Math147Lecture28.pdf - Math 147 Lecture Notes Lecture 28...

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Math 147 Lecture Notes: Lecture 28 Walter Carlip March, 2018 In solving a problem of this sort, the grand thing is to be able to reason backwards. That is a very useful accomplishment, and a very easy one, but people do not practise it much. In the every-day affairs of life it is more useful to reason forwards, and so the other comes to be neglected. There are fifty who can reason synthetically for one who can reason analytically . . . . Let me see if I can make it clearer. Most people, if you describe a train of events to them, will tell you what the result would be. They can put those events together in their minds, and argue from them that something will come to pass. There are few people, however, who, if you told them a result, would be able to evolve from their own inner consciousness what the steps were which led up to that result. This power is what I mean when I talk of reasoning backwards, or analytically. Sherlock Holmes in “A Study in Scarlet” Copyright c circlecopyrt 2018 by Walter Carlip 1 Lecture 28
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Today’s Topics Inference & Confidence Intervals Understanding Confidence Intervals Averages Copyright c circlecopyrt 2018 by Walter Carlip 2 Lecture 28
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Last Lecture: Confidence Intervals We left off last time discussing confidence intervals , and looking at an example. I will remind you of the example and pick up where we left off. Copyright c circlecopyrt 2018 by Walter Carlip 3 Lecture 28
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Second Example: Students Living at Home Example: A university has 25 , 000 registered students and wishes to estimate the percentage who are living at home. A simple random sample of 400 students is examined, and it is determined that 317 of the students in the sample live at home. Estimate the percentage of students in the university who live at home and attach a confidence interval to the estimate. Solution: The sample percentage is 317 400 × 100% = . 7925 79% . To compute the standard error , we use a box model: To estimate the standard deviation of this box , we use the bootstrap method: we approximate the fraction of 1 ’s and 0 ’s in the box using the fractions from the sample: SD of Box . 79 × . 21 0 . 41 . Then the standard error for the number of students living at home in the sample is SE for number 400 × 0 . 41 = 20 × 0 . 41 = 8 . 2 8 . Copyright c circlecopyrt 2018 by Walter Carlip 4 Lecture 28
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Example: Students Living at Home Finally, we compute the SE for percentage : SE for percentage 8 400 = 2% . Thus, we estimate that about 79% of the students live at home, give or take about 2%. Copyright c circlecopyrt 2018 by Walter Carlip 5 Lecture 28
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Confidence Intervals In the example 79% of the students in the sample were living at home, and we used this value to estimate that 79% of the students in the population were living at home. How far can the percentage in the actual population be from our 79% estimate?
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