1. The Hypothesis of Difference Chapter 10

2. Sampling Distribution of Differences Use a Sampling Distribution of Differences when we want to examine a hypothesis of difference We want to compare sets of scores from two samples EX: We want to find the average IQ of all students at Omega College (pop=6000) -put all 6000 student names in a fishbowl -randomly select 30 students with our left hand & 30 students with our right hand -give them each an IQ test -find a mean of 118 for students selected with our left hand (M 1 =118) -find a mean of 115 for students selected with our right hand (M 2 =115) -then we calculate the difference between these two means (118-115= 3) -then, we randomly select another 30 students with the left hand and 30 with the right hand, give them the IQ test & calculate the difference between the means….so on and so forth. Mean of the Distribution of Differences: μM 1 -M 2 -theoretically, if you add the differences between the means together and divide by the number of differences, you could calculate the mean of the distribution of differences -if we did that we should get a mean of approximately zero since all of the samples come from a single population **WHY? Because there should be the same number of positive differences as there are negative differences so they will cancel each other out

3. Standard Error of Difference Standard Error of Difference: allows us to predict the value of the standard deviation of the entire distribution of differences between the means of successively drawn pairs of random samples. -ie. the estimated standard deviation of the sampling distribution of differences -based on information contained in just two samples *Symbolized as SE D Formula: Step 1: Calculate the means for M 1 and M 2 Step 2: Calculate the standard deviations for SD 1 and SD 2 Step 3: Calculate the standard error of the mean for SE M1 and SE M2 Step 4: Calculate the standard error of the difference

4. Independent t-test Two-sample t-test for Independent Samples -allows us to make a probability statement regarding whether two independently selected samples represent a single population. -“independent” means the samples are not dependent on each other *ie: we can’t use this test if we are doing a repeated measure or matched subjects design -Research Question: Are these two samples from the same population? Ho: μ 1 = μ 2 The samples are from the same population Ha: μ 1 ≠ μ 2 The samples are not from the same population (they are significantly different) Formula:

5. Evaluation of t In order to Reject Ho the t-value must fall within the.05 or.01 critical areas *critical areas=rejection region We use the Levels of Significance (critical areas) to reduce the probability of committing an alpha error *the smaller the critical area, the stronger the conclusion to Reject Ho is After we calculate t, we need to determine the t-values at the.05 and.01 levels so we can make a comparison *ie. Does our observed t-value fall within the critical areas? We must calculate the degrees of freedom: In order to Reject Ho, the calculated t-value must be GREATER than the the tabled t-value (Table C & D)

6. Evaluation of t

7. One Tail vs. Two-Tail Advantage of the one-tail test: do not have to obtain as high of a calculated t in order to reject the null, as we do with a two-tailed Disadvantage of the one-tail test: the sign of the t-value matters (whether it is positive or negative) thus limiting the ability to reject the null -if you predict a positive t-value (saying M 1 is greater than M 2 ) then you must calculate a positive t-value to be able to reject the null -if you predict a negative t-value (saying M 1 is less than M 2 ) then you must calculate a negative t-value to be able to reject the null Evaluation of t

8. Type I & Type II Errors α β

9. Power Power is our ability to Reject Ho -we want power!! -the more power we have, the more likely we are to reject Ho -power is 1-β (β=probability that you won’t reject Ho) Q: if β=.15 then what is the power of the test? *conceptually, we want to pull this curve to the left *researchers generally want power to get to.80 How can we increase power? -increase our alpha levels **Problem is, while you will reject more nulls this includes true nulls—increase Type 1 errors -control for extraneous variables **if we can get rid of variability that we don’t want, it can’t interfere with the variability we want -reduce measurement error **ie. Be precise in your measurement -increase your sample size **the more people you have, the more likely to find what you are looking for **However, if it’s too large then you are likely to detect variability that isn’t meaningful