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STAT 3701 random variable
创建日期：2024-07-16 14:43:53
所属分类：统计学statistics
标签： STAT 3701
random variable

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STAT 3701 Homework 3

Show all work. Submit your solutions in a pdf document on Canvas. Include your R code (which must be

commented and properly indented) in the pdf file. Also submit one text file with all your R code (comments

and all) clearly labeled with the problem it goes with. This must be properly indented. Before every solution

with random sampling use set.seed(3701).

Question 1 (15 points).

Let’s consider a slightly different data generating model for paired data. Suppose we want to compare

students scores of a standardized test before and after taking a prep course. There are n students that will be

taking the test. Let X be the yet-to-be observed random variable of the score before the course and Y be the

yet-to-be observed random variable of the score after the course. Moreover, we let X and Y be the linear

combinations of two common latent variable, i.e., the model for the ith student is Xi = µ1 + a1Ai + b1Bi

Yi = µ2 + a2Ai + b2Bi,

where µj , aj , bj , j = 1, 2 are constants. Ai

, Bi are independent both within the same subject and across

different subjects. Ai’s are iid ∼ N(0, σ2A), Bi

is are iid N(0, σ2B).

(a) (4 points) What are the distributions of Xi and Yi

, respectively? If we let Zi = Xi −Yi

, does this data

generating model still satisfy the assumptions for paired t-test? Why?

(b) (5 points) Let µ1 = 68, µ2 = 70, σ2

A = 2, σ2

B = 1, a1 = 2, b1 = 1, a2 = 1, b2 = 2. It can

be show that E[X] = 68, E[Y ] = 70. Use simulation to produce a 99% confidence interval for

Var(X) = E[(X−68)2

] and a 99% confidence interval for Var(Y ) = E[(Y −70)2

]. Set reps = 105

.

(c) (3 points) Now we are interested in paired t-test for

H0 : µ1 = µ2

Ha : µ1 ̸= µ2,

with data assumed to be coming from our data generating model. Create a function named mypaired.pval

that generate realizations of the random p-value of this test. Your function should have the following

arguments

• mu1, mean of X

• mu2, mean of Y

• a1,b1,a2,b2 as defined in our data generating model

• sigmaA, standard deviation of A

• sigmaAB, standard deviation of B

1

• n, sample size (number of pairs)

• reps, number of replications.

The function should output a vector containing the realizations of the random p-value.

(d) (4 points) Use µ1 = 68, µ2 = 68.5, σ2

A = 2, σ2

B = 1, a1 = 2, b1 = 1, a2 = 1, b2 = 2, α = 0.05. Set

reps = 1000. Test your function by creating a simulation based estimation of the power curve with

the sample size n ∈ {20, 25, . . . , 200}. Produce a pointwise 95% confidence interval for the power

curve with the Clopper-Peason CI and include the curves for the lower bound and upper bound of CI

in the same plot of the estimated power curve.

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