Research Methods I

1. Research Methods I T-tests

2. Different Kinds of T-tests One sample t-test When population variance is not known Two sample t-test for independent samples Compare two sample means Samples are independent of each other Two sample t-test for dependent samples Compare two sample means Samples are paired / dependent

3. One-Sample T-test One-sample t-test when � is known but s is not known. For the one-sample t-test sM is being substituted with sM (standard error with estimated standard error)

4. T-distribution The sampling distributions of z and t are different. For z-tests the sampling distribution of sample means is a normal curve For t-tests the sampling distribution changes according to sample size (above n=120 almost like z distribution)

5. Degrees of Freedom Degrees of freedom are the number of scores in a sample that are free to vary. Because the sample mean places a restriction on the value of one score in the sample, there are n-1 degrees of freedom for the sample The critical value for t depends on the degrees of freedom (df) Degrees of freedom depend on the sample size df = n-1

6. Two-sample t-tests In two-sample t-tests two sample means are being compared. What is the probability that the difference in the means that is observed is due to chance? Population parameters are no longer known Normal distribution of population must be assumed unless sample size is very large There are basically two kinds of t-tests Dependent Independent

7. Dependent vs Independent Samples Dependent samples Repeated-measures, or within-subjects design Members of one sample are matched or paired in some way with members of the other sample Independent samples Independent-measures, or between-subjects design Members of one sample are in no way related to matched to members in the other sample

8. Independent Samples t-test Assumes that composition of one sample is independent from composition of other sample. Both samples reflect different populations H0: �1= �2 H1: �1? �2 or �1< �2 or �1> �2 Sample size (n), sample mean, and sample variance have to be known for each sample If two random samples are collected and one has had a treatment exposure only the treatment exposure should make the difference between the two groups.

9. T-test for Independent Samples T-test for independent samples is based on distribution of differences between means. It is like the t-distribution except using the differences between means In order to calculate t we get the difference between the two sample means, and divide it by the standard deviation of the sampling distribution (the standard error) However, standard error for this distribution is based on differences of means and is therefore different.

10. T-test for Independent Samples 1. Write out null and alternative hypothesis for the original problem. 2. For each sample determine its sample size, its mean, and its variance 3. To determine which t formula to use for the F test for homogeneity of variance 4. Perform the appropriate F test

11. T-tests for Independent Samples The standard error for our distribution is based on variances of two samples. If both variances are close in magnitude we can calculate the pooled estimate of common variance. This is based on a weighted average of our two sample variances. If both variances are not close in magnitude we have to use a different formula for the t-test

12. F-test for homogeneity of variance The F-test for homogeneity of variance tells us whether the variance between both samples is similar or different Write out the null and alternative hypothesis for the F test Calculate F and its two degrees of freedom Compare the F value with Fcritical If your F value is smaller than Fcritical then assume equal population variances. If your F value is larger than Fcritical then assume unequal population variances.

13. F-test for homogeneity of variance H0 : variance of sample 1 is equal to variance of sample 2 H1 : variance of sample 1 is not equal to variance of sample 2 F tests are always non-directional F = larger of the two sample variances divided by the smaller of the two sample variances Degrees of freedom for both numerator and denominator: df = n � 1 F-table gives critical values n1 = numerator degrees of freedom

14. T-test for SES 15:04 Wednesday, October 22, 2003 4 The TTEST Procedure Statistics Lower CL Upper CL Lower CL Upper CL Variable group N Mean Mean Mean Std Dev Std Dev Std Dev Std Err ses 53 3.7184 4.1698 4.6212 1.3746 1.6377 2.0263 0.225 0 ses 46 3.7534 4.2391 4.7248 1.3565 1.6355 2.06 0.2411 1 ses Diff (1-2) -0.724 -0.069 0.5853 1.4352 1.6367 1.9045 0.3298 T-Tests Variable Method Variances DF t Value Pr > |t| ses Pooled Equal 97 -0.21 0.8340 ses Satterthwaite Unequal 95.1 -0.21 0.8340 Equality of Variances Variable Method Num DF Den DF F Value Pr > F ses Folded F 52 45 1.00 0.9981

15. SAS for Paired T-tests data new; set work.julia20; diff = final_exam - mid_term_exam; run; proc print; var diff; run; proc means n mean stderr t prt; var diff; run;

16. The MEANS Procedure Analysis Variable : diff N Mean Std Error t Value Pr > |t| -4.0000000 1.1742180 -3.41 0.0059 For dependent sample t-test we run a proc means with the options n, mean, stderr, t, and prt These statistics will be computed for the difference variable T will give the t-value and its probability, testing the null hypothesis that the variable DISS comes from a population whose mean is zero. The mean gives the average difference score. If p<.05 we can say that the two groups are significantly different from one another.