Inferences Involving The Mean When  Is Not Known: One- And Two-Sample Designs Chapter 11 SHARON LAWNER WEINBERG SARAH KNAPP ABRAMOWITZ StatisticsSPSS An Integrative Approach SECOND EDITION Using

Steps for Conducting a Hypothesis Test Step 1: State Hypotheses. Step 2: Evaluate underlying assumptions or indicate why it is not necessary to do that. Step 3: Gather evidence (p-value or CI). Step 4: State conclusion. Step 5: If result is statistically significant, report and interpret effect size.

Effect Size for a One-Sample t-Test

One Sample t-test: Is N>30? Yes: Test robust to normality violations No: Need to check Normality Is skewness ratio less than 2? Yes: Normality Tenable No: Normality not Tenable Result is stat. sig. Report result, calculate and interpret effect size. Result is not stat. sig. Report result. Result is not stat. sig. Report result. Is N > 30? Does the sample mean differ in the direction Indicated in the alternative hypothesis? Yes No 1-tailed Stop. Test result may not be valid. Use a non-linear transformation or a non-parametric test. Result is stat. sig. Look at sample mean to determine the direction of the difference. Report result, calculate and interpret effect size. Result is not stat. sig. Report result. Is the test 1-tailed (i.e., H 1 :  c ) or 2-tailed (i.e., H 1 :  ≠ c)? 2-tailed For p-value, is p <.05? For confidence interval, is the null hypothesized value For  outside of the CI? YesNo

Effect Size for Independent Samples t-Test Correct formula: Easier formula we will use in Math 3 that gives a good-enough approximation:

Independent Samples t-test: Yes: Test robust To normality violations No: Need to check normality Is skewness ratio less than 2? Yes: Normality tenable No: Normality not tenable Result is stat. sig. Report results. Calculate and interpret effect size. Result is not stat.sig. Report results. Result is not stat. sig. Report result. Are N 1 and N 2 > 30? Do sample means differ in the direction specified in the alternative hypothesis? Is (Two tailed sig./2) <.05? Yes No 1-Tailed Test result may not be valid. Perform a non-linear transformation Or use a non-parametric test. Result is stat.sig. Look at sample mean to determine the direction of the difference. Report result. Calculate and interpret effect size. Result is not stat.sig. Report result. Is the test 1-tailed (i.e. H 1  1  2 or H 1  1  2  or 2-tailed (i.e. H 1 :  1 ≠  2 )? 2-Tailed For p-values is p <  ? For confidence intervals, is 0 outside of the range of values? YesNo Is for both groups? Yes: The equal and unequal variance output are the same Need to check homogeneity of variances. From Levene’s test, if p < α, use unequal variances line for t-test. Otherwise, use equal variances line. Is N 1 =N 2 ? No

Effect Size for Paired Samples Can use formula for independent samples Or for one sample

Paired Samples t-test: Yes: Test robust To normality violations No: Need to check normality Is skewness ratio less than 2? Yes: Normality tenable No Result is stat. sig. Report results. Calculate and interpret effect size. Result is not stat.sig. Report result. Result is not stat. sig. Report result. Is N 1 = N 2 > 30? Do sample means support the direction of the alternative hypothesis? Is (p/2) <.05? Yes No 1-tailed Test result may not be valid. Perform a non-linear transformation or use a non-parametric test. Result is stat. sig. Look at sample mean to determine the direction of the difference. Report results. Calculate and interpret effect size. Result is not stat.sig. Report result. Is the test 1-tailed (i.e. H 1  1  2 or H 1  1  2  or 2-tailed (i.e. H 1 :  1 ≠  2 )? 2-tailed For p-values is p <  ? For confidence intervals Is 0 outside of the range of values? YesNo Is for both groups? Because N 1 =N 2, the test is robust to violations of the homogeneity of variances assumption.

Questions One Sample Is the average SAT total score of Drew seniors 1200? Do students rate their entire experience at Drew as better than good (3), on average? Independent Samples Is there a difference between commuter and residential students in the average rating of their entire experience at Drew? Is there a difference between males and females in the average rating of the extent to which Drew contributed to their understanding of people of other racial and ethnic backgrounds? Paired Samples Is there a difference between average student ratings of the degree to which Drew contributed to their ability to write and to analyze quantitative problems?