The Hypothesis of Difference Chapter 10

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 ( = 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

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

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:

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)

Evaluation of t

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

Type I & Type II Errors α β

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