1. Chapter 6 Preview – Generalizing from Samples to Populations  Sampling error always occurs.  Standard error of the mean allows us to construct confidence intervals for means/proportions.  If the population standard deviation is known, use Z-scores.  However, this is rarely the case which is why we use T-ratios.

2. Chapter 7 Testing Differences between Means

3. Introduction  We’ve learned that a population mean or proportion can be estimated.  Researchers really want to test hypotheses.  Hypotheses typically refer to differences between groups.  In this chapter, we’ll learn how test hypotheses about differences between sample means and proportions.

4. 4 Testing Differences Between Means  Establish hypothesis about populations, collect sample data, and see how likely the sample results are, given the hypothesis. Example: Memory enhancement  N = 10 Method AMethod B 8278 8377 8276 8078 8376 Mean = 82Mean = 77

5. 5 Method AMethod B 9070 9890 6391 7456 8578 Mean = 82Mean = 77  Now suppose instead that the following sets of scores produced the two sample means of 82 and 77.

6. 6 The Null Hypothesis  No difference between means An obtained difference between two sample means does not represent a true difference between their population means  Mean of the first population = mean of the second population  Example: Germans are no more obedient to authority than Americans.  Example: Female and Male teachers have the same disciplinary styles Retain or reject the null hypothesis Null hypothesis shown as H 0 :

7. 7 The Research Hypothesis  Differences between groups, whether expected on theoretical or empirical grounds, often provide the rationale for research Mean of the first population does not equal the mean of the second population  If we reject the null hypothesis, we automatically accept the research hypothesis that a true population difference does exist. Different means Example: Germans differ from Americans with respect to obedience to authority. Research hypothesis shown as H 1 :

8. 8 Testing Hypothesis with the Distribution of Differences Between Means Seek to make statements of probability about the difference scores in the sampling distribution of differences between means.

9. 9 Testing Hypothesis with the Distribution of Differences Between Means Cont.  The sampling distribution of differences provides a sound basis for testing hypotheses about the difference between two sample means Example: Suppose for interest, that a social researcher wanted to test the child-rearing practices of parents, at least a variation of it, in a realistic way  Is there a gender difference in child-rearing attitudes?  At random, selects 30 mothers and 30 fathers.  Mean of the mothers is a score of 45, and the mean for the fathers is a score of 40.

10. 10 Levels of Significance  To establish whether our obtained sample difference is statistically significant – the result of a real population difference and not just sampling error – it is customary to set up a level of significance Denoted by the Greek letter alpha (α)  The alpha value is the level of probability at which the null hypothesis can be rejected with confidence and the research hypothesis can be accepted with confidence.

11. 11 Levels of Significance Cont.  Significance levels can be set up for any degree of probability  The alpha =.05 level of significance is the most commonly used.  The.05 level of significance is found in the small areas of the tails of the distribution mean differences  Other common values are.01 and in exploratory research,.10 is used occasionally

12. 12 Levels of Significance Cont. .01 level of significance is a more stringent level, referred to as beta  The null hypothesis is rejected if there is less than 1 chance out of 100 that the obtained sample difference could occur by sampling error represented by the area that lies 2.58 standard deviations in both directions from a mean difference of zero.  However, the research must also take into consideration the implications of committing a Type I and Type II error

13. 13 Type I and II Errors Correct DecisionType I Error P (Type I Error) = alpha Type II Error P (Type II Error) = beta Correct Decision DECISION Retain Null Reject Null Null is true REALITY Null is false

14. Type I and Type II Errors 14 Rejecting the null hypothesis when we should have retained it is known as a Type I error. The more stringent our level of significance, the less likely we are to make Type I errors. Retaining the null hypothesis when we should reject is known as a Type II error Best solution – increase the sample size so that a true population difference is more likely to be represented.

15. 15 Choosing a Level of Significance  Type I error and Type II error are inversely related: the larger one error is, the smaller the other The larger the chosen level of significance (say,.05 or even.10), the larger the chance of Type I error and the smaller the chance of Type II error. The smaller the chosen significance level (say,.01 or even.001), the smaller the chance of Type I error, but the greater the likelihood of Type II error.

16. 16 Choosing a Level of Significance Suppose for example that a researcher were doing research on gender differences in SAT performance for which she administered an SAT to a sample of males and a sample of females. What would be worse? Type I error or Type II error? Suppose that a researcher is testing the effects of marijuana smoking on SAT performance, and he compares a sample of smokers with a sample of nonsmokers. What would be worse? Type I error or Type II error?

17. 17 What is the Difference Between P and Alpha?  The difference between P and alpha can be a bit confusing P is the exact probability that the null hypothesis is true in light of the sample data The alpha value is the threshold below which is considered so small that we decide to reject the null hypothesis A =.001 A =.01 A =.05 P <.001 P <.01 P <.05

18. Standard Error of the Difference between Means  Standard deviation of the distribution of differences can be estimated.  The standard error of the differences between means is shown as:

19. Testing the Difference between Means  Why use t instead of z? We don’t know the true population standard deviation.  Test differences between means using t:  This is referred to as our T computed

20. Comparing our T value  Using Table C, we find our T critical value.  To calculate the degrees of freedom (df) when testing the difference between means we use the following formula df = N1 + N2 – 2  Alpha value is given (.05 or.01)  If T computed > T critical, reject null  If T computed < T critical, accept null 20

21. 21 Testing the Difference between Means  Suppose that we obtained the following data for a sample of 25 liberals and 35 conservatives on the permissiveness scale.  Calculate the estimate of the standard error of the differences between means.  Then, translate the difference between sample means into a t ratio. LiberalsConservatives N 1 = 25N 2 = 35 Mean 1 = 60Mean 2 = 49 S 1 = 12S 2 = 14

22. 22 Continued. If necessary, find the mean and standard deviation first. Otherwise:  Step 1: Find the standard error of the difference between means.  Step 2: Compute the t ratio.  Step 3: Determine the critical value for t.  Step 4: Compare the calculated and table t values.

23. End Day 1  http://www.people- press.org/files/legacy-pdf/10-15- 12%20Debates%20Release%20.p df http://www.people- press.org/files/legacy-pdf/10-15- 12%20Debates%20Release%20.p df 23

24. 24 Comparing the Same Sample Measured Twice  So far, we have discussed making comparisons between two independently drawn samples  Before-after or panel design: the case of a single sample measured at two different points in time (time 1 vs. time 2)  For example, a polling organization might interview the same 1,000 Americans both in 1995 and 2000 in order to measure their change in attitude over time.  Numerous uses for this type of test

25. 25 Test of Difference between Means for Same Sample Measured Twice  Suppose that several individuals have been forced by a city government to relocate their homes to make way for highway construction.  As researchers, we are interested in determining the impact of forced residential mobility on feelings of neighborliness.  What would the null and research hypotheses state?  We interview a random sample of 6 individuals about their neighbors both before and after they are forced to move.

26. Testing the Difference Between Means for the Same Sample Measured Twice  To obtain the standard error of the difference between means use the following formula:  Where: S D = Standard deviation of the distribution of before-after difference scores. 26

27. Finding the t ratio 27 Computed T ratio: Critical T: df = N – 1 α =.05 or.01 Use Table C Compare the computed T with the critical T. If |T| > critical T, reject null hypothesis. If |T| < critical T, retain null hypothesis.

28. Their Scores RespondentBeforeAfter Stephanie21 Leon12 Carol31 Jake31 Julie12 David41 28

29. 29 Test of Difference between Means for Same Sample Measured Twice RespondentBeforeAfterDifferenceDifference 2 Stephanie2111 Leon121 Carol3124 Jake3124 Julie121 David4139 ΣX = 14ΣX = 8 ΣD 2 = 20 N = 6

30. 30 Test of Difference between Means for Same Sample Measured Twice Make sure to first state the null and research hypothesis! After finishing the table:  Step 1: Find the mean for each point in time (Before = X 1 ; After = X 2 )  Step 2: Find the standard deviation for the difference between time 1 and time 2  Step 3: Find the standard error of the difference between means  Step 4: Calculate the t score  Step 5: Find the critical t score  Step 6: Compare the obtained t ratio with the critical t score

31. Two Sample Test of Proportions 31

32. Two Sample Test of Proportions Formulas 32

33. Two Sample Test Example A criminal justice researcher is interested in marijuana usage and driving while high of upper level undergraduates in her particular school. After taking a random sample of 300 students, she discards any surveys of students who have not smoked marijuana. She is left with the following data: Test the research hypothesis at the alpha level of.05. What do your results indicate? 33 MaleFemale Sample Size127149 Driven high5636

34. 34 One-Tailed Tests 1. A one-tailed test rejects the null hypothesis at only one tail of the sampling distribution. 2. It should be emphasized, however, that the only changes are in the way the hypotheses are stated and the place where the t table is entered. 3. Used when the researcher anticipates the direction of change.

35. 35 One-Tailed Tests Cont. 1. A one-tailed test is appropriate when a researcher is only concerned with a chance (for a sample tested twice) or difference (between two independent samples) in one pre-specified direction or when a researcher anticipates the DIRECTION of the change or difference. 1. Example: An attempt to show that Black defendants receive harsher sentences (mean sentence) than Whites indicates the need for a one-tailed test. 1. Null: u1 ≥ u2 White defendants receive equal or harsher sentences. 2. Research: u1 < u2 Black defendants receive harsher sentences 2. If we were only attempting to show that there are differences in sentencing by race indicates the need for a two tailed test. 2. If however, a researcher is just looking for differences in sentencing by race, whether it is Blacks or Whites who get harsher sentences, he or she would instead use a two-tailed test.

36. 36 Requirments for Testing the Differences between Means 1. A comparison between two means 2. Interval data 3. Random sampling 4. A normal distribution 5. Equal population variances

37. Summary  Testing hypotheses about differences between sample means  Null / Research hypothesis  Type I and Type II errors  Alpha levels used to reject or retain the null hypothesis