Refer to the matched-pairs experiment of Section 12.3 and

Chapter 12, Problem 31SE

(choose chapter or problem)

Refer to the matched-pairs experiment of Section 12.3 and assume that the measurement receiving treatment i, where \(i=1,2\), in the j th pair, where \(j=1,2, \ldots, n\), is

\(Y_{i j}=\mu_{i}+P_{j}+\varepsilon_{i j}\)

where  \(\mu_{i}\) =  expected response for treatment i, for \(i=1,2\),

        \(P_{j}\) = additive random effect (positive or negative) contribution by the j th pair of experimental units, for \(j=1,2, \ldots, n\),

        \(\varepsilon_{i j}\) = random error associated with the experimental unit in the j th pair that receives treatment i.

Assume that the \(\varepsilon_{i j}\)'s are independent normal random variables with \(E\left(\varepsilon_{i j}\right)=0, V\left(\varepsilon_{i j}\right)=\sigma^{2}\); and assume that the \(P_{j}\)'s are independent normal random variables with \(E\left(P_{j}\right)=0, V\left(P_{j}\right)=\sigma_{p}^{2}\). Also, assume that the \(P_{j}\)'s and \(\varepsilon_{i j}\)'s are independent.

a. Find \(E\left(Y_{i j}\right)\).

b. Find \(\left(\bar{Y}_{i}\right)\) and \(V\left(\bar{Y}_{i}\right)\), where \(\bar{Y}_{i}\) is the mean of the n observations receiving treatment i, where \(i=1,2\).

c. Let \(\bar{D}=\bar{Y}_{1}-\bar{Y}_{2}\). Find \(E(\bar{D}), V(\bar{D})\), and the probability distribution for \(\bar{D}\).

Equation Transcription:

 

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Text Transcription:

i = 1,2

j = 1,2, ... ,n

Y_ij = mu_i + P_j + epsilon_ij

mu_1

P_j

epsilon_ij

E(epsilon_ij) = 0, V(epsilon_ij) = sigma^2

E(P_j) = 0, V(P_j)  = sigma_p^2

E(Y_ij)

(Y bar_i)

D bar = Y bar_1 - Y bar_2

E(D bar), B(D bar)

D bar