lecture11 - ISYE 2028 A and B Lecture 11 Condence Intervals...

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ISYE 2028 A and B Lecture 11 Confidence Intervals and Hypothesis Testing Dr. Kobi Abayomi March 13, 2009 1 A confidence interval is an interval estimate for a population parameter First, an interval estimate is a range of possible values. If I say ” μ is between 8 and 12” (or μ [8 , 12]) I am presenting an interval estimate for μ . 1.1 An Extraterrestrial Example We set this illustration on another world for emphasis. There is what really exists - the population, and there is what we see - the sample. We can only suggest a model for the population and then seek to gain insight about it via the data. In this example, we of course can never see the true population. But we can take the time and expense to observe some data, and then make a statement about an assumed model for the population. A good Biologist goes to Mars and collects a sample of Martians. She records how many eyes each martian had in her sample, and then the average number of eyes per Martian. At a local bar I ask her ”How many eyes does a Martian have” She responds: ”Oh, I don’t known for sure. My sample average, my estimate, is 10. I’m 95 percent sure the true mean is between 8 and 12. Between 8 and 12, that’s my 95 percent confidence interval.” 1
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This means: if she were able to repeat her experiment 100 times, 95 out of one 100 times she would generate a interval estimate that covered the true mean. Notice that this does not mean that she will get a sample mean of 10 on 95 out of 100 experiments. Notice also that this does not mean that her interval estimate will be [8 , 12] on 95 out of 100 experiments. Notice also that implicit in her answer is the model assumption: Martians have eyes and there is an expected number, or population mean, number of eyes for every Martian. Somewhere God, who loves the Martians too, knows the true number of eyes each Martian has, and as well (of course) the mean number of eyes for all of his Martian children. The Biologist, in most belief systems, cannot know the mind of God. However, the Biologist does know the sampling distribution and can make an inference about what He knows ( μ ) from what she saw x . It would be expensive to go back to Mars so we use the distribution of the sample mean to make statements about the true, unknown, population parameter. A confidence interval is our interval estimate attached to some probability. From one experiment we get one confidence interval which suggests a range of values for the true population parameter. We feel pretty good about this. Think about this example more if you have to. Ask me to draw you a picture. I will. 1.2 The general setup Take a random variable X μ,σ 2 and with n very very large.
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lecture11 - ISYE 2028 A and B Lecture 11 Condence Intervals...

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