1 Chapter 8 Introduction to Hypothesis Testing

2 Name of the game… Hypothesis testing Statistical method that uses sample data to evaluate a hypothesis about a population

3 Experimental Hypotheses Experimental hypotheses describe the predicted outcome we may or may not find in an experiment.

4 Order of Procedure State hyp about pop Before selecting a sample..use hyp to predict characteristics that sample should have Obtain random sample Compare sample data w/ prediction made in hyp

5 Of interest to the researcher… Did the treatment have any effect on the individuals Must be large (significant) differences in means

6 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 ≠.

7 The Role of Inferential Statistics in Research

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

9 Setting up Inferential Procedures

10 Null Hypothesis H 0 The null hypothesis describes the population parameters that the sample data represent if the predicted relationship does not exist.

11 Alternative Hypothesis H 1 The alternative hypothesis describes the population parameters that the sample data represent if the predicted relationship exists.

12 A Graph Showing the Existence of a Relationship

13 A Graph Showing That a Relationship Does Not Exist

14 Interpreting Significant Results When we reject H 0 and accept H 1, we do not prove that H 0 is false While it is unlikely for a mean that lies within the rejection region to occur, the sampling distribution shows that such means do occur once in a while

15 Failing to Reject H 0 When the statistic does not fall beyond the critical value, the statistic does not lie within the region of rejection, so we do not reject H 0 When we fail to reject H 0 we say the results are nonsignificant. Nonsignificant indicates that the results are likely to occur if the predicted relationship does not exist in the population.

16 Interpreting Nonsignificant Results When we fail to reject H 0, we do not prove that H 0 is true Nonsignificant results provide no convincing evidence—one way or the other—as to whether a relationship exists in nature

17 Errors in Statistical Decision Making

18 Type I Errors A Type I error is defined as rejecting H 0 when H 0 is true In a Type I error, there is so much sampling error that we conclude that the predicted relationship exists when it really does not The theoretical probability of a Type I error equals 

19 Alpha  Probability that the test will lead to a Type I error Alpha level determines the probability of obtaining sample data in the critical region even though the null hypo is true

20 Type II Errors A Type II error is defined as retaining H 0 when H 0 is false (and H 1 is true) In a Type II error, the sample mean is so close to the  described by H 0 that we conclude that the predicted relationship does not exist when it really does The probability of a Type II error is 

21 Power The goal of research is to reject H 0 when H 0 is false The probability of rejecting H 0 when it is false is called power

22 Possible Results of Rejecting or Retaining H 0

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

24 Nonparametric Procedures Nonparametric statistics are inferential procedures that do not require stringent assumptions about the populations being represented.

25 Robust Procedures Parametric procedures are robust. If the data don’t meet the assumptions of the procedure perfectly, we will have only a negligible amount of error in the inferences we draw.

26 Predicting a Relationship A two-tailed test is used when we predict that there is 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.

27 The One-Tailed Test

28 One-Tailed Hypotheses In a one-tailed test, if it is hypothesized that the independent variable causes an increase in scores, then the null hypothesis is that the population mean is less than or equal to a given value and the alternative hypothesis is that the population mean is greater than the same value. For example: H 0 :  ≤ 50 H a :  > 50

29 A Sampling Distribution Showing the Region of Rejection for a One-tailed Test of Whether Scores Increase

30 One-Tailed Hypotheses In a one-tailed test, if it is hypothesized that the independent variable causes a decrease in scores, then the null hypothesis is that the population mean is greater than or equal to a given value and the alternative hypothesis is that the population mean is less than the same value. For example: H 0 :  ≥ 50 H a :  < 50

31 A Sampling Distribution Showing the Region of Rejection for a One-tailed Test of Whether Scores Decrease

32 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.

33 Performing the z-Test

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

35 Assumptions of the z-Test 1. We have randomly selected one sample 2. The dependent variable is at least approximately normally distributed in the population and involves an interval or ratio scale 3. We know the mean of the population of raw scores under some other condition of the independent variable 4.We know the true standard deviation of the population described by the null hypothesis

36 Setting up for a Two-Tailed Test 1. Choose alpha. Common values are 0.05 and Locate the region of rejection. For a two- tailed test, this will involve defining an area in both tails of the sampling distribution. 3. Determine the critical value. Using the chosen alpha, find the z crit value that gives the appropriate region of rejection.

37 A Sampling Distribution for H0 Showing the Region of Rejection for  = 0.05 in a Two-tailed Test

38 In a two-tailed test, the null hypothesis states that the population mean equals a given value. For example, H 0 :  = 100. In a two-tailed test, the alternative hypothesis states that the population mean does not equal the same given value as in the null hypothesis. For example, H a :  100. Two-Tailed Hypotheses

39 Computing z The z-score is computed using the same formula as before where

40 Rejecting H 0 When the z obt falls beyond the critical value, the statistic lies in the region of rejection, so we reject H 0 and accept H a When we reject H 0 and accept H a we say the results are significant. Significant indicates that the results are too unlikely to occur if the predicted relationship does not exist in the population.

41 Interpreting Significant Results When we reject H 0 and accept H a, we do not prove that H 0 is false While it is unlikely for a mean that lies within the rejection region to occur, the sampling distribution shows that such means do occur once in a while

42 Failing to Reject H 0 When the z obt does not fall beyond the critical value, the statistic does not lie within the region of rejection, so we do not reject H 0 When we fail to reject H 0 we say the results are nonsignificant. Nonsignificant indicates that the results are likely to occur if the predicted relationship does not exist in the population.

43 Interpreting Nonsignificant Results When we fail to reject H 0, we do not prove that H 0 is true Nonsignificant results provide no convincing evidence—one way or the other—as to whether a relationship exists in nature

44 1. Determine the experimental hypotheses and create the statistical hypothesis Summary of the z-Test 2.Compute and compute z obt 3.Set up the sampling distribution 4.Compare z obt to z crit