Solution Found!
Refer to Exercise 7.13. Suppose that n = 20 observations are to be taken on ln(LC50)
Chapter 7, Problem 7.21(choose chapter or problem)
Refer to Exercise 7.13. Suppose that n = 20 observations are to be taken on ln(LC50) measurements and that \(\sigma^{2}=1.4\). Let \(S^2\) denote the sample variance of the 20 measurements.
a. Find a number b such that \(P\left(S^{2} \leq b\right)=.975\).
b. Find a number a such that \(P\left(a \leq S^{2}\right)=.975\).
c. If a and b are as in parts (a) and (b), what is \(P\left(a \leq S^{2} \leq b\right)\)?
Questions & Answers
QUESTION:
Refer to Exercise 7.13. Suppose that n = 20 observations are to be taken on ln(LC50) measurements and that \(\sigma^{2}=1.4\). Let \(S^2\) denote the sample variance of the 20 measurements.
a. Find a number b such that \(P\left(S^{2} \leq b\right)=.975\).
b. Find a number a such that \(P\left(a \leq S^{2}\right)=.975\).
c. If a and b are as in parts (a) and (b), what is \(P\left(a \leq S^{2} \leq b\right)\)?
ANSWER:Step 1 of 3
a)
Given that
\(\begin{array}{c} \sigma^{2}=1.4 \\ n=20 \\ P\left(S^{2} \leq b\right)=0.975 \end{array}\)
Multiply all sides of the inequality by \(\frac{(n-1)}{\sigma^{2}}\)
\(P\left(\frac{(n-1)}{\sigma^{2}} S^{2} \leq \frac{(n-1)}{\sigma^{2}} b\right)=0.975\)
Fill in the known values
\(P\left(\frac{(n-1)}{\sigma^{2}} S^{2} \leq \frac{(20-1)}{1.4} b\right)=0.975\)
Determine the critical value using table 6 with \(d f=n-1=20-1=19\)
\(P\left(\frac{(n-1)}{\sigma^{2}} S^{2} \leq 32.8523\right)=0.975\)
The two boundaries have to be the same
\(\frac{(20-1)}{1.4} b=32.8523\)
Solve to b
\(b=\frac{32.8523(1.4)}{20-1}\approx2.42\)