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03 Confidence Intervals and Hypothesis Testing

# 03 Confidence Intervals and Hypothesis Testing - Economics...

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Economics 140A Confidence Intervals and Hypothesis Testing inference because it is highly unlikely that the population value of a parameter is equal to the estimate. We wish instead to know how close the population value is likely to be to an estimate. To determine this, we must estimate an interval, which uses more of the information in the data (the variance of the estimator) than does point estimation. Let f Y i g n i =1 be a sequence of independent identically distributed N ( ± 2 ) ran- dom variables. Hence Y n N ± 2 n ± ; so Y n ± ±= p n N (0 ; 1) : (Diagram density) From tabulated values of N (0 ; 1) P ± 1 : 96 ² ² Y n ± ³ ±= p n ² 1 : 96 ! = : 95 : Do the algebra one step at a time on the board. Step 1 P ± 1 : 96 ± p n ² Y n ± ² 1 : 96 ± p n ± = : 95 : Step 2 P ± Y n ± 1 : 96 ± p n ² ± ² ± Y n + 1 : 96 ± p n ± = : 95 : Step 3 P Y n ± 1 : 96 ± p n ² ² Y n + 1 : 96 ± p n ± = : 95 :

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Thus, the random interval Y n 1 : 96 p n ; Y n + 1 : 96 p n ± contains ± 95 percent of the time. Remark: Note the interval is random and ± percent of all intervals contain ± rather than ± falls in the interval 95 percent of the time. For a given sample, we have the estimate y n . If we replace the estimator with the estimate we have y n 1 : 96 p n ; y n + 1 : 96 p n ± ; ± or does not, so we cannot refer to the probability that the interval contains ± . To overcome the di¢ - as probability. We say ± lies in y n 1 : 96 p n ; y n + 1 : 96 p n ± is .95 or ± is y n 1 : 96 p n ; y n + 1 : 96 p n ± .
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03 Confidence Intervals and Hypothesis Testing - Economics...

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