Understanding Statistical Estimators and Confidence Intervals
School
University of Illinois, Urbana Champaign**We aren't endorsed by this school
Course
STAT 400
Subject
Statistics
Date
Dec 11, 2024
Pages
4
Uploaded by CommodoreGoosePerson594
e3keyjulie2024-11-18EstimatorsAny question/subquestion marked with a * indicates a bonus question that is slightly out ofthe scope of STAT400, but is still do-able with your current knowledge1.a.12nQQQQQQQX2ib.¯X√π2c.(4−π)nπσ2d.R∼Exp(2σ2)2. Suppose we have a PDF:f(x;β) =βx2, β < xa.min(xi)b. It is biased ;ˆβ(n−1)n3. Consider the standard Poisson distribution with a rate ofλf(x;λ) =λke−λk!, k∈N0a.¯Xb. 04. Consider the table below.xf(x)0λ12λ−12λ32λ1
a.n0+n2+n32nb.n0+n2+n32n−λc.¯X−220d. 02
Confidence Intervals and Transformations5. Suppose we have a function:f(y) =1θexp(−yθ), y≥0Suppose we then have the pivotal quantityU=g(Y, θ) =Yθa.U∼Exp(1)b.[Y3.6889,Y0.0253]6.Suppose I was doing an unethical experiment where I was trying to measure the average hours a studentis awake on an exam week. Given the following normally distributed data:¯x= 20, σ2= 4, n= 6, s2= 7a.(18.954,21.046)b. 19.186c.(17.23,22.77)7.It’s election time! In some state in the Midwest, a random poll shows thatx= 225voters preferredBaby Eater #1 over Baby Eater #2 out of then= 500voters that went to this certain poll.a.(0.4133,0.4867)8.Suppose I was observing some fish and the waiting time until they inevitably died of pollution. Let’scall this random variableF. Suppose:F∼Gamma(α= 8, θ=β)It is a well-known result in statistics that ifF∼Gamma(α, θ=1λ), then:2λF∼χ2(2α)Using the pivot ofG=2Fβ,a.β≤2F10.8b.[2F28.869,2F3.4338]c.β≥2F9.978d.β≥.6353
9.Suppose I was refreshing the page to buy tickets to Charli XCX’s and Troye Sivan’s SWEAT tour showin Chicago. LetBbe the random variable where I measure the rate at which the tickets get sold in a12 hour period. SupposeB∼Poisson(36000)For large values ofλ, we can approximate the Poisson distribution with the normal distributionB∼N(λ, λ)because of the central limit theorem. With this new-found knowledge,a.(35687.08,36312.92)b. 35871.84c. 36312.92d.[6.939,43.760]4