1. 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

2. 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

3. t-TestsSlide #3 Student’s t-distribution Compared to a standard normal (Z): Similarities –symmetric about 0 –approximately bell-shaped Differences –more probability in the tails –less probability in the center –Exact shape depends on degrees-of-freedom (df) See HO for R work -4-2024 Z or t 1 df 2 df 5 df10 df

4. t-TestsSlide #4 1-sample t-test H o :  =  o (where  o = specific value) Statistic: Test Statistic: Assume: –  is UNknown – n is large (so that the test stat follows a t-distribution) n > 40, OR n > 15 and histogram is not strongly skewed, OR Histogram is approximately normal When: Quantitative variable, one population sampled,  is UNknown. df = n-1

5. t-TestsSlide #5 A Full Example In Health magazine reported (March/April 1990) that the average saturated fat in one pound packages of butter was 66%. A food company wants to determine if its brand significantly differs from this overall mean. They analyzed a random sample of 96 one pound packages of its butter. Test the company’s hypothesis at the 1% level. Variable n Mean St. Dev. Min... %SatFat 96 65.6 1.41 60.2...

6. t-TestsSlide #6 Practical Significance Is there a real difference between 66% and 65.6% saturated fat? If the sample size is large enough, any hypothesis can be rejected.

7. R Handout t-TestsSlide #7

8. t-TestsSlide #8 Example Data – Cottonmouths Researchers have determined that a population of cottonmouth snakes must have an average litter size greater than 5.8 snakes in order for the population to grow. A sample of snake litters from this population was taken and the number of snakes in the litter was recorded in Cottonmouth.txt. Test, at a very conservative level, if the average litter size is large enough for the population to grow. Based on data from Blem, X. and X. Blem. 1995. Journal of Herpetology 29:391-398.