Problem
Let $Y_1 \sim N(μ_1,1),Y_2 \sim N(μ_2,1), Y_3 \sim N(μ_3,1)$ and also assume that these three random variables are mutually independent. The observed sample values are $y_1 = 2, y_2 = 0 \ \mathrm{and} \ y_3 = −1$, respectively. We are interested in doing inference on the parameter $$\theta = 0.1\mu_1 + 0.2\mu_2 + 0.7\mu_3,$$ where $\theta$ is a weighted average of the population means. Now, suppose that $μ_2 = 0$ and $μ_3 = 0$ are known. Derive an exact upper range $90\%$ confidence interval for $\theta$ taking this additional information into account.
My answers
Method 1
Let $$\overline{Y} = 0.1Y_1 + 0.2Y_2 + 0.7Y_3.$$ We have $$\begin{aligned} Var(\overline{Y}) & = 0.1^2 Var(Y_1) + 0.2^2 Var(Y_2) + 0.7^2 Var(Y_3) \\ & = \frac{27}{50} \\ \end{aligned}$$ and $$\begin{aligned} \mathbb{E}(\overline{Y}) & = 0.1 \mathbb{E} (Y_1) + 0.2 \mathbb{E} (Y_2) + 0.7 \mathbb{E} (Y_3)\\ & = 0.1 \mu_1 + 0.2 \mu_2 + 0.7 \mu_3 \\ & = \theta \\ \implies \overline{Y} & \sim N \left( \theta , \frac{27}{50} \right) \end{aligned}$$ Thus, the exact upper range $90\%$ confidence interval for $\theta$ is $$\left( 0.1 \mu_1 + 0.2 \mu_2 + 0.7 \mu_3 - Z \frac{\sigma}{\sqrt{n}} , \infty \right),$$ but since we know $\mu_2 = 0 = \mu_3$, and from the sample, an estimate of $\mu_1 = 2$, we have $$\left(0.2 - 1.282 \sqrt{\frac{27}{50}}, \infty \right)$$
Method 2
Since $\mu_2 = 0 = \mu_3$, so $\theta$ reduces to $0.1 \mu_1$.
Redefine $$\overline{Y} = 0.1Y_1.$$
We have $$\begin{aligned} Var(\overline{Y}) & = 0.1^2 Var(Y_1) \\ & = 0.01 \\ \end{aligned}$$ and $$\begin{aligned} \mathbb{E}(\overline{Y}) & = 0.1 \mathbb{E} (Y_1) \\ & = 0.1 \mu_1 \\ & = \theta \\ \implies \overline{Y} & \sim N (\theta , 0.01) \end{aligned}$$ Thus, the exact upper range $90\%$ confidence interval for $\theta$ is $$\left( 0.1 \mu_1 - Z \frac{\sigma}{\sqrt{n}}, \infty \right)$$ and, since we know, from the sample, an estimate of $\mu_1 = 2$, we have $$(0.2 - 1.282 \sqrt{0.01}, \infty)$$
As I have just covered confidence intervals, I am uncertain whether method $1$ or method $2$ is correct. In particular, the standard error I obtained from each method is different. Any intuitive explanations will be greatly appreciated!