Question

Consider the following linear combinations of interest in a single-factor study involving four factor levels: (i) $\quad \mu_{1}+3 \mu_{2}-4 \mu_{3}$ (ii) $\quad .3 \mu_{1}+.5 \mu_{2}+.1 \mu_{3}+.1 \mu_{4}$ (iii) $\frac{\mu_{1}+\mu_{2}+\mu_{3}}{3}-\mu_{4}$ a. Which of the linear combinations are contrasts? State the coefficients for each of the contrasts. b. Give an unbiased estimator for each of the linear combinations, Also give the estimated variance of each estimator assuming that $n_{i}=n$

          Consider the following linear combinations of interest in a single-factor study involving four factor levels:
(i) $\quad \mu_{1}+3 \mu_{2}-4 \mu_{3}$
(ii) $\quad .3 \mu_{1}+.5 \mu_{2}+.1 \mu_{3}+.1 \mu_{4}$
(iii) $\frac{\mu_{1}+\mu_{2}+\mu_{3}}{3}-\mu_{4}$
a. Which of the linear combinations are contrasts? State the coefficients for each of the contrasts.
b. Give an unbiased estimator for each of the linear combinations, Also give the estimated variance of each estimator assuming that $n_{i}=n$

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Added by Norma R.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

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Solved by Expert Kirsty Gledhill

01/30/2024

Step 1

A linear combination is a contrast if the sum of its coefficients is equal to zero. Let's check each linear combination: (i) $\mu_{1}+3 \mu_{2}-4 \mu_{3}$: The coefficients are 1, 3, and -4. Their sum is 1 + 3 - 4 = 0, so this is a contrast. (ii) $.3 \mu_{1}+.5 Show more…

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Consider the following linear combinations of interest in a single-factor study involving four factor levels: (i) $\quad \mu_{1}+3 \mu_{2}-4 \mu_{3}$ (ii) $\quad .3 \mu_{1}+.5 \mu_{2}+.1 \mu_{3}+.1 \mu_{4}$ (iii) $\frac{\mu_{1}+\mu_{2}+\mu_{3}}{3}-\mu_{4}$ a. Which of the linear combinations are contrasts? State the coefficients for each of the contrasts. b. Give an unbiased estimator for each of the linear combinations, Also give the estimated variance of each estimator assuming that $n_{i}=n$

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