Chapter 7 For Explaining Psychological Statistics, 4th ed. by B. Cohen 1 Chapter 7: The t Test for Two Independent Sample Means To conduct a t test for two independent sample means, we need to know what the sampling distribution of the difference between two means looks like. We assume that it is a normal distribution. It will be centered on zero if the null hypothesis is that the two population means are the same (i.e., H 0 : µ 1 = µ 2 ). The SD of this distribution is called the standard error of the difference. For large samples, a z test can be used:

Chapter 7For Explaining Psychological Statistics, 4th ed. by B. Cohen 2 Estimating the Standard Error of the Difference When the Sample Sizes Are Not Large –Pooled-variances estimate: The two sample variances can be pooled together to form a single estimate of the population variance called s 2 pooled or s 2 p –s 2 pooled can then be used to estimate the standard error of the difference by means of the following formula: –The use of s 2 p is based on the assump- tion that the two populations have the same variance.

Chapter 7For Explaining Psychological Statistics, 4th ed. by B. Cohen 3 The Pooled-Variances t Test Formula The denominator of this formula is the estimated standard error of the difference (SED): Inserting the formula for estimating the SED from the pooled variances yields the following formula: Note that the use of s 2 pooled is based on the assumption that the two populations have the same variance, and that df= N 1 + N 2 – 2.

Chapter 7For Explaining Psychological Statistics, 4th ed. by B. Cohen 4 Alternative Versions of the Pooled-Variances t Test Formula Here, the formula for s 2 pooled is included in the t formula and it is assumed that the null hypothesis is µ 1 - µ 2 = 0: where the critical t is based on df = N 1 + N 2 – 2 The formula for equal sample sizes reduces to: where the critical t is based on df = 2n – 2.

Chapter 7 For Explaining Psychological Statistics, 4th ed. by B. Cohen 5 A researcher is studying the effect of drinking Red Bull on the number of errors a partici- pant makes in a motor skills test. Test the null hypothesis that the population mean for Red Bull drinkers is the same as for placebo drinkers. t.05 (32) = 2.04 (approx.) < 5.51, so the null hypothesis can be rejected. Try this example….

Chapter 7 For Explaining Psychological Statistics, 4th ed. by B. Cohen 6 Limitations of Statistical Conclusions –Our significant Red Bull result could be a Type I error. If this is a novel result, replication is recommended. –Statistical significance is no guarantee that you are dealing with a large or important difference. –If you have not assigned participants to the two conditions (e.g., you are comparing habitual coffee drinkers with those who do not like coffee), you cannot make causal conclusions from your significant results. –More information can be obtained by constructing a confidence interval for the difference of the means of the two populations.

Chapter 7 For Explaining Psychological Statistics, 4th ed. by B. Cohen 7 Confidence Intervals for the Difference Between Two Population Means –The formula for the CI for the difference of two population means is obtained by solving the t test formula for µ 1 – µ 2. –For a 95% CI, t crit is the critical t for a.05, two-tailed significance test. For the Red Bull example, the 95% CI is: –Thus, the CI extends from a difference of 2.52 errors up to a difference of 5.48 errors. –Because zero is not in the interval, we know that the usual H 0 can be rejected at the.05 level, with a two-tailed test.

Chapter 7 For Explaining Psychological Statistics, 4th ed. by B. Cohen 8 Assumptions for the Two-Group t Test and CI for the Difference of Population Means – The dependent variable has been meas- ured on an interval or ratio scale (there are nonparametric tests available if an ordinal scale was used). – Independent random sampling Ideal: Both groups should be random samples. Each individual selected for one sample should be independent of all the individuals in the other sample. However, typically, the two groups are formed by random assignment from one sample of convenience. – Normal distributions The DV should follow a normal distribution in both groups. However, the Central Limit Theorem implies that, if the samples are not very small, the two- group t test will still be valid.

Chapter 7For Explaining Psychological Statistics, 4th ed. by B. Cohen 9 Assumptions (cont.) –To use the pooled-variance t test, an additional assumption must be made: Homogeneity of Variance. –However, this assumption is usually ignored if: Both samples are quite large, or The two samples are the same size, or The sample variances are not very different (e.g., one variance is no more than twice as large as the other) –If it is not reasonable to assume HOV, consider performing the separate- variances t test: –The df for the critical t are best found by statistical software.