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Tests with two+ groups. We have examined tests of means for a single group, and for a difference if we have a matched sample (as in husbands and wives) Now we consider differences of means between two or more groups. Two sample t test. Compare means on a variable for two different groups.
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Tests with two+ groups • We have examined tests of means for a single group, and for a difference if we have a matched sample (as in husbands and wives) • Now we consider differences of means between two or more groups
Two sample t test • Compare means on a variable for two different groups. • Income differences between males and females • Average SAT score for blacks and whites • Mean time to failure for parts manufactured using two different processes
New Test - Same Logic • Find the probability that the observed difference could be due to chance factors in taking the random sample. • If probability is very low, then conclude that difference did not happen by chance (reject null hypothesis) • If probability not low, cannot reject null hypothesis (no diff. between groups)
Sampling Distributions Note in this case each mean is not in the critical region of other sampling dist. Mean 1 Mean2
Sampling Distributions Note each mean is well into the critical region of other sampling distribution. Mean 1 Mean 2
Sampling Dist. of Difference Big Differences Hypothesize Zero Diff. Difference of Means
Procedure • Calculate means for each group • Calculate difference • Calculate standard error of difference • Test to see if difference is bigger than • “t” standard errors (small samples) • z standard errors (large samples) • t and z are taken from tables at 95 or 99 percent confidence level.
Standard error of difference Pooled estimate of standard deviation Divide by sample sizes
t test Difference of Means Standard error of difference of means If t is greater than table value of t for 95% confidence level, reject null hypothesis
Three or more groups • If there are three or more groups, we cannot take a single difference, so we need a new test for differences among several means. • This test is called ANOVA for ANalysis Of VAriance • It can also be used if there are only two groups
Analysis of Variance • Note the name of the test says that we are looking at variance or variability. • The logic is to compare variability between groups (differences among the means) and variability within the group (variability of scores around the mean) • These are call the between variance and the within variance, respectively
The logic • If the between variance is large relative to the within variance, we conclude that there are significant differences among the means. • If the between variance is not so large, we accept the null hypothesis
Examples Large Between Both examples have same Within Small Between
Variance • Calculate sum of squares and then divide by degrees of freedom • Three ways to do this
Total, Within, and Between • Total variance is the mean squared deviation of individual scores around the overall (total) mean • Within variance is the mean squared deviation of individual scores around each of the group means • Between variance is the mean squared deviation of group means around the overall (total) mean
Total, Within, and Between Total = SST/dfT Within = SSW/dfW Between = SSB/dfB
F test for ANOVA • The F statistic has a distribution somewhat like the chi-square. It made of the ratio of two variances. • For our purpose, we will compare the between and within estimates of variance • Create a ratio of the two -- called an F ratio. Between variance divided by the within variance
F-ratio • Table in the back of the book has critical values of the F statistic. Like the t distribution, we have to know degrees of freedom • Different than the t distribution, there are two different degrees of freedom we need • Between (numerator) and within (denominator)
Decision • If F-ratio for our sample is larger than the critical value, we reject the null hypothesis of no differences among the means • If F-ratio is not so large, we accept null hypothesis of no differences among the means
Example (three groups) Observations 1 2 3 4 5 6 7 8 9 Overall mean is 5 60
Example (within) Observations 1 2 3 4 5 6 7 8 9 2 5 8 Group Means 6
Example (between) Observations 1 2 3 4 5 6 7 8 9 2 5 8 Group Means Overall mean is 5 54
F-ratio • Between variance divided by within variance. • Between= 54 / 2 = 27 (remember k-1 degrees of freedom, so df = 3-1 • Within = 6 / 6 = 1 (remember n-k degrees of freedom, so df = 9-3 • F-ratio is 27/1 with 2 and 6 df • Critical value (95%) of F is 5.14