Chapter 7 Notes  Inference for Single Samples. You know already for a large sample, you can invoke the CLT so:


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1 Chapter 7 Notes  Inference for Single Samples You know already for a large sample, you can invoke the CLT so: X N(µ, ). Also for a large sample, you can replace an unknown σ by s. You know how to do a hypothesis test for the mean, either: calculate zstatistic x µ 0 z = and compare it with z α or z α/. calculate pvalue and compare with α or α/. calculate CI and see whether µ 0 is within it. Let s add two more calculations. 1) Determine n to achieve a certain width for a sided confidence interval. Of course, small width large n. Derivation of Sample Size Calculation for CI z α/ σ n = E where E is the halfwidth of the CI. (Sample Size Calculation) ) Power Calculation For upper 1sided ztests: H 0 : µ µ 0 H 1 : µ > µ 0, in fact, we ll take µ = µ 1. The calculation only makes sense if µ 1 > µ 0. We want to know what the power of the test is to detect mean µ 1. We ll compute power as a function of µ 1. Derivation of Power Calculation for Upper 1sided ztests µ 1 µ 0 π(µ 1 ) = P (test rejects H 0 in favor of H 1 H 1 ) = Φ z α +. 1
2 Now we can consider π(µ 1 ) as a function of µ 1. Again, the alternative hypothesis only make sense if µ 1 > µ 0. As µ 1 increases, what happens to π(µ 1 )? For lower 1sided tests, ( µ 0 µ ) 1 π(µ 1 ) = Φ z α +. The alternative hypothesis only makes sense when µ 1 < µ 0. As µ 1 increases (and gets closer to a µ 0 ), what happens to π(µ 1 )? For sided tests, ( ) ( ) σ σ π(µ 1 ) = P X < µ 0 z α/ µ = µ 1 + P X > µ 0 + z α/ µ = µ 1 n n ( ) ( ) µ 0 µ 1 µ 1 µ 0 = Φ z α/ + + Φ z α/ + As µ 1 changes, what happens to π(µ 1 )?
3 3) Sample size calculation for power. Want to find the n required to guarantee a certain power, 1 β, for an αlevel ztest. Let := µ 1 µ 0 so that µ 1 = µ 0 +. For upper 1sided, we have (look up at the power calculation we did for upper 1sided): ( ) π(µ 1 ) = π(µ 0 + ) = Φ z α + = 1 β. Since our notation says that z β is defined as the number where Φ(z β ) = 1 β: Now solve that for n: z α + = z β. (z α + z β )σ n =. For lower 1sided, n is the same by symmetry. For sided, turns out one of the two terms of π(µ 1 ) can be ignored to get an approximation: (z α/ + z β )σ n. Remember to round up to the next integer when doing samplesize calculations! 3
4 7. Inferences on Small Samples If n < 30, we often need to use the tdistribution rather than zdistribution N(0, 1) since s doesn t approximate σ very well. Need X 1,..., X n N(µ, ). The bottom line is that we replace: X µ X µ Z = by T = S/ n for a ttest on the mean. Replace z α by t n 1,α. Replace σ by S. There s a chart in your book on page 53 that summarizes this. Note that the power calculation is harder for ttests, so for this class, just say S σ and use the normal distribution power calculation. You ll get an approximation. 7.3 Inferences on Variances Assume X 1,..., X n N(µ, ). Inferences on variance are very sensitive to this assumption, so inference only with caution! The bottom line is that we replace: X µ (n 1)S Z = by χ = (and test for not µ). Replace z α by χ and/or χ n 1,1 α n 1,α. Hypothesis tests on variance are not quite the same as on the mean. Let s do some of the computations to show you. First, we ll compute the CI. 4
5 sided CI for. As usual, start with what we know: ( ) (n 1)S (n 1)S 1 α = P χ χ and remember χ =, n 1,1 α/ n 1,α/ (*1) (*) and we want: 1 α = P (L U) for some L and U. Let s solve it on the left for (*1) and on the right for (*): Putting it together we have: 1 α = P (n 1)S (n 1)S χn 1,1 α/ χ n 1,α/ (n 1)S χ n 1,α/ (n 1)S χ n 1,1 α/ 1 α = P L U. The 100(1 α)% confidence interval for is then Similarly, 1sided CI s for are: (n 1)s (n 1)s. χ n 1,α/ χ n 1,1 α/ (n 1)s (n 1)s and. χ χ n 1,α n 1,1 α Hypothesis tests on Variance (a chisquare test) To test H = = 0 : 0 vs H 1 : 0, we can either: Compute χ statistic: χ = (n 1)s and reject H 0 when either χ > χ n 1,α/ or χ < χ n 1,1 α/. Compute pvalue: First we calculate the probability to be as extreme in either direction: 0 5
6 P U = P (χ n 1 χ ) or P L = P (χn 1 χ ) depending on which is smaller (more extreme). The probability to obtain a χ at least as extreme under H 0 is: min(p U, P L ). This accounts for being extreme in either direction. Compute CI (already done) Table 7.6 on page 57 summarizes the chisquare hypothesis test on variance. Note that this is not the most commonly used chisquare test! See Wikipedia: A chisquare test is any statistical hypothesis test in which the sampling distribution of the test statistic is a chisquare distribution when the null hypothesis is true... (n 1)S (In this case, we have normal random variables, so the distribution of the test statistic is chisquare.) 6
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