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l12 - Math 127B Notes for lecture 12 Mrinal Raghupathi...

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Math 127B, Notes for lecture 12 Mrinal Raghupathi Wednesday, March 2nd, 2011 Administrivia Today we will discuss the t -test and tests of significance with percentages. Reminders 1. Homework 7 is due Thursday, 3/17 in recitation. 2. Quiz in recitation over chapter 26 on 3/17. 3. This week’s recitation will be a discussion of problems from chapter 26. Today we will see how to use the z -test when dealing with 0-1 boxes. This will give us a way to deal with null hypotheses stated in terms of percentages. Unlike the case of the average we will see examples where there is no suitable alternative hypothesis. After this we will consider Student’s distribution and the corresponding test, which is used when the sample is small and the normal curve does not apply. 1 0-1 boxes Example 1 . A certain town of 100,000 adults is electing a mayor. It is a close election and Smith and Jones are neck-and-neck. A sample of 400 adults indicates that 52% of will vote for Smith. Is Smith really ahead? The box is a 0-1 box, with a 1 for Smith and a 0 for Jones. The null hypothesis is that there is a split in the town and that the percentage of people who favor Smith is 50%. The alternative is that Smith has an advantage/majority. With 400 draws we can use the normal curve approximation for percentages. The SD of the box is approximated by 0 . 48 × 0 . 52 0 . 5. The SE of the sum is 400 × 0 . 5 = 10 and the SE of the percentage is 10 / 400 × 100 = 2 . 5%. The test statistic is 52 - 50 2 . 5 = 0 . 8 . 1
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The P -value associated to this statistic is 21%. The difference is chance error and we can not reject the null hypothesis. We can not say whether Smith has an advantage. o Sometimes there is no suitable alternative hypothesis. The reason could be that the alternative hypothesis does not fit the box model. In Tart’s ESP experiment the null hypothesis is that the data are like 7,500 draws from the following box 1 0 0 0 . this would be the case if the answers were the result of guessing. However, if the subjects do have ESP then the answers
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