PSY 307 – Statistics for the Behavioral Sciences

1. PSY 307 – Statistics for the Behavioral Sciences Chapter 14 – t-Test for Two Independent Samples

2. Independent Samples • Observations in one sample are not paired on a one-to-one basis with observations in the other sample. • Effect – any difference between two population means. • Hypotheses: • Null H0: m1 – m2 = 0 ≤ 0 • Alternative H1: m1 – m2 ≠ 0 > 0

3. The Difference Between Two Sample Means Effect Size X1 minus X2 The null hypothesis (H0) is that these two means come from underlying populations with the same mean m (so the difference between them is 0 and m1 – m2 = 0).

4. Sampling Distribution of Differences in Sample Means All possible x1-x2 difference scores that could occur by chance x1-x2 Critical Value m1 – m2 Critical Value Does our x1-x2 exceed the critical value? YES – reject the null (H0)

5. What if the Difference is Smaller? All possible x1-x2 difference scores that could occur by chance x1-x2 Critical Value m1 – m2 Critical Value Does our x1-x2 exceed the critical value? NO – retain the null (H0)

6. Distribution of the Differences • In a one-sample case, the mean of the sampling distribution is the population mean. • In a two-sample case, the mean of the sampling distribution is the difference between the two population means. • The standard deviation of the difference scores is the standard error of this distribution.

7. Formulas for t-test (independent) Estimated standard error

8. Estimated Standard Error • Pooled variance – the variance common to both populations is estimated by combining the variances. • The variance average is computed by weighting the group variance by the degrees of freedom (df) then dividing by combined df. • Df for pooled variance: n1 + n2 - 2

9. Confidence Intervals for t • The confidence interval for two independent samples is: • Find the appropriate value of t in the t table using the formula for df. • The true difference in population means will lie between the upper and lower limits some % of the time

10. Assumptions • Both populations are normally distributed with equal variance. • With equal sample sizes > 10, valid results will occur even with non-normal populations. • Equate sample sizes to minimize effects of unequal variance. • Increase sample size to minimize non-normality.

11. Population Correlation Coefficient • Two correlated variables are similar to a matched sample because in both cases, observations are paired. • A population correlation coefficient (r) would represent the mean of r’s for all possible pairs of samples. • Hypotheses: • H0: r = 0 • H1: r ≠ 0

12. t-Test for Rho (r) • Similar to a t–test for a single group. • Tests whether the value of r is significantly different than what might occur by chance. • Do the two variables vary together by accident or due to an underlying relationship?

13. Formula for t Standard error of prediction

14. Calculating t for Correlated Variables • Except that r is used in place of X, the formula for calculating the t statistic is the same. • The standard error of prediction is used in the denominator to calculate the standard deviation. • Compare against the critical value for t with df = n – 2 (n = pairs).

15. Importance of Sample Size • Lower values of r become significant with greater sample sizes: • As n increases, the critical value of t decreases, so it is easier to obtain a significant result. • Cohen’s rule of thumb • .10 = weak relationship • .30 = moderate relationship • .50 = strong relationship