Suppose that we take a sample of size n1 from a normally

Chapter 8, Problem 128SE

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Suppose that we take a sample of size \(n_{1}\) from a normally distributed population with mean and variance \(\mu_{1}\) and \(\sigma_{1}^{2}\) and an independent of sample size \(n_{2}\) from a normally distributed population with mean and variance \(\mu_{2}\)and \(\sigma_{2}^{2}\). If it is reasonable to assume that \(\sigma_{1}^{2}=\sigma_{2}^{2}\), then the results given in Section 8.8 apply.

What can be done if we cannot assume that the unknown variances are equal but are fortunate enough to know that \(\sigma_{2}^{2}=k \sigma_{1}^{2}\) for some known constant \(k \neq 1\)? Suppose, as previously, that the sample means are given by \(\bar{Y}_{1}\) and \(\bar{Y}_{2}\)and the sample variances by \(S_{1}^{2}\) and \(S_{2}^{2}\), respectively.
a Show that \(Z^{\star}\) given below has a standard normal distribution.

$$Z^{\star}=\frac{\left(\bar{Y}_{1}-\bar{Y}_{2}\right)-\left(\mu_{1}-\mu_{2}\right)}{\sigma_{1} \sqrt{\frac{1}{n_{1}}+\frac{k}{n_{2}}}}$$

b Show that \(W^{\star}\) given below has a \(\chi^{2}\) distribution with \(n_{1}+n_{2}-2\) df.

$$W^{\star}=\frac{\left(n_{1}-1\right) S_{1}^{2}+\left(n_{2}-1\right) S_{2}^{2} / k}{\sigma_{1}^{2}}$$

c Notice that \(Z^{\star}\) and \(W^{\star}\) from parts (a) and (b) are independent. Finally, show that

                  \(T^{\star}=\frac{\left(\overline{Y_{1}}-\overline{Y_{2}}\right)-\left(\mu_{1}-\mu_{2}\right)}{S_{p}^{\star} \sqrt{\frac{1}{n_{1}}+\frac{k}{n_{2}}}}\), where \(S_{p}^{2 \star}=\frac{\left(n_{1}-1\right) S_{1}^{2}+\left(n_{2}-1\right) S_{2}^{2} / k}{n_{1}+n_{2}-2}\) has a \(t\) distribution with \(n_{1}+n_{2}-2\)df.

d Use the result in part (c) to give a \(100(1-\alpha) \%\) confidence interval for \(\mu_{1}-\mu_{2}\), assuming that \(\sigma_{2}^{2}=k \sigma_{1}^{2}\)

e What happens if \(k=1\) in parts (a)-(d)?

Equation Transcription:

𝝌2

Text Transcription:

n_1

mu_1

sigma_{1}^{2}

n_1

mu_2

sigma_{2}^{2}

sigma_{1}^{2} = sigma_{2}^{2}

sigma_{2}^{2}=k sigma_{1}^{2}

k neq 1

bar Y_1

bar Y_2

S_{1}^{2}

S_{2}^{2}

Z^star

Z^star = frac(bar Y_{1} - bar Y_ {2}) - (mu _1 - mu _2)}{sigma_1 sqrt{frac{1}{n_1} + frac{k}{n_2}

W^star

chi^2

n_1 +n_2 -2

W^star = frac(n_{1}-1) S_{1}^{2} + (n_{2}-1) S_{2}^{2} / k}{sigma_{1}^{2}}

Z^star

W^star

T^star = frac{(bar Y_{1} - bar Y_{2}}) - (mu_1 -mu_2)}{S_{p}^{star} sqrt {frac{1}{n_{1}} + {frac{k}{n_{2}}

S_{p}^{2star} = frac{(n_{1} -1) S_{1}^{2} + (n_{2}-1) S_{2}^{2}/k}{n_{1}+n_{2} - 2}

t

n_1 + n_{2}-2

100(1 - alpha) %

mu_1 - mu_2

sigma_{2}^{2} = k sigma_{1}^{2}

k = 1

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