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Chapter Ten Introduction to Hypothesis Testing

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 2 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 ≠.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 4 Sampling Error Remember: Sampling error results when random chance produces a sample statistic that does not equal the population parameter it represents.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 5 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.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 7 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.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 9 Experimental Hypotheses Experimental hypotheses describe the predicted outcome we may or may not find in an experiment.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 10 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.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 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.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 12 Alternative Hypothesis The alternative hypothesis describes the population parameters that the sample data represent if the predicted relationship exists.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 13 Null Hypothesis The null hypothesis describes the population parameters that the sample data represent if the predicted relationship does not exist.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 17 The z-Test The z-test is the procedure for computing a z-score for a sample mean on the sampling distribution of means.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 18 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

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 19 Setting up for a Two-Tailed Test 1.Choose alpha. Common values are 0.05 and 0.01. 2.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.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 20 A Sampling Distribution for H0 Showing the Region of Rejection for  = 0.05 in a Two-tailed Test

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 21 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

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 23 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.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 24 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

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 25 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.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 26 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

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 27 1.Determine the experimental hypotheses and create the statistical hypothesis 2.Compute and compute z obt 3.Set up the sampling distribution 4.Compare z obt to z crit Summary of the z-Test

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 29 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

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 31 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

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 32 A Sampling Distribution Showing the Region of Rejection for a One-tailed Test of Whether Scores Decrease [Insert Figure 10.8 here]

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 33 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.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 35 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 

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 36 Type II Errors A Type II error is defined as retaining H 0 when H 0 is false (and H a 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 

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 37 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

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 39 14 13151115 131012131413 14151714 15 Example Use the following data set and conduct a two-tailed z-test to determine if  = 11 if the population standard deviation is known to be 4.1

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 40 Example 1.H 0 :  = 11; H a :  ≠ 11 2.Choose  = 0.05 3.Reject H 0 if z obt > +1.965 or if z obt < -1.965.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 10 - 41 Example Since z obt lies within the rejection region, we reject H 0 and accept H a. Therefore, we conclude that  ≠ 11.