability of rejecting H0 when it is false is called power Chapter 10 - 38

  39. Possible Results of Rejecting or Retaining H0 Chapter 10 - 39

  40. Example Use the following data set and conduct a two-tailed z-test to determine if m = 11 if the population standard deviation is known to be 4.1 Chapter 10 - 40

  41. Example H0: m = 11; Ha: m ≠ 11 Choose a = 0.05 Reject H0 if zobt > +1.965 or if zobt < -1.965 Chapter 10 - 41

  42. Example Since zobt lies within the rejection region, we reject H0 and accept Ha. Therefore, we conclude that m ≠ 11. Chapter 10 - 42

  43. Key Terms • one-tailed test • parametric statistics • power • significant • statistical hypotheses • two-tailed test • Type I error • Type II error • z-test alpha alternative hypothesis beta experimental hypotheses inferential statistics nonparametric statistics nonsignificant null hypothesis Chapter 10 - 43

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