Statistical Exam Questions: Confidence Intervals, Hypothesis Testing, Poisson, Exams of Statistics

Download Statistical Exam Questions: Confidence Intervals, Hypothesis Testing, Poisson and more Exams Statistics in PDF only on Docsity! 1 Stat 447 Final Exam May 6, 2002 Prof. Vardeman 1. An example in Moore’s Basic Practice of Statistics concerns the deterioration of synthetic fiber when buried in a landfill. A total of 10 strips of a polyester fabric were buried in well-drained soil. 1 5n = strips were then dug up and strength tested after 2 weeks, and the other 2 5n = were dug up and strength tested after 16 weeks. The mean and standard deviation of the first 5 measured strengths were respectively 123.8 lbs and 4.6 lbs. The mean and standard deviation of the last 5 measured strengths were respectively 116.4 lbs and 16.1 lbs. a) Give 95% two-sided confidence limits for the mean strength of strips of this polyester buried for 16 weeks based on a normal distribution assumption. (Plug in correctly, but there is no need to simplify.) b) There is some hint in the data summaries above that the longer polyester is buried, the less consistent is its breaking strength. How strong do you judge this evidence to be? Give some quantitative analysis to support your answer. Henceforth ignore any misgivings about change in consistency of breaking strength raised in b). c) Can the null hypothesis of equality of mean strengths after 2 and after 16 weeks be rejected in favor of a decrease in mean strength with .05α = ? Show appropriate work to support your answer. 2 2. A data set in Moore’s Basic Practice of Statistics gives heating degree days ix and natural gas consumption iy (in 100 cu ft) for 16n = months for a particular household. These produced 0 1.0892b = , 1 .188999b = and /14 .3389SSE = . The 16n = months had 22.3x = and ( ) 16 2 1 4,719i i x x = − =∑ . a) Give 90% two-sided confidence limits for the standard deviation of gas consumption for any fixed number of heating degree days based on the fact that 2/SSE σ has a 2χ distribution under the simple linear regression model. (Plug in, but you need not simplify.) b) Give 90% two-sided confidence limits for the increase in mean gas consumption per 1 degree day increase in x (that is, for 1β ). (Plug in, but you need not simplify.) c) Give 90% two-sided prediction limits for the next gas consumption for this household in a month in which 30x = . (Pug in, but you need not simplify.) 5 5. The decay of a radioactive material is such that the number of emitted particles detected by a certain Geiger counter in 1 second is a Poisson random variable with mean λ (and the number detected in k seconds is Poisson with mean kλ ). a) Suppose that I plan to make a count X of particles detected in 1 second, intending to test 0H : 1λ = versus aH : 1λ > . I determine to reject 0H if 4X ≥ . Use the book’s table of Poisson probabilities and evaluate the type I error probability (α ) I am using. b) In the context of part a), suppose that in fact 2λ = . What is the type II error probability ( β )? c) Suppose that “a priori” I model λ as Uniform (0, 2) . What is my “prior” probability that 1.5λ > ? d) If I use the “prior” distribution of part c) and in 10 seconds detect 20 particles, the posterior distribution of λ (that is NOT of a standard type) has a pdf proportional to 20exp( 10 )λ λ− on the interval (0, 2) . This is pictured below. Approximate the “posterior” probability that 1.5λ > . 6 6. Consider the continuous distribution with pdf ( ) 2 21 1 ( 1) | (1 ) exp exp 2 22 2 x x f x p p p π π    − = − − + −        for 0 1p≤ ≤ . This pdf is appropriate if an observable is ( )N 1,1 with probability p and is ( )N 0,1 with probability 1 p− . The distribution has mean p and variance 21 p p+ − . This problem concerns inference for p based on observations 1 2, , , nX X X… that are iid according to ( | )f x p . Attached is a printout made using MATHCAD for the analysis of a particular set of 25n = data values. The function l is the loglikelihood, the function lprime is the score function, and the function lprimeprime is the derivative of the score function. The 25 observations have .070x = and 1.094s = and the printout shows (.104) 0 lprime = and (.104) 29.106lprimeprime = − . GIVE NUMERICAL ANSWERS TO THE QUESTIONS BELOW BASED ON THIS SAMPLE. a) Give both the maximum likelihood estimate p̂ and the method of moments estimate p% for p . p̂ = __________ p =% __________ b) Use the maximum likelihood estimate and find approximate 90% two-sided confidence limits for p based on it. c) Make 90% two-sided confidence limits for p based on and x s . d) Give an approximate observed value for the likelihood ratio statistic for testing 0H : .9p = . x .5 .14 .82 .86 2.05 2.29 .97 .37 .07 .71 1.39 .63 1.33 .05 1.65 1.33 .52 .1 .57 .71 1.32 1.03 1.08 .33 1.53 l p( ) 0 24 i ln 1 p( ) dnorm xi 0, 1, . p dnorm xi 1, 1, . = lprime p( ) p l p( )d d lprimeprime p( ) 2p l p( )d d 2