Suppose that X1, X2, …, Xn1, Y1, Y2, …, Yn1, and W1, W2, …, Wn1 are independent random samples f

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Suppose that X1, X2, …, Xn1, Y1, Y2, …, Yn1, and W1, W2, …,...

Suppose that $X_{1}, X_{2}, \ldots, X_{n_{1}}, Y_{1}, Y_{2}, \ldots, Y_{n_{1}},$ and $W_{1}, W_{2}, \ldots, W_{n_{1}}$ are independent random samples from normal distributions with respective unknown means $\mu_{1}, \mu_{2},$ and $\mu_{3}$ and common variances $\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma_{3}^{2}=\sigma^{2} .$ Suppose that we want to estimate a linear function of the means: a. What is the standard error of the estimator $\hat{\theta}$ ? b. What is the distribution of the estimator $\hat{\theta}$ ? c. If the sample variances are given by $S_{1}^{2}, S_{2}^{2},$ and $S_{3}^{2},$ respectively, consider $$S_{p}^{2}=\frac{\left(n_{1}-1\right) S_{1}^{2}+\left(n_{2}-1\right) S_{2}^{2}+\left(n_{3}-1\right) S_{3}^{2}}{n_{1}+n_{2}+n_{3}-3}$$ i. What is the distribution of $\left(n_{1}+n_{2}+n_{3}-3\right) S_{p}^{2} / \sigma^{2} ?$ ii. What is the distribution of $$T=\frac{\hat{\theta}-\theta}{S_{p} \sqrt{\frac{a_{1}^{2}}{n_{1}}+\frac{a_{2}^{2}}{n_{2}}+\frac{a_{3}^{2}}{n_{3}}} ?} ?$$ d. Give a confidence interval for $\theta$ with confidence coefficient $1-\alpha$ e. Develop a test for $H_{0}: \theta=\theta_{0}$ versus $H_{a}: \theta \neq \theta_{0}$ $\theta=a_{1} \mu_{1}+a_{2} \mu_{2}+a_{3} \mu_{3}$. Because the maximum-likelihood estimator (MLE) of a function of parameters is the function of the MLEs of the parameters, the MLE of $\theta$ is $\hat{\theta}=a_{1} X+a_{2} Y+a_{3} W$

Suppose that $X_{1}, X_{2}, \ldots, X_{n_{1}}, Y_{1}, Y_{2}, \ldots, Y_{n_{1}},$ and $W_{1}, W_{2}, \ldots, W_{n_{1}}$ are independent random samples from normal distributions with respective unknown means $\mu_{1}, \mu_{2},$ and $\mu_{3}$ and common variances $\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma_{3}^{2}=\sigma^{2} .$ Suppose that we want to estimate a linear function of the means:
a. What is the standard error of the estimator $\hat{\theta}$ ?
b. What is the distribution of the estimator $\hat{\theta}$ ?
c. If the sample variances are given by $S_{1}^{2}, S_{2}^{2},$ and $S_{3}^{2},$ respectively, consider
$$S_{p}^{2}=\frac{\left(n_{1}-1\right) S_{1}^{2}+\left(n_{2}-1\right) S_{2}^{2}+\left(n_{3}-1\right) S_{3}^{2}}{n_{1}+n_{2}+n_{3}-3}$$
i. What is the distribution of $\left(n_{1}+n_{2}+n_{3}-3\right) S_{p}^{2} / \sigma^{2} ?$
ii. What is the distribution of
$$T=\frac{\hat{\theta}-\theta}{S_{p} \sqrt{\frac{a_{1}^{2}}{n_{1}}+\frac{a_{2}^{2}}{n_{2}}+\frac{a_{3}^{2}}{n_{3}}} ?} ?$$
d. Give a confidence interval for $\theta$ with confidence coefficient $1-\alpha$
e. Develop a test for $H_{0}: \theta=\theta_{0}$ versus $H_{a}: \theta \neq \theta_{0}$
$\theta=a_{1} \mu_{1}+a_{2} \mu_{2}+a_{3} \mu_{3}$. Because the maximum-likelihood estimator (MLE) of a function of parameters is the function of the MLEs of the parameters, the MLE of $\theta$ is $\hat{\theta}=a_{1} X+a_{2} Y+a_{3} W$

Key Concepts

Propagation of Variance

This concept deals with how the variance of a linear combination of independent random variables can be computed from the variances of the individual variables. When forming an estimator that is a linear combination of sample means, the overall variance is the sum of the variances of the individual terms weighted by the squares of the coefficients. This is fundamental in determining the standard error of the estimator, which quantifies its variability.

Sampling Distribution of Linear Combinations

When independent normal random variables are combined linearly, the resulting estimator also follows a normal distribution. This concept is essential for constructing the distribution of the estimator, where the mean is the linear combination of the true means and the variance is derived from the individual sample variances scaled appropriately. This property underlies many inferential procedures in the normal model.

Pooled Variance Estimator

In situations where several independent samples are assumed to have a common unknown variance, the pooled variance estimator is used to combine data from all samples. It is a weighted average of the individual sample variances, with weights proportional to their degrees of freedom. The pooled variance follows a scaled chi-square distribution, which is crucial for further inference procedures involving the common variance.

t Distribution

When the common variance is unknown and estimated by a pooled variance estimator, subsequent inference about the mean involves a t statistic rather than a z statistic. The t distribution arises because the standardization process includes an estimated standard error, and it has heavier tails compared to the normal distribution. This is particularly important for constructing confidence intervals and performing hypothesis tests in small samples.

Confidence Intervals

A confidence interval provides a range of plausible values for an unknown parameter, constructed using the sampling distribution of the estimator. For a linear combination of means with an unknown common variance, the interval is typically based on the t distribution and incorporates the standard error computed from the pooled variance estimator. This methodology ensures that the interval has the desired coverage probability.

Hypothesis Testing in Normal Models

Hypothesis testing in the context of linear functions of means from normal distributions involves formulating a null hypothesis about the parameter and then testing it using a t statistic derived from the estimator and its associated standard error. The test compares the observed t value to critical values from the t distribution to determine whether to reject the null hypothesis. This framework is central to making decisions about parameter values based on sample data.

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