1. CHAPTER 7: TESTING HYPOTHESES Leon-Guerrero and Frankfort-Nachmias, Essentials of Statistics for a Diverse Society

2. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Chapter 7: Testing Hypotheses  Overview  Assumptions of Statistical Hypothesis Testing  Stating the Research and Null Hypotheses  Determining What is Sufficiently Improbable: Probability Values and Alpha  Five Steps in Hypothesis Testing: A Summary  Errors in Hypothesis Testing  Testing Hypotheses About Two Samples  The Sampling Distribution of the Difference Between Means  The t Statistic

3. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Overview Interval Nominal Dependent Variable Independent Variables Nominal Interval Considers the distribution of one variable across the categories of another variable Considers the difference between the mean of one group on a variable with another group Considers how a change in a variable affects a discrete outcome Considers the degree to which a change in one variable results in a change in another

4. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Overview Interval Nominal Dependent Variable Independent Variables Nominal Interval Logistic Regression Regression Correlation Lambda Confidence Intervals T-Test Overview

5. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications General Examples Is one group scoring significantly higher on average than another group? Is a group statistically different from another on a particular dimension? Is Group A ’ s mean higher than Group B ’ s?

6. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Specific Examples Do people living in rural communities live longer than those in urban or suburban areas? Do students from private high schools perform better in college than those from public high schools? Is the average number of years with an employer lower or higher for large firms (over 100 employees) compared to those with fewer than 100 employees?

7. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Statistical hypothesis testing – A procedure that allows us to evaluate hypotheses about population parameters based on sample statistics. Research hypothesis (H 1 ) – A statement reflecting the substantive hypothesis. It is always expressed in terms of population parameters, but its specific form varies from test to test. Null hypothesis (H 0 ) – A statement of “ no difference, ” which contradicts the research hypothesis and is always expressed in terms of population parameters. Testing Hypotheses

8. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Research and Null Hypotheses One Tail — specifies the hypothesized direction Research Hypothesis: H 1 :  1  2, or  1  2 > 0 Null Hypothesis: H 0 :  1  2, or  1  2 = 0 Two Tail — direction is not specified (more common) Research Hypothesis: H 1 :  1  2, or  1  2 = 0 Null Hypothesis: H 0 :  1  2, or  1  2 = 0

9. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications One-Tailed Tests  One-tailed hypothesis test – A hypothesis test in which the alternative is stated in such a way that the probability of making a Type I error is entirely in one tail of a sampling distribution.  Right-tailed test – A one-tailed test in which the sample outcome is hypothesized to be at the right tail of the sampling distribution.  Left-tailed test – A one-tailed test in which the sample outcome is hypothesized to be at the left tail of the sampling distribution.

10. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Two-Tailed Tests  Two-tailed hypothesis test – A hypothesis test in which the region of rejection falls equally within both tails of the sampling distribution.

11. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Probability Values  Z statistic (obtained) – The test statistic computed by converting a sample statistic (such as the mean) to a Z score. The formula for obtaining Z varies from test to test.  P value – The probability associated with the obtained value of Z.

12. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Probability Values

13. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Probability Values  Alpha ( ) – The level of probability at which the null hypothesis is rejected. It is customary to set alpha at the.05,.01, or.001 level.

14. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Five Steps to Hypothesis Testing (1) Making assumptions (2) Stating the research and null hypotheses and selecting alpha (3) Selecting the sampling distribution and specifying the test statistic (4) Computing the test statistic (5) Making a decision and interpreting the results

15. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Type I error (false rejection error)  the probability (equal to  ) associated with rejecting a true null hypothesis. Type II error (false acceptance error)  the probability associated with failing to reject a false null hypothesis. Based on sample results, the decision made is to… reject H 0 do not reject H 0 In thetrueType Icorrect populationerror (  )decision H 0 is... false correctType II error decision Type I and Type II Errors

16. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications t-Test  t statistic (obtained) – The test statistic computed to test the null hypothesis about a population mean when the population standard deviation is unknown and is estimated using the sample standard deviation.  t distribution – A family of curves, each determined by its degrees of freedom (df). It is used when the population standard deviation is unknown and the standard error is estimated from the sample standard deviation.  Degrees of freedom (df) – The number of scores that are free to vary in calculating a statistic.

17. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications t Distribution

18. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications t Distribution Table

19. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications t-Test for Difference Between Two Means Is the value of  2  1 significantly different from 0? This test gives you the answer: If the t value is greater than 1.96, the difference between the means is significantly different from zero at an alpha of.05 (or a 95% confidence level).  The difference between the two means  the estimated standard error of the difference The critical value of t will be higher than 1.96 if the total N is less than 122. See Appendix C for exact critical values when N < 122.

20. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Estimated Standard Error of the Difference Between Two Means Assuming Unequal Variances

21. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications t-Test and Confidence Intervals The t-test is essentially creating a confidence interval around the difference score. Rearranging the above formula, we can calculate the confidence interval around the difference between two means: If this confidence interval overlaps with zero, then we cannot be certain that there is a difference between the means for the two samples.

22. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Why a t Score and not a Z Score? Use of the Z distribution has assumes the population standard error of the difference is known. In practice, we have to estimate it and so we use a t score. When N gets larger than 50, the t distribution converges with a Z distribution so the results would be identical regardless of whether you used a t or Z. In most sociological studies, you will not need to worry about the distinction between Z and t.

23. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Example (1): t-Test What can we conclude about the difference in wages? Men (N=54)Women (N=46) Mean pay$10.06$10.29 Standard Deviation.9051.8766

24. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications What can we conclude about the difference in wages? Example (2):t-Test Men (N=51)Women (N=57) Mean Pay$10.32$9.68 Standard Deviation.94611.0550

25. Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society © 2012 SAGE Publications Using these GSS income data, calculate a t-test statistic to determine if the difference between the two group means is statistically significant. In-Class Exercise