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# Hw11 - ≥ 3” a Find the significance level of this test...

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STAT 410 Homework #11 Spring 2008 (due Friday, April 11, by 3:00 p.m.) 1. Let X 1 , X 2 , … , X 25 be a random sample from a N ( μ , σ 2 = 100 ) population, and suppose the null hypothesis H 0 : μ = 100 is to be tested. a) If the alternative hypothesis is H 1 : μ 100, compute the power of the appropriate test at μ 1 = 101. Use α = 0.05. b) If the alternative hypothesis is H 1 : μ > 100, compute the power of the appropriate test at μ 1 = 101. Use α = 0.05. c) For the test in (a) compute the p-value associated with X = 103.5. d) For the test in (b) compute the p-value associated with X = 103.5. 2. 5.1.3 “Hint”: Table II ( p. 673 ) gives quantiles (percentiles) of χ 2 distribution. 3. 5.4.13 4. 8.2.3 Hint: Do NOT look at the answer at the back of the textbook. 5. 5.5.4 6. 5.5.9 7. 5.5.12

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8. Let X have a Binomial distribution with the number of trials n = 5 and with p either 0.20 or 0.30. To test H 0 : p = 0.20 vs. H 1 : p = 0.30, we will use the rejection ( critical ) region “Reject H 0 if X 3”. a) Find the significance level of this test.
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Unformatted text preview: ≥ 3”. a) Find the significance level of this test. b) Find the power of this test. c) Suppose the observed value of X is x = 2. Find the p-value. 9. 8.2.2 Hint: Consider three cases: 0 < θ < ½ , ½ < θ < 1, and θ > 1. _________________________________________________________________________ If you are registered for 4 credit hours: ( to be handed in separately ) 10. Let X have a Binomial distribution with parameters n and p . Recall that ( ) 1 X p p n p n--has an approximate Standard Normal N ( 0, 1 ) distribution, provided that n is large enough, and ( ) α-& & ± ² ³ ³ ´ µ <--<-≈ 1 1 X P 2 2 z p p n p n z . Show that an approximate 100 ( 1 – ) % confidence interval for p is ( ) n z n z n p p z n z p 2 2 2 2 2 2 2 2 1 4 1 2 ˆ ˆ ˆ + +-± + , where n p X ˆ = . This interval is called the Wilson interval. Note that for large n , this interval is approximately equal to ( ) n p p z p 2 ˆ ˆ ˆ 1-± ....
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