1. Getting Started with Hypothesis Testing The Single Sample

2. Outline  Remembering the binomial situation and z-score basics  Hypothesis testing with the normal distribution  When σ is unknown – the t distribution  One vs. Two-tails  Problems

3. Recall the binomial  We were able to do hypothesis testing regarding a proportion (of ‘success’)  We created a probability distribution with respect to the expected probability of success, and then calculate the observed p-value for our specific result  For example:  H 0 : π =.5  Probability if obtained 9/10 or 10/10  p = ~.01

4. Continuous measures  If we know the population mean and standard deviation, or just want to speak about our sample, for any value of X we can compute a z-score  Z-score tells us how far above or below the mean a value is in terms of standard deviation

5. Hypothesis testing using the normal  If we were to test a hypothesis regarding our sample mean we must consult the sampling distribution and now are dealing with the standard error  Our formula is the same as before, but substitutes our sample mean for an individual score and the standard error (regarding the sampling distribution) for the population standard deviation  The tail probability is our observed p-value, and based on that we can decide whether our sample comes from a population suggested by the null hypothesis

6. Conceptual summary thus far  H 0 : μ = some value  Sample mean does not equal H 0  But how different is it?  Is it what we would typically expect due to sampling variability or extreme enough to think that our sample does not come from such a population suggested by the null hypothesis?

7. Z to t  In most situations we do not know   However the sample standard deviation has properties that make it a good estimate of the population value  We can use our sample standard deviation to estimate the population standard deviation  However, if we use the normal distribution probabilities, they would be incorrect

8. t-test  Which leads to:  where  And degrees of freedom (n-1)

9. Interpretation  How many standard deviations away from the population mean is my sample mean in terms of the sampling distribution of means

10. What’s the difference?  Why a “t” now not a “z”?  The difference involves using our sample standard deviation to estimate the population standard deviation  Standard deviation is positively skewed, and so slightly underestimates the population value  As we have discussed it is actually a biased estimate  Our standard error part of the formula will also be smaller than it should  larger value of z than should be  Increased type I error

11. Estimating   Because we are trying to estimate , how well s does this depends on the sample size  When n is larger, s is closer to   When degrees of freedom =  then t = z  As N gets larger and larger the t distribution more closely approximates the normal distribution

12. Example  The UNT Psychology Department claims in its recruiting literature that its graduate students get an average of 8 hours of sleep a night  Collected sleep data from 25 grad students, this sample has a mean of 7.2 hours sleep, s = 1.5

13. Plug in the numbers Formula where t = (7.2 - 8)/(1.5/sqrt(25)) = -0.8/0.3 = -2.667 What else do we need to know?

14. Critical value of t  One approach  df = n-1  t.05 (24) = +2.064 1  The t obtained [-2.667] falls beyond the critical value  Therefore p <.05  Whose hypothesis testing approach is this?  Or better, go by specific p-value provided by statistical software  p =.007 2  Whose hypothesis testing approach is this?  Conclusion?

15. One vs. two-tailed probability  Note that just about every time you see a probability for zs, ts, and correlations it is a two-tailed probability  In other words, it’s the probability of that difference of that size greater or less than the null hypothesis value  This reflects complete ignorance about the research situation, which is rare  Unfortunately most test this way. Why?  Truly exploratory work  Poor estimation or ignorance of prior research  Habit.025

16. One vs. two-tailed probability  A one-tailed test suggests that one expects a result of a certain type that you expect e.g. your result to be greater than the null hypothesis value  The informed situation is more statistically powerful, as here you can see the difference seen (and associated t-statistic) would not have to be as extreme to reach.05 probability.05

17. Your turn...  The average grade from year to year in undergrad statistics courses at UNT is an 81. This year the stats students (200 thus far) have an average of 83 w/ s = 10. Is this unusual? t.05 (199) 1 = ? 2-tailed, i.e. probability associated with scores higher or lower than null. Before going in to this we would not have known whether they would be better or worse than the previous. t = ? p = ? Conclusion?

18. Problems with t  Wilcox and others note that when we sample from a non- normal population, assuming normality of the sampling distribution may be optimistic with small samples  Furthermore outliers have an influence on both the mean and sd used to calculate t  Has a larger effect on variance, increasing type II error due to std error increasing more so than the mean  This is not to say we throw the t-distribution out the window  If normality can be assumed, it is appropriate  However, if we cannot meet the normality assumption we may have to try a different approach  E.g. bootstrapping