511_Chap2

511_Chap2 - 511 Chap 2 1 Inference Using t-Distributions...

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Unformatted text preview: 511 Chap 2 1 Inference Using t-Distributions Chapter 2 511 Chap 2 2 Bumpus Data -- Case Study I ■ Humerus length of adult sparrows in 1898 ■ 2 groups: perished, survived (after a storm) ■ surviving sparrows were stressed ■ unequal group sizes (24 and 35) ■ observational study ■ no random selection ■ is there evidence of natural selection? 511 Chap 2 3 Bumpus Data -- Case Study I ■ Explain in words what a p-value of 0.08 means. ■ Even if we had an extremely low p-value, why would it be difficult to make general inferences? 511 Chap 2 4 Case Study II ■ 15 pairs of monozygotic twins -- one of which was schizophrenic ■ volume of left hippocampus was measured ■ 2 groups – but the individuals from the groups are paired , not independent ■ observational study, no random selection ■ is left hippocampus volume associated with schizophrenia? 511 Chap 2 5 Case Study II ■ 15 pairs of twins -- one of which was schizophrenic ■ We seem to find a difference between the sizes of a certain brain region -- the p-value was 0.0061. ■ Explain in English what this value of 0.0061 means. ■ Did the size difference cause the disease? 511 Chap 2 6 These case studies serve to review tools you are familiar with ■ The twin study can be analyzed using a paired t-test (or paired t tools) . ■ The Bumpus data can be approached using a 2-sample t-test (or 2- sample t tools). 511 Chap 2 7 Start with twin example ■ pretend that the selected sets of twins were a random sample of the population of such sets ■ hence the unaffected-affected differences would be a random sample of the population of such differences ■ consider the average difference in our sample ( ) to be the test statistic Y 511 Chap 2 8 twin example (continued) ■ if schizophrenia made no difference, should be ‘close’ to 0 ■ we can see if it is ‘close’ to zero by looking at the distribution of all possible ’s for samples of this size from the population of differences with an assumed mean of 0 ■ but how can we do that on the limited information we have about that population? Y Y 511 Chap 2 9 Some remarkable mathematical results come to the rescue ■ Distributions of sample averages are surprisingly quite predictable ■ Example: weight of male students at BYU. Population average ( μ ) is 185 pounds. ■ We take a random sample of size n=100 and calculate . Will we get 185? ■ No, but we will probably be close 511 Chap 2 10 A Second Sample of Size 100 ■ Will this second sample have the same as our first sample? ■ No ■ Will it be 185? ■ No, but again it will probably be pretty close. 511 Chap 2 11 A 3 rd , 4 th , Etc. Samples ■ Each of these samples will differ from each other....
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511_Chap2 - 511 Chap 2 1 Inference Using t-Distributions...

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