Chapter 10: The t Test For Two Independent Samples

Independent-Measures Designs The independent-measures hypothesis test allows researchers to evaluate the mean difference between two populations using the data from two separate samples. The identifying characteristic of the independent-measures or between-subjects design is the existence of two separate or independent samples. Thus, an independent-measures design can be used to test for mean differences between two distinct populations (such as men versus women) or between two different treatment conditions (such as drug versus no-drug).

Population A vs. Population B µ = 450  = 100 µ = ?  = 100 z

Z-formula broken down Population Sample Statistic Parameter Standard Error (Sampling Error)

Population A vs. Population B based on sample data µ = 30  = ? µ = 30  = ? t

Estimated Standard Error t-formula broken down Population Parameter Sample Statistic Estimated Standard Error (Sampling Error)

Different populations using different teaching methods Population A Taught by method A Population B Taught by method B Unknown µ = ? Unknown µ = ? Sample A Sample B

Goal: Evaluate Mean difference Between 2 Population Means Ho: µ1 - µ2 = 0 (No difference between the population means) H1: µ1 - µ2 ≠ 0 (There is a difference between the pop means)

Procedure for obtaining empirical sampling distribution of differences between two means Population 1 too large to measure in its entirety; in fact: Population 2 too large to measure in its entirety; in fact: Normal Normal 12 = 22 22 = 12 µ1 = 80 µ2 = 80 Difference (X1 - X2) ( ≠ 0 due to sampling error) Random Random Sample 1 N1 = 30 X1 = 81.6 Sample 2 N1 = 30 X1 = 79.7 Trial 1 +1.9 Sample 1 N1 = 30 X1 = 78.2 Sample 2 N1 = 30 X1 = 79.4 Trial 2 -1.2 Sample 1 N1 = 30 X1 = 80.1 Sample 2 N1 = 30 X1 = 79.5 Trial 1000 +0.6

Table of Mean Differences

Frequency polygon for mean differences table 150 100 Frequency (f) 50 -10 -5 5 10 Difference between sample means (X1 - X2)

t-formula for comparing sample means Sample Statistic Population Parameter Estimated Standard Error (Sampling Error)

t-formula (for sample means) broken down Sample Statistic Population Parameter Sampling Error (Standard Error of the Mean Difference)

Standard Error of the Means formula

Population II vs. Population I 10 20 30 40 50 60 70 80

Pooled Variance Formulas

Final Formula for Independent t df = df 1st sample + df 2nd sample = df1 + df2 OR df = (n1 - 1) + (n2 - 1)

5 step process for t statistic 1. Ho: µ1 - µ2 = 0 (No effect of…) H1: µ1 - µ2 ≠ 0 (Effect…)  2. Critical region for t statistic df = df1 + df2 p <  tcritical =  = 3. Collect data and compute test statistic 4. Make a decision 5. Conclusion

Do mental images help memory? 40 pairs of nouns (dog / bicycle, lamp / piano, etc.) 2 groups (or samples) 1. Memorization (control) 2. Imagery (experimental) Subsequent memory test

Data (Number of words recalled) Example 10.2 p.274 Data (Number of words recalled) Group 1 Group 2 (No images) (Images) 24 13 18 31 23 17 19 29 16 20 26 15 21 30

Data (Number of words recalled) Example 10.2 p.274 Data (Number of words recalled) Group 1 Group 2 (No images) (Images) 24 13 18 31 23 17 19 29 16 20 26 15 21 30 n = 10 n = 10 X = 19 X = 25 SS = 160 SS = 200

Critical regions Reject Ho Reject Ho Fail to reject Ho -2.101 2.101

t formula and standard error formula

Table demonstrating long Sum of Squares computation X1 (X1 - X1) (X1 - X1)2 24 19 +5 25 23 +4 16 -3 9 17 -2 4 13 -6 36 20 +1 1 15 -4 26 +7 49

Table demonstrating short SS computation X1 X12 24 576 23 529 16 256 17 289 19 361 13 169 20 400 15 225 26 676 ∑x = 190 ∑x2 = 3770

Figure 10.4 p. 276 df = 18 Reject H0 -1.734

Assumptions Underlying the Independent-Measures t statistic The observation within each sample must be independent The two populations from which the samples come must be normal The two populations from which the samples are selected must have equal variances - homogeneity of variance

Homogeneity Assumption: How do we know when we have violated it? For small samples (n < 10): If one of the sample variances is more than 4 times larger than the other For larger samples: If one of the sample variances is more than 2 times larger than the other

Frequency Histogram of Attitude Scores: SS = 4 Women Men n = 7 n = 7 X = 8.00 X = 12.00 SS = 4 SS = 4 4 3 Frequency 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Attitude Scores

Frequency Histogram of Attitude Scores: SS = 242 Women Men n = 7 n = 7 X = 8.00 X = 12.00 SS = 242 SS = 242 4 3 Frequency 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Attitude Scores