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Independent Samples Ttest • With previous tests, we were interested in comparing a single sample with a population • With most research, you do not have knowledge about the population -- you don’t know the population mean and standard deviation INDEPENDENT SAMPLES T-TEST: • Hypothesis testing procedure that uses separate samples for each treatment condition (between subjects design) • Use this test when the population mean and standard deviation are unknown, and 2 separate groups are being compared Example: Do males and females differ in terms of their exam scores' • Take a sample of males and a separate sample of females and apply the hypothesis testing steps to determine if there is a significant difference in scores between the groups 1 Formula: t= (x1 − x2 )− (µ1 − µ 2 ) s x1 − x2 • We are interested in a difference between 2 populations (females, µ1, and males, µ2) and we use 2 samples (females, x1, and males, x2) to estimate this difference ESTIMATED STANDARD ERROR OF THE DIFFERENCE: • Gives us the total amount of error involved in using 2 sample means to estimate 2 population means. It tells us the average distance between the sample difference (x1-x2) and the population difference (µ1-µ2) • As we’ve done previously, we have to estimate the standard error using the sample standard deviation or variance and, since there are 2 samples, we must average the two sample variances. 2 POOLED VARIANCE: The average of the two sample variances, allowing the larger sample to weighted more heavily Formulae: s 2 pooled (df1 ) s 21 + (df 2 ) s 2 2 = df1 + df 2 SS1 + SS 2 OR s 2 pooled = df1 + df 2 df1=df for 1st sample; n1-1 df2=df for 2nd sample; n2-1 Estimated Standard Error of the Difference s x1 − x2 = sx1 −x 2 s2 pooled n1 + s2 pooled n2 book formula  SS1 + SS2  1 1  =   +   n1 + n 2 − 2  n1 n 2  € Degrees of freedom (df) for the Independent t statistic is n1 + n2 - 2 or df1+df2 3 Hypothesis testing using an Independent Samples t-Test: Example: Do males and females differ in their test scores for exam 2' The mean test score for females is 27.1 (s=2.57, n=19), and the mean test score for males is 26.7 (s=3.63, n=20) Step 1: State the hypotheses H0: µ1-µ2=0 (µ1=µ2) H1: µ1-µ2≠0 (µ1 ≠µ2) • This is a two-tailed test (no direction is predicted) Step 2: Set the criterion • α=' • df= n1+n2-2=' • Critical value for the t-test ' Step 3: Collect sample data, calculate x and s From the example we know the mean test score for females is 27.1 (s=2.57, n=19), and the mean test score for males is 26.7 (s=3.63, n=20) 4 Step 4: Compute the t-statistic t= (x1 − x2 )− (µ1 − µ 2 ) s x1 − x2 where s x1 − x2 = s2 pooled n1 + s2 pooled n2 • Calculate the estimated standard error of the difference s 2 pooled (df1 ) s 21 + (df 2 ) s 2 2 = df1 + df 2 (18)2.57 2 + (19)3.63 2 18 + 19 s2 pooled = = = (18)6.61 + (19)13.18 37 118.98 + 250.36 37 = 369.34 = 9.98 37 5 • Compute the standard error (continued) s x1 − x2 = s x1 − x2 = s2 pooled n1 + s2 pooled n2 9.98 9.98 = .525 + .499 = 1.01 + 19 20 •Calculate the t statistic t= t= (x1 − x2 )− (µ1 − µ 2 ) s x1 − x2 *This always defaults to 0 (27.1 − 26.7) .4 = = .396 1.01 1.01 Step 5: Make a decision about the hypotheses • The critical value for a two-tailed t-test with df=37 (approx. 40) and α=.05 is 2.021 • Will we reject or fail to reject the null hypothesis' 6 Assumptions for the Independent t-Test: • Independence: Observations within each sample must be independent (they don’t influence each other) • Normal Distribution: The scores in each population must be normally distributed • Homogeneity of Variance: The two populations must have equal variances (the degree to which the distributions are spread out is approximately equal) Repeated Measures T-test • Uses the same sample of subjects measured on two different occasions (within-subjects design) • Use this when the population mean and standard deviation are unknown and you are comparing the means of a sample of subjects before and after a treatment • We are interested in finding out how much difference exists between subjects’ scores before the treatment and after the treatment 7 DIFFERENCE SCORE (or D) • The difference between subjects’ scores before the treatment and after the treatment • It is computed as x2-x1, where x2 is the subjects’ score after the treatment and x1 is the subjects’ score before the treatment • We use the sample of difference scores to estimate the population of difference scores (µD) Example: Does alcohol affect a person’s ability to drive' A researcher selects a sample of 5 people and sets up an obstacle course. – Each subject drives the course and the number of cones he or she knocks over is counted. – Next, the researcher has each subject drink a six-pack of beer, then drive the course again, counting the number of cones each subject knocks over. NOTE: Theory has shown that alcohol decreases motor and cognitive skills 8 Step 1: State the hypotheses H0: µD=0 H1: µD ≠0 Step 2: Set the criterion • One-tail test or two-tail test' • α=' • df=n-1 • Critical value for t ' Step 3: Collect sample data, calculate D • Once the difference scores are obtained, all further statistics are calculated using these scores instead of the pretest / posttest or before / after scores Subject 1 2 3 4 5 Before (x1) 2 0 4 2 3 After (x2) 8 4 11 5 8 D (x2 - x1) 6 4 7 3 5 ∑ D = 25 9 • Find the mean (average) difference score (D) D= ∑D n D= 6 + 4 + 7 + 3 + 5 25 = =5 5 5 • The average difference of the number of cones knocked down from before drinking to after drinking is 5 cones. Remember, we are hypothesizing the difference to be zero. Step 4: Calculate the t-statistic Formula: estimated std. deviation of diff. scores t= D sD where sD = sD n and D= ∑D n estimated std.error of mean diff. scores the mean difference score 10 D 6 4 7 3 5 • Compute the estimated standard deviation of the difference scores (sD) 2 D− D 1 -1 2 -2 0 (D − D) 1 1 4 4 0 SSD = 10 σ = SS = 1.58 n -1 sD = SS 10 = = 1.58 n −1 4 • The average deviation of the difference scores (D) about the mean difference score (D) is 1.58 cones € • Compute the estimated standard error of the mean difference scores sD = sD n sD = 1.58 5 = .707 – The average deviation of the sample mean difference scores (D) from the population mean difference score (µD) is .707 cones • Compute the t-statistic t= D sD t= 5 = 7.07 .707 11 Step 5: Make a decision • The critical value for a one-tailed t-test with df=4 and α=.05 is 2.132 • Will we reject or fail to reject the null hypothesis' Advantages and Disadvantages of the Repeated Measures t-Test: Advantages: • Controls for pre-existing individual differences between samples (because only 1 sample of people are being used • More economical (fewer subjects are needed) Disadvantages: • Subject to practice effects - the subjects are performing the measurement task (i.e. driving the obstacle course, taking an exam) twice - scores may improve due to the practice 12 Assumptions of the Repeated Measures tTest: Independent Observations: The scores from before and after the treatment must not be related (no practice effects) Normal Distribution: The population of difference scores must be normally distributed Summary of Hypothesis Testing through t-statistic • We have looked at four inferential statistics: – – – – z-score statistic single sample t-statistic independent samples t-statistic repeated measures t-statistic, or matched subjects • the generic formula for these statistics is: z or t = sample statistic - population parameter standard error 13 Summary of Hypothesis Testing through t-statistic • z-score statistic compares a sample to a population when the population s.d. is known • t-statistic compares a sample to a population when the population s.d. is unknown • independent samples t-statistic compares 2 independent samples • repeated measures t-statistic compares 1 sample measured on 2 occasions 14