Advanced Psychological Statistics: Two-Sample Z-Test and T-Test Overview
Explore the concepts of two-sample Z-test and t-test in advanced psychological statistics, covering assumptions, calculations, and hypotheses. Understand the importance of pooled variance estimates and Satterthwaite's approximation for degrees of freedom.
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Psychology 202a Advanced Psychological Statistics October 8, 2019
The plan for today The two-sample, independent-groups Z test The two-sample, independent-groups t test
The two-sample Z test ( ) ( ) M M = . 1 2 1 2 Z M M 1 2 2 1 2 2 = + . M M n n 1 2 1 2
Two-sample Z test (cont.) = = : , 0 05 . , two - tailed test. H 0 1 2 2 2 10 10 = + = . 8 M M 25 25 1 2 103 108 = = . 1 768 . Z 8
Assumptions of the two-sample Z test Independent observations within groups. Independent observations between groups. is known for both populations. Distribution is normal or sample is sufficiently large in both populations.
t tests for differences between means The two-sample Z test: = M M + . 1 2 Z 2 1 2 2 n n 1 2 The two-sample t test: can t just substitute estimated standard deviation.
The pooled variance estimate Weighted average of the two individual variance estimates: + ) 1 2 1 + 2 2 ( ) 1 n + ( n s n s = 2 1 2 sP 2 n 1 2 2 1 2 2 df s df s = . 1 df 2 + df 1 2 df = n1+n2 - 2
The two-sample independent-groups t test M M = , 1 2 t s M M 1 2 where 2 2 ?? ?1 +?? ??1 ?2= . ?2
Whats the null hypothesis? = : 0 H 0 1 2 : 0 H 1 2 A
What if it doesnt make sense to pool the variances? Satterthwaite s approximation for degrees of freedom: 2 2 2 ?1 ?1+?2 ?2 ?? . 2 2 2 2 ?1 ?1 ?2 ?2 /(?1 1) + /(?2 1) Use unpooled variances for the standard error with adjusted degrees of freedom. The t test in R.
Assumptions of the t test Independence within each population. Independence between populations. Equal variances in the two populations. Also known as homoscedasticity. Both populations normally distributed.
Evaluating the assumptions Independence within populations: examine the data collection procedure. Independence between populations: examine the process that created the groups. Random assignment guarantees independence between populations.
Evaluating the assumptions Homoscedasticity: Graphical comparisons of the two groups Comparison of the two sample standard deviations Normality: Graphical examination of each group Q-Q plots
The illogic of auxiliary hypothesis tests Auxiliary hypothesis tests: Involve confirmation of the null hypothesis. Are least likely to detect a problem under precisely the circumstances where the problem matters the most. Involve assumptions of their own (implying infinite recursion).
