STAT3110 2020W Assignment 4 Due: Friday, March 20, 2020 ——————————————————————————————————————– 1. Suppose a random sample of size 2, X1 and X2, is from a population given the probability distribution...

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STAT3110 2020W Assignment 4 Due: Friday, March 20, 2020 ——————————————————————————————————————– 1. Suppose a random sample of size 2, X1 and X2, is from a population given the probability distribution function f(x; θ) = θxθ−1, 0 < x="">< 1="" to="" test="" the="" hypothesis="" h0="" :="" θ="1" against="" ha="" :="" θ="2" we="" reject="" the="" h0="" if="" x1x2="" ≥="" 34="" .="" find="" the="" probability="" of="" type="" i="" error="" and="" the="" power="" of="" this="" test.="" 2.="" a="" single="" observation,="" x,="" having="" a="" geometric="" distribution="" is="" used="" to="" test="" h0="" :="" θ="θ0" against="" ha="" :="" θ="θ1"> θ0. The H0 will be rejected if and only if the observed value of X ≥ k where k is a positive integer, find expressions for the probabilities of type I and type II errors. 3. Assume the SAT mathematic score of students in a give population followsN(µ, 8100). With a random sample of size n = 36, we want to test H0 : µ = 530 against the Ha : µ < 530.="" a)="" determine="" the="" critical="" region="" such="" that="" the="" significance="" level="" α="0.05." b)="" find="" the="" power="" function="" for="" the="" test="" based="" on="" the="" critical="" region="" found="" in="" (a)="" d)="" sketch="" a="" graph="" of="" the="" power="" function="" over="" the="" region="" of="" 510="" ≤="" µ="" ≤="" 530.="" 4.="" a="" random="" sample="" of="" size="" n="" from="" a="" normal="" population="" with="" unknown="" mean="" and="" variance="" is="" to="" be="" used="" to="" test="" h0="" :="" µ="µ0" against="" ha="" :="" µ="" 6="µ0." finding="" the="" mles="" of="" µ="" and="" σ2="" and="" show="" that="" the="" likelihood="" ratio="" test="" statistic="" can="" be="" written="" in="" the="" form="" λ="(1" +="" t2="" n−="" 1="" )−n/2="" where="" t="x̄−µ0" s/="" √="" n="" .="" 5.="" suppose="" a="" random="" sample="" x1,="" ...,="" x25="" from="" n(µ,="" 100)="" is="" used="" to="" test="" h0="" :="" µ="60" against="" ha="" :="" µ=""> 60. Suppose we obtain the sample mean of x̄ = 62.75. At the significance level of α = 0.05, determine the critical region and the p-value for the test, make your conclusion about the test. (Hint, use pnorm() in R to find the p-value for the test statistic.) 1