t-tests Quantitative Data One group  1-sample t-test Two independent groups  2-sample t-test Two dependent groups  Matched Pairs t-test t-TestsSlide #1

t-TestsSlide #2 A Full Reality No longer know what  is!!!!! What should be used instead? –Our best guess at   s Changes details, not the big picture t

t-TestsSlide #3 2-Sample t-test -- Examples Do test scores differ between sections of a class? Is the average number of yew per m 2 different between areas exposed to and areas protected from moose browsing? Does the time from ingesting a pill until a subject claims no more headache pain less for subjects given an experimental drug as compared to those given a placebo?

t-TestsSlide #4 2-sample t-test H o :   =  2 (where the subscript is a sample index) because  1 =  2 same as  1 -  2 =0 Statistic: Test Statistic: where s p 2 is the pooled sample variance - 0 df = n 1 + n 2 - 2

t-TestsSlide #5 2-sample t-test Assume: – n 1 + n 2 is large (to use a t-distribution) n 1 + n 2 > 40, OR n 1 + n 2 > 15 and both histograms are not strongly skewed, OR both histograms are approximately normal – the two samples are independent –  1 2 =  2 2 Are  1 2 &  2 2 parameters or statistics? Hypothesis Test -- Levene’s Test –H o :  1 2 =  2 2 vs H a :  1 2   2 2 –What do you do with the p-value?

t-TestsSlide #6 Levene’s Test Summary A hypothesis test within a hypothesis test. Small p-values mean the variances are unequal. If Levene’s test is a reject, can not continue with the 2-sample t-test (as presented here).

t-TestsSlide #7 Are You Surprised by the SE for the difference in means? Let’s do some algebra on the SE for a sample mean It is “similar” to the SE for a single sample mean

t-TestsSlide #8 2-sample t-test When: quantitative data, samples from two populations, samples are independent

t-TestsSlide #9 A study of the effect of caffeine on muscle metabolism used 36 male volunteers who each underwent arm exercise tests. Eighteen of the men were randomly selected to take a capsule containing pure caffeine one hour before the test. The other men received a placebo capsule. During each exercise the subject's respiratory exchange ratio (RER) was measured. [RER is the ratio of CO 2 produced to O 2 consumed and is an indicator of whether energy is being obtained from carbohydrates or fats]. The question of interest to the experimenter was whether, on average and at the 5% level, caffeine changed mean RER. A Full Example

t-TestsSlide #10 Group n Mean StDev Min 1 st Qu Median 3 rd Qu Max Caffeine Placebo Levene’s Test p-value = A Full Example

R Handout t-TestsSlide #11

t-TestsSlide #12 Example Data – Sex & Direction A sample of 30 males and 30 female was taken to an unfamiliar wooded park and given spatial orientation tests, including pointing to the south. The absolute pointing error, in degrees, was recorded. The results are in the SexDirection.txt file on the webpage. Do men have a better sense of direction than women, at the 1% level? from Sholl, X, et al The relation of sex and sense of direction to spatial orientation in an unfamiliar environment. Journal of Environmental Psychology. 20:17-28.