Correlational Studies of Differences between Means Review 1 Correlation relation between variables Prediction predict the value of one variable predicted variable from the value of another variable predictor variable Predict how far a value on one variable differs from the mean of that variable based on how far the value on the other variable differs from its mean Prediction based on regression line Regression coefficient Regression constant Do Humans Have Abnormally Large Brains Comparing the brains of a number of species humans do seem to have larger brains But hardly the largest But humans also have larger bodies How do brain sizes correlate with body sizes across species 1 Correlations and Allometry Allometry correlates the size of parts of organisms brains with overall size Useful for determining whether the part is unusually larger in a given species Human brains only slightly larger than expected Review 2 You have found a correlation in a sample How likely is it that there is a correlation in that direction in the actual population At what level is the correlation statistically significant p 05 less than 1 chance in 20 that the result is due to chance factors in drawing the sample p 01 less than 1 chance in 100 that the result is due to chance factors in drawing the sample Choose significance level p value according to how important it is not to be wrong Review 3 Statistical significance and importance are very different things Statistical significance simply tells you whether the sample is informative about the real population Some surprising results are simply sampling phenomena and so don t require special explanation Regression to the mean Extreme scores are likely to be in large part due to chance factors in sampling There is no sophomore slump Streaks in samples are expected TTTHTHHHHTTT 2 What to base predictions on In trying to make predictions we very often ignore the most important variable for making a prediction Frank was drawn at random from a group of 30 lawyers and 70 engineers He spends most of his free time hanging around his country club At the bar he often talks about his regrets at having tried to follow in his esteemed father s footsteps The long hours he spent slaving in school could have been better spent learning to be less quarrelsome in his relationships with other people Is Frank a lawyer or an engineer What to base predictions on Would you answer this one any differently Frank was drawn at random from a group of 30 engineers and 70 lawyers He spends most of his free time hanging around his country club At the bar he often talks about his regrets at having tried to follow in his esteemed father s footsteps The long hours he spent slaving in school could have been better spent learning to be less quarrelsome in his relationships with other people Is Frank a lawyer or an engineer What to base legal decisions on In a city in which two cab companies Blue and Green operate a taxicab was involved in a hit and run accident one night 85 of the cabs in the city are Green 15 Blue A witness identified the cab as Blue The Court tested the ability of the witness to identify cab colors under appropriate visibility conditions He she made the correct identification 80 of the time What is the probability that the cab involved was Blue Confident enough to convict 3 What to base legal decisions on Said Blue Said Green Totals Blue 12 3 15 Green 17 68 85 Totals 29 71 100 Of the times he she said it was Blue it was blue 12 29 or 41 Is 50 accuracy good enough to convict Base Rates Base rates are often the best predictor It matters greatly whether the population was 70 30 lawyers or 70 30 engineers It matters greatly that 85 of the cabs were Green This trumps the witness s 80 accuracy But humans almost universally ignore base rates if there is anything else on which to base the decision Police lawyers scientists doctors Even philosophy professors Dealing with probabilities rationally Fred is a 40 something college professor active in social causes and very concerned about the environment How likely is it that Fred is a Republican Fred recycles his trash Fred is a Republican who recycles his trash Fred is a Republican and recyles his trash cannot be more likely than the other two 4 Combining probabilities If two events are independent the probability of one does not depend on the probability of the other you determine their joint probability by multiplying Since probabilities have values in the range 0 to 1 the product will never be larger than the probability of the more likely event If the events are not independent one has to adjust the formula but multiplication is still the central operation and the above point still holds Comparing two populations Populations defined in terms of nominal variables Men women Gay straight Taking Phil 12 not taking Phil 12 Compare the two populations on another variable If this variable is a score variable ask Do the distributions differ Do the means differ Diagramming differences between means Difference between heights of columns reflects differences in means When the whole population is tabulated very straightforward Hours of TV Watching Use bar graph 3 0 Difference 1 3 hours 2 0 1 0 No College Degree College Degree 5 Using samples to assess differences between means 3 0 Difference 1 3 hours 2 0 take sample 1 0 Hours of TV Watching Hours of TV Watching You take a sample and there is a difference in means Where did this difference come from A difference in the real population 3 0 Difference 1 3 hours 2 0 1 0 No College College Degree Degree No College College Degree Degree Sample Real population Using samples to assess differences between means 2 3 0 Difference 0 hours 2 0 take sample 1 0 No College College Degree Degree Real population Hours of TV Watching Hours of TV Watching But it could also arise from a real population in which there was no difference 3 0 Difference 1 3 hours 2 0 1 0 No CollegeCollege Degree Degree Sample In this case the result is due to who happened to get chosen for the sample How to tell whether a sample difference is real What is the probability that the difference in the sample could have resulted by chance had there been no difference in the population The hypothesis that there is no difference between the means of the two groups is known as the null hypothesis Strategy try to reject the null hypothesis Accept that the difference is found in the real population when the sample you get would be very