confidence intervals

confidence intervals - Confidence Intervals Let us start by...

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Confidence Intervals: Let us start by examining problem 7 from problem set 10. 7. Understanding the nice symmetry of the Normal distribution can helpful when dealing with percentiles and probabilities. Suppose Z ~N(0, 1) and assume z > 0. (a) Let p = P( Z ≤ z). Write an expression for P( z Z z) in terms of p. (b) Use (a) to find an interval [ z, z] such that P( z Z z) = 0.95. 7a)We let p=P(Z≤ -z). ( ) ( ) ( ) (1 ( )) ( ) ( )) ( ) P z Z z P Z z P Z z P Z z P Z z P Z z P Z z           =1-2p. 7b) 1-2p=.95 means that 2p=.05, so p=.025. If we look up this z-value we get -1.96. So, P(- 1.96 ≤ Z ≤ 1.96) = .95 Similarly, if you defined p as P(Z z) when z>0, then your equation for ) ( z Z z P would be 2p-1. For problem 7b this means you would have 2p-1 =.95, so p=.975. So, we obtained .975 and .025. If you subtract .025 from .975 you get .95. This is exactly what we want with a 95% confidence interval. It represents the MIDDLE 95% OF THE DATA. What else do you notice about .025 and .975? Do we need to look up both of these? The answer to the second question is no. We only need one of these because of the answer to the first question. They have the same numerical value; the only difference is that one is a positive z-value whereas the other is a negative z-value.

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confidence intervals - Confidence Intervals Let us start by...

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