Chapter 9 Three Tests of Significance Winston Jackson and Norine Verberg Methods: Doing Social Research, 4e

9-2© 2007 Pearson Education Canada Inferential Statistics Inferential statistics do two things: 1. Allow us to judge the accuracy of generalizing from a limited sample to the larger population We can be 95% certain that the sample mean will be 30%, plus or minus 4% 2.Conduct hypothesis testing Indicate whether study outcome is a fluke or reflects a true difference in the population i.e., say if the findings are statistically significant

9-3© 2007 Pearson Education Canada What Does Statistically Significant Mean? A test of significance reports the probability that an observed difference or association is a result of sampling fluctuations and not reflective of a “true” difference in the population from which the sample was selected Three tests of statistical significance introduced in Chapter 9 Chi-square test, t-test, and F-test

9-4© 2007 Pearson Education Canada Preliminary Considerations 1. Research and null hypothesis 2. The sampling distribution Standard error of the means 3. One- and two-tailed tests of significance

9-5© 2007 Pearson Education Canada 1. Research and Null Hypothesis Tests of significance are used to test hypotheses Set up in the form of a “research hypothesis” and “null hypothesis” Research Hypothesis (or Alternative Hypothesis) States one’s prediction of the relationship between the variables Null Hypothesis States the prediction that there is no relation between the variables

9-6© 2007 Pearson Education Canada Research and Null Hypothesis (cont’d) Research hypothesis 1: The greater the participation, the higher the self-esteem Null hypothesis 1: There is no relation between levels of participation and levels of self-esteem Research hypothesis 2: Male university faculty members earn more money than their female counterparts after controlling for qualifications, achievements, and experience Null hypothesis 2: There is no relation between gender and earnings of faculty members, after controls

9-7© 2007 Pearson Education Canada Research and Null Hypothesis (cont’d) It is the null hypothesis that is tested Leads us to accept or reject the null hypothesis If the null hypothesis is accepted: Conclude that the association or difference may simply be the result of sampling fluctuations and may not reflect an association or difference in the population being studied Research hypothesis deemed to therefore be false

9-8© 2007 Pearson Education Canada Research and Null Hypothesis (cont’d) If the null hypothesis is rejected: Argue that there is an association between the variables in the population, and that this association is of a magnitude that probably has not occurred because of chance fluctuations in sampling Would then examine the data to see if the association is in the predicted direction i.e., consistent with prediction (it could be different than predicted)

9-9© 2007 Pearson Education Canada Findings and Probability When the results of a study lead to the rejection of the null hypothesis, this only means that there is probably a relationship between the variables under examination It is one piece of evidence that the relationships exists Other researchers will test it again, and either confirm or disconfirm the past findings Research conclusions are therefore treated as tentative, open to disconfirmation

9-10© 2007 Pearson Education Canada Did they fail if they accept the null? Some researchers believe they failed if they accept the null hypothesis (i.e., find no relation between the variables rather than support for the predicted relationship) Not so: it is just as important to show that two variables are not associated as it is to find out they are associated

9-11© 2007 Pearson Education Canada 2. The Sample Distribution Tests of significance report whether an observed relationship could be the result of sample fluctuations or reflect a “real” difference in the population from which the sample has been taken Sample fluctuation is the idea that each time we select a sample we will get somewhat different results If we draw 1,000 samples of 50 cases, each will be slightly different from the first sample

9-12© 2007 Pearson Education Canada The Sample Distribution (cont’d) If the means of the same variable for each of the samples were plotted, a normal curve would results, but it would be peaked (or leptokurtic) Example: The means of weights of respondents are plotted The weights range from 70 to 80 kg, but the majority of samples would cluster around the true mean weight of 75 kg Note: We are plotting the mean weights of the respondents in each of the 1000 samples drawn

9-13© 2007 Pearson Education Canada The Sample Distribution (cont’d) The distribution is quite peaked because we are plotting the mean weights for each sample To measure the dispersion of the means of the samples, we use a statistic called the standard error of the means

9-14© 2007 Pearson Education Canada The Sample Distribution (cont’d) Relevance to hypothesis testing? In doing tests of significance, we are assessing whether the results of one sample fall within the null hypothesis acceptance zone (usually 95% of the distribution) or outside the zone, in which we reject the null hypothesis Four key points that can be made about probability sampling procedures where repeated measures are taken

9-15© 2007 Pearson Education Canada Four Key Points: Repeated Samples 1. Plotting the means of repeated samples will produce a normal distribution: it will be more peaked than when raw data are plotted (as shown in Figure 9.1)

9-16© 2007 Pearson Education Canada Four Key Points (cont’d) 2. The larger the sample sizes, the more peaked the distribution and the closer the means of the samples to the population mean (shown in Figure 9.2)

9-17© 2007 Pearson Education Canada Four Key Points (cont’d) 3. The greater the variability in the population, the greater the variations in the samples 4. When sample sizes are above 100, even if a variable in the population is not normally distributed, the means will be normally distributed when repeated samples are plotted E.g., weight of population of males and females will be bimodal, but if we did repeated samples, the weights would be normally distributed

9-18© 2007 Pearson Education Canada 3. One- and Two-Tailed Tests If the direction of a relationship is predicted, the appropriate test will be one-tailed If the direction of the relationship is not predicted, conduct a two-tailed test Example: One tailed: Females are less approving of violence than are males Two-tailed: There is a gender difference in the acceptance of violence [Note: No prediction about which gender is more approving]

9-19© 2007 Pearson Education Canada One- and Two-Tailed Tests (cont’d) Figure 9.3 (next slide) shows two normal distribution curves The first one has the 5% rejection area split between the two tails—this would be a two-tailed test The second one has the 5% rejection area all in one tail, indicating a one tailed test Same principle applies to 1% level

9-20© 2007 Pearson Education Canada Figure 9.3 Five Percent Probability Rejection Area: One- and Two-Tailed Tests

9-21© 2007 Pearson Education Canada Chi-Square: Red and White Balls The Chi-Square test (X 2 ) is used primarily in contingency table analysis, where the dependent variable is nominal level The formula is:

9-22© 2007 Pearson Education Canada One Sample Chi-Square Test Suppose the following incomes: INCOMESTUDENT SAMPLE % OF SAMPLEGENERAL POPULATION Over $100, $40,000 to $99, Under $40, TOTAL

9-23© 2007 Pearson Education Canada The Computation Chi-squares compare expected frequencies (assuming the null hypothesis is correct) to the observed frequencies. To calculate the expected frequencies, simply multiply the proportion in each category of the general population times the total number of cases (e.g., 200 students) Why do you do this?

9-24© 2007 Pearson Education Canada Why? If the student sample is drawn equally from all segments of society, then they should have the same income distribution (this is assuming the null hypothesis is correct) So what are the expected frequencies in this case?

9-25© 2007 Pearson Education Canada Expected Frequencies f e Frequency Observed Expected (200 x.078) (200 x.689) (200 x.233) Degrees of Freedom = 2

9-26© 2007 Pearson Education Canada Decision Look up critical value: Table 9.2, p. 260 Need to know: 2 degrees of freedom.05 level of significance 1 tailed test (i.e., column one) Find the critical value = 4.61 Compare to the Chi-Square calculated = Decision: Calculated value exceeds critical value so reject null hypothesis Inspect the data, conclude university students from higher SES background

9-27© 2007 Pearson Education Canada Standard Chi-Square Test Drug use by Gender (Box 9.4, p. 261) 3 categories of drug use (no experience, once or twice, three or more times) row marginal x column marginal ÷ total N of cases = expected frequencies Degrees of freedom = (row – 1)(columns – 1) = 2

9-28© 2007 Pearson Education Canada Decision With 2 degrees of freedom, 2-tailed test,.05 level of significance, the critical value is 5.99 Calculated Chi-Square is 5.69 Does not equal or exceed the critical value So, your decision is what? Accept the null hypothesis

9-29© 2007 Pearson Education Canada The t Distribution: t-Test Groups and Pairs Used often for experimental data t-test used when: Sample size is small (e.g.,< 30) Dependent variable measured at ratio level Random assignment to treatment/control groups Treatment has two levels only Population normally distributed

9-30© 2007 Pearson Education Canada The t Distribution: t-Test Groups and Pairs The t-test represents the ratio between the difference in means between two groups and the standard error of the difference. Thus: t = difference between the means standard error of the difference

9-31© 2007 Pearson Education Canada Two t-Tests: Between- and Within-Subject Design Between-subjects: used in an experimental design, with an experimental and a control group, where the groups have been independently established Within-subjects: In these designs the same person is subjected to different treatments and a comparison is made between the two treatments.

9-32© 2007 Pearson Education Canada The F Distribution: Means, ANOVA Box 9.7 provides an illustration of one-way analysis of variance (“Egalitarianism by Country,” p. 269) Concern is with how much variation there is within columns compared to variation between columns The F represents the ratio of between variation divided by within variation Probabilities looked up on Table 9.4, p. 272

9-33© 2007 Pearson Education Canada When Are Tests of Significance Not Appropriate? Total populations studied Non-probability sampling procedures used High nonparticipation rates Nonexperimental research tests for intervening variables Research is not guided by formal hypotheses