Ch 7.3 - 37E

Chapter 7, Problem 37E

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QUESTION:

\(\bar{X}_{1}\) and \(S_{1}^{2}\) are the sample mean and sample variance from a population with mean \(\mu_{1}\) and variance \(\sigma_{1}^{2}\). Similarly, \(\bar{X}_{2}\) and \(S_{2}^{2}\) are the sample mean and sample variance from a second independent population with mean \(\mu_{2}\) and variance \(\sigma_{2}^{2}\). The sample sizes are \(n_{1}\) and \(n_{2}\), respectively.

(a) Show that \(\bar{X}_{1}-\bar{X}_{2}\) is an unbiased estimator of \(\mu_{1}-\mu_{2}\).

(b) Find the standard error of \(\bar{X}_{1}-\bar{X}_{2}\). How could you estimate the standard error?

(c) Suppose that both populations have the same variance; that is \(\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma^{2}\). Show that

\(S_{p}^{2}=\frac{\left(n_{1}-1\right) S_{1}^{2}+\left(n_{2}-1\right) S_{2}^{2}}{n_{1}+n_{2}-2}\)

is an unbiased estimator of \(\sigma^{2}\).

Questions & Answers

QUESTION:

\(\bar{X}_{1}\) and \(S_{1}^{2}\) are the sample mean and sample variance from a population with mean \(\mu_{1}\) and variance \(\sigma_{1}^{2}\). Similarly, \(\bar{X}_{2}\) and \(S_{2}^{2}\) are the sample mean and sample variance from a second independent population with mean \(\mu_{2}\) and variance \(\sigma_{2}^{2}\). The sample sizes are \(n_{1}\) and \(n_{2}\), respectively.

(a) Show that \(\bar{X}_{1}-\bar{X}_{2}\) is an unbiased estimator of \(\mu_{1}-\mu_{2}\).

(b) Find the standard error of \(\bar{X}_{1}-\bar{X}_{2}\). How could you estimate the standard error?

(c) Suppose that both populations have the same variance; that is \(\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma^{2}\). Show that

\(S_{p}^{2}=\frac{\left(n_{1}-1\right) S_{1}^{2}+\left(n_{2}-1\right) S_{2}^{2}}{n_{1}+n_{2}-2}\)

is an unbiased estimator of \(\sigma^{2}\).

ANSWER:

Step 1 of 4

Given data:

\({\overline X _1}\)  and \(S_1^2\) are the sample mean and variance from a population with mean \({\mu _1}\)  and variance \(\sigma _1^2\). Similarly, \({\overline X _2}\) and \(S_2^2\) are the sample mean and sample variance from a second independent population with mean \({\mu _2}\) and variance \(\sigma _2^2\). The sample sizes are \({n_1}\) and \({n_2}\), respectively.

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