1 Hypothesis testing for a single mean


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1 BST Hypothesis Testig Review otes 1 Hypothesis testig for a sigle mea 1. The ull, or status quo, hypothesis is labeled H 0, the alterative H a or H 1 or H.... A type I error occurs whe we falsely reject the ull hypothesis. The probability of a type I error is usually labeled α. 3. A type II error occurs whe we falsely fail to reject the ull hypothesis. A type II error is usually labeled β. 4. A Power is the probability that we correctly reject the ull hypothesis, 1 β. 5. The Z test for H 0 : µ = µ 0 versus H 1 : µ < µ 0 or H : µ µ 0 or H 3 : µ > µ 0 costructs a test statistic TS = X µ 0 S/ ad rejects the ull hypothesis whe H 1 TS Z 1 α H TS Z 1 α/ H 3 TS Z 1 α respectively. 6. The Z test requires the assumptios of the CLT ad for to be large eough for it to apply. 7. If is small, the a Studet s T test is performed exactly i the same way, with the ormal quatiles replaced by the appropriate Studet s T quatiles ad 1 df. 8. Tests defie cofidece itervals by cosiderig the collectio of values of µ 0 for which you fail to reject a two sided test. This yields exactly the T ad Z cofidece itervals respectively. 9. Coversely, cofidece itervals defie tests by the rule where oe rejects H 0 if µ 0 is ot i the cofidece iterval. 10. A Pvalue is the probability of gettig evidece as extreme or more extreme tha we actually got uder the ull hypothesis. For H 3 above, the Pvalue is calculated as PZ TS obs µ = µ 0 where TS obs is the observed value of our test statistic. To get the Pvalue for H, calculate a oe sided Pvalue ad double it. 11. The Pvalue is equal to the attaied sigificace level. That is, the smallest α value for which we would have rejected the ull hypothesis. Therefore, rejectig the ull hypothesis if a Pvalue is less tha α is the same as performig the rejectio regio test. 1
2 1. The power of a Z test for H 3 is give by the formula kow how this is obtaied PTS > Z 1 α µ = µ 1 = P Z µ 0 µ 1 σ/ + Z 1 α. Notice that power required a value for µ 1, the value uder the ull hypothesis. Correspodigly for H 1 we have P Z µ 0 µ 1 σ/ Z 1 α. For H, the power is approximately the appropriate oe sided power usig α/. 13. Some facts about power. a. Power goes up as α goes up. b. Power of a oe sided test is greater tha the power of the associated two sided test. c. Power goes up as µ 1 gets further away from µ 0. d. Power goes up as goes up. 14. The prior formula ca be used to calculate the sample size. For example, usig the power formula for H 1, settig Z 1 β = µ 0 µ 1 σ/ Z 1 α yields = Z 1 β + Z 1 α σ µ 0 µ 1, which gives the sample size to have power = 1 β. This formula applies for H 3 also. For the two sided test, H, replace α by α/. 15. Determiats of sample size. a. gets larger as α gets smaller. b. gets larger as the power you wat gets larger. c. gets lager the closer µ 1 is to µ 0. Biomial cofidece itervals ad tests 1. Biomial distributios are used to model proportios. If X Biomial, p the ˆp = X/ is a sample proportio.. ˆp has the followig properties. a. It is a sample mea of Beroulli radom variables. b. It has expected value p. c. It has variace p1 p/. Note that the largest value that p1 p ca take is 1/4 at p = 1/.
3 d. Z = ˆp p follows a stadard ormal distributio for large by the CLT. The p1 p/ covergece to ormality is fastest whe p = The Wald test for H 0 : p = p 0 versus oe of H 1 : p < p 0, H : p = p 0, ad H 3 : p > p 0 uses the test statistic ˆp p TS = ˆp1 ˆp/ which is compared to stadard ormal quatiles. 4. The Wald cofidece iterval for a biomial proportio is ˆp ± Z 1 α/ ˆp1 ˆp/. The Wald iterval is the iterval obtaied by ivertig the Wald test ad vice versa. 5. The Score test for a biomial proportio is TS = ˆp p p0 1 p 0 /. The score test has better fiite sample performace tha the Wald test. 6. The Score iterval is obtaied by ivertig the score test ad vice versa ˆp + 1 Z1 α/ +Z1 α/ +Z1 α/ ] Z1 α/ ±Z 1 α/ ˆp1 ˆp. 1 +Z1 α/ +Z 1 α/ Z 1 α/ 7. A approximate score iterval for α =.05 ca be obtaied by takig p = X+ +4 ad calculatig the Wald iterval usig p istead of ˆp. 8. A exact biomial test for H 3 ca be performed by calculatig the exact Pvalue PX x obs p = p 0 = k=x obs k p k 01 p 0 k. where x obs is the observed success cout. For H 1 the correspodig exact Pvalue is x obs PX x obs p = p 0 = k k=0 p k 01 p 0 k. These cofidece itervals are exact, which meas that the actual type oe error rate is o larger tha α. The actual type oe error rate is geerally smaller tha α. Therefore these tests are coservative. For H, calculate the appropriate oe sided Pvalue ad double it. 3
4 9. Occasioally, someoe will try to covice you to obtai a exact Type I error rate usig supplemetal radomizatio. Igore them. 10. Ivertig the exact test, choosig those value of p 0 for which we fail to reject H 0, yields a exact cofidece iterval. This iterval has to be calculated umerically. The coverage of the exact biomial iterval is o lower tha 1001 α%. 3 Group comparisos 1. For group comparisos, make sure to differetiate whether or ot the observatios are paired or matched versus idepedet.. For paired comparisos for cotiuous data, oe strategy is to calculate the differeces ad use the methods for testig ad performig hypotheses regardig a sigle mea. The resultig tests ad cofidece itervals are called paired Studet s T tests ad itervals respectively. 3. For idepedet groups of iid variables, say X i ad Y i, with a costat variace σ across groups Z = X Ȳ µ x µ y S 1 p + 1 limits to a stadard ormal radom variable as both ad get large. Here S p = 1S x + 1S y + is the pooled estimate of the variace. Obviously, X, Sx, are the sample mea, sample stadard deviatio ad sample size for the X i ad Ȳ, S y ad are defied aalogously. 4. If the X i ad Y i happe to be ormal, the Z follows the Studet s T distributio with + degrees of freedom. 5. The test statistic TS = X Ȳ S p q 1 x + 1 y ca be used to test the hypothesis that H 0 : µ x = µ y versus the alteratives H 1 : µ x < µ y, H : µ x µ y ad H 3 : µ x > µ y. The test statistic should be compared to Studet s T quatiles with + df. 6. S x/σ x S y /σ y follows what is called the F distributio with 1 umerator degrees of freedom ad 1 deomiator degrees of freedom. 7. To test the hypothesis H 0 : σ x = σ y versus th hypotheses H 1 : σ x < σ y, H : σ x σ y ad H 3 : σ x > σ y compare the statistic TS = S 1/S to the F distributio. We reject H 0 if: H 1 if TS < F x 1, 1,α, H if TS < F x 1, 1,α/ or TS > F x 1, 1,1 α/, 4
5 H 3 if TS > F x 1, 1,1 α. 8. The F distributio satisfies the property that F x 1, 1,α = F y 1, 1,1 α. So that, it turs out, that our results are cosistet whether we put S x o the top or bottom. 9. Usig the fact that 1 α = P F x 1, 1,α/ S x/σx Sy/σ y we ca calculate a cofidece iterval for σ y as σx course, the cofidece iterval for σ x is σy 10. F tests are ot robust to the ormality assumptio. 11. The statistic F x 1, 1,1 α/ F S x x 1, 1,α,F S Sy x x 1, 1,1 α/ Sy ] Sy F Sy y 1, 1,α,F Sx y 1, 1,1 α/. Sx X Ȳ µ x µ y S x + S y ]. Of follows a stadard ormal distributio for large ad. It follows a approximate Studets T distributio if the X i ad Y i are ormally distributed. The degrees of freedom are give below. 1. For testig H 0 : µ x = µ y i the evet where there is evidece to suggest that σ x σ y, the test statistic TS = X Ȳ follows a approximate Studet s T distributio uder r S x x + S y y the ull hypothesis whe X i ad Y i are ormally distributed. The degrees of freedom are approximated with S x/ + S y/ S x/ / 1 + S y/ / The power for a Z test of H 0 : µ x = µ y versus H 3 : µ x > µ y is give by P Z Z 1 α µ x µ y while for H 1 : µ x < µ y it is P Z Z 1 α 14. Sample size calculatio assumig = = σ x + σ y µ x µ y σ x + σ y. = Z 1 α + Z 1 β σ x + σ y µ x µ y. 5
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