Question

3. Prove that if X ~ t(n), then X^2 ~ F(1, n) 4. Prove that if X ~ F(m, n), then 1/X ~ F(n, m) 5. Let X1, ... , Xm be iid N(?1, ?^2) and Y1, ... , Yn be iid N(?2, ?^2) where Xi and Yj are independent for any i and j. When ??^2 = (?_{i=1}^m (Xi - X?)^2 + ?_{j=1}^n (Yi - Y?)^2) / (n + m - 2), show (i) that (n + m - 2) * (??^2 / ?^2) ~ ?^2(m + n - 2), (ii) that E(??^2) = ?^2, (iii) and that ((X? - Y?) - (?1 - ?2)) / (?? * ?(1/m + 1/n)) ~ t(m + n - 2).

          3. Prove that if X ~ t(n), then X^2 ~ F(1, n)

4. Prove that if X ~ F(m, n), then 1/X ~ F(n, m)

5. Let X1, ... , Xm be iid N(?1, ?^2) and Y1, ... , Yn be iid N(?2, ?^2) where Xi and Yj are independent for any i and j. When

??^2 = (?_{i=1}^m (Xi - X?)^2 + ?_{j=1}^n (Yi - Y?)^2) / (n + m - 2),

show (i) that

(n + m - 2) * (??^2 / ?^2) ~ ?^2(m + n - 2),

(ii) that

E(??^2) = ?^2,

(iii) and that

((X? - Y?) - (?1 - ?2)) / (?? * ?(1/m + 1/n)) ~ t(m + n - 2).

3. Prove that if X t(n), then X^2 F(1, n)

4. Prove that if X F(m, n), then 1/X F(n, m)

5. Let X1, ... , Xm be iid N(?1, ?^2) and Y1, ... , Yn be iid N(?2, ?^2) where Xi and Yj are independent for any i and j. When

??^2 = (?i=1^m (Xi - X?)^2 + ?j=1^n (Yi - Y?)^2) / (n + m - 2),

show (i) that

(n + m - 2) * (??^2 / ?^2) ?^2(m + n - 2),

(ii) that

E(??^2) = ?^2,

(iii) and that

((X? - Y?) - (?1 - ?2)) / (?? * ?(1/m + 1/n)) t(m + n - 2).

Added by Magdalena O.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Dominador Tan

09/14/2023

Step 1

Prove that if X ~ t(n), then x² ~ F(1, n) We know that if X ~ t(n), then X = Z / sqrt(W/n), where Z ~ N(0, 1) and W ~ χ²(n) are independent. Now, let's square X: x² = (Z / sqrt(W/n))² = Z² / (W/n) We know that Z² ~ χ²(1) since Z ~ N(0, 1). Therefore, x² has the Show more…

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3. Prove that if X ~ t(n), then X^2 ~ F(1, n) 4. Prove that if X ~ F(m, n), then 1/X ~ F(n, m) 5. Let X1, ... , Xm be iid N(μ1, σ^2) and Y1, ... , Yn be iid N(μ2, σ^2) where Xi and Yj are independent for any i and j. When σ̂^2 = (∑_{i=1}^m (Xi - X̄)^2 + ∑_{j=1}^n (Yi - Ȳ)^2) / (n + m - 2), show (i) that (n + m - 2) * (σ̂^2 / σ^2) ~ χ^2(m + n - 2), (ii) that E(σ̂^2) = σ^2, (iii) and that ((X̄ - Ȳ) - (μ1 - μ2)) / (σ̂ * √(1/m + 1/n)) ~ t(m + n - 2).