Statistics and Hypothesis Testing

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Why is Y a good estimator of µ Y Unbiased We showed last time that E(Y ) = µ Y (“on average, the sample mean is equal to the population mean”), which is the definition of unbiasedness. (Note, however, that unbiasedness also holds for using the first observation Y 1 as the estimator of µ Y : “On average, the first observation is equal to the population mean.”) Consistent Y becomes closer and closer to (a better and better estimate of) µ Y as the sample size grows. (Note, by the way, that Y 1 does not become a better estimate of µ Y as the sample size grows.) Most efficient Y has variance σ2 Y n , which turns out to be the lowest possible variance among unbiased estimators of µ Y . (Note that Y 1 has variance σY 2 , which is terrible by comparison.) The book demonstrates that Y , which equally weights each observations in ∑ n i=1 1 n Y i, has lower variance than alternative weightings of the data. Alternative weighted averages of all the data are better (lower variance) than Y 1 but not as good as Y
Y is the least squares estimator of µ Y Suppose that the data Y 1 , Y 2 , . . . , Y n are spread along the number line, and you can make one guess, m, about where to put an estimate of µ Y . Y i The criterion for judging the guess will be to make n∑ (Y i − m) 2 i=1 as small as possible. (Translation: square the gap between each observations Y i and the guess and add up the sum of squared gaps.) If the guess m is too high, then the small values of Y i will make the sum of squared gaps get big. If the guess m is too low, then the big values of Y i will make the sum of squares gaps get big. If m is just right, then the sum of squared gaps will be as Y i
Page 1 and 2: Statistics and Hypothesis Testing M
Page 3 and 4: Summary of Main Points ◮ We will
Page 5: What would be a good way to guess
Page 9 and 10: Random Sampling The consequences of
Page 11 and 12: Hypothesis Testing 1. Gather a samp
Page 13 and 14: The p-Value p-value ≡ Pr H 0 [
Page 15 and 16: How likely is the observed value In
Page 17 and 18: Sample Variance, Sample Standard De
Page 19 and 20: Sample Standard Deviation The sampl
Page 21 and 22: Summary: Why standard error ◮ Can
Page 23 and 24: The Prespecified Significance Level
Page 25 and 26: An example Is the mean wage among r
Page 27 and 28: Average wage example, continued ◮
Page 29 and 30: Type I and Type II errors Null Hypo
Page 31 and 32: From Hypothesis Testing to Confiden
Page 33 and 34: Example: 95 percent CI In a sample
Page 35: Next time we do something from the

Statistics and Hypothesis Testing

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