Let S21 denote the sample variance for a random sample of

Chapter 7, Problem 36E

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QUESTION:

Let \(S_{1}^{2}\) denote the sample variance for a random sample of ten ln(LC50) values for copper and

let  \(S_{2}^{2}\) denote the sample variance for a random sample of eight ln(LC50) values for lead, both

samples using the same species of fish. The population variance for measurements on copper is assumed to be twice the corresponding population variance for measurements on lead. Assume \(S_{1}^{2}\) to be independent of \(S_{2}^{2}\).

Find a number b such that

\(P\left(\frac{S_{1}^{2}}{s_{2}^{2}} \leq b\right)=.95\)

Find a number a such that

\(P\left(a \leq \frac{s_{1}^{2}}{s_{2}^{2}}\right)=.95\)

[Hint: Use the result of Exercise 7.29 and notice that \(\mathrm{P}\left(\mathrm{U}_{1} / \mathrm{U}_{2} \leq \mathrm{k}\right)=\mathrm{P}\left(\mathrm{U}_{2} / \mathrm{U}_{1} \geq 1 / \mathrm{k}\right)\).]

If a and b are as in parts (a) and (b), find

\(\mathrm{P}\left(a \leq \frac{s_{1}^{2}}{S_{2}^{2}} \leq b\right)=.95\)

Equation Transcription:

Text Transcription:

S_1^2

S_2^2

S_1^2

S_2^2

P(S_1^2 s_2^2 \leq =.95

P(a \leq s_1^2 s_2^2=.95

P(U1/U2 \leq k)=P(U2/U1\geq 1/k)

P(a \leq \s_1^2 S_2^2 \leq b =.95

Questions & Answers

QUESTION:

Let \(S_{1}^{2}\) denote the sample variance for a random sample of ten ln(LC50) values for copper and

let  \(S_{2}^{2}\) denote the sample variance for a random sample of eight ln(LC50) values for lead, both

samples using the same species of fish. The population variance for measurements on copper is assumed to be twice the corresponding population variance for measurements on lead. Assume \(S_{1}^{2}\) to be independent of \(S_{2}^{2}\).

Find a number b such that

\(P\left(\frac{S_{1}^{2}}{s_{2}^{2}} \leq b\right)=.95\)

Find a number a such that

\(P\left(a \leq \frac{s_{1}^{2}}{s_{2}^{2}}\right)=.95\)

[Hint: Use the result of Exercise 7.29 and notice that \(\mathrm{P}\left(\mathrm{U}_{1} / \mathrm{U}_{2} \leq \mathrm{k}\right)=\mathrm{P}\left(\mathrm{U}_{2} / \mathrm{U}_{1} \geq 1 / \mathrm{k}\right)\).]

If a and b are as in parts (a) and (b), find

\(\mathrm{P}\left(a \leq \frac{s_{1}^{2}}{S_{2}^{2}} \leq b\right)=.95\)

Equation Transcription:

Text Transcription:

S_1^2

S_2^2

S_1^2

S_2^2

P(S_1^2 s_2^2 \leq =.95

P(a \leq s_1^2 s_2^2=.95

P(U1/U2 \leq k)=P(U2/U1\geq 1/k)

P(a \leq \s_1^2 S_2^2 \leq b =.95

ANSWER:

Step 1 of 4

Given data

For n=10 samples of ln(LC50) values of copper

Sample variance is

For n=8 samples of ln(LC50) values of lead

Samples variance is

We are also given that

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