Chapter 9: Problem 4
Show that the square of a noncentral \(T\) random variable is a noncentral \(F\) random variable.
Chapter 9: Problem 4
Show that the square of a noncentral \(T\) random variable is a noncentral \(F\) random variable.
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Get started for freeLet \(Q_{1}\) and \(Q_{2}\) be two nonnegative quadratic forms in the observations of a random sample from a distribution which is \(N\left(0, \sigma^{2}\right) .\) Show that another quadratic form \(Q\) is independent of \(Q_{1}+Q_{2}\) if and only if \(Q\) is independent of each of \(Q_{1}\) and \(Q_{2}\) Hint: \(\quad\) Consider the orthogonal transformation that diagonalizes the matrix of \(Q_{1}+Q_{2}\). After this transformation, what are the forms of the matrices \(Q, Q_{1}\) and \(Q_{2}\) if \(Q\) and \(Q_{1}+Q_{2}\) are independent?
Show that \(\sum_{j=1}^{b} \sum_{i=1}^{a}\left(X_{i j}-\bar{X}_{i .}\right)^{2}=\sum_{j=1}^{b} \sum_{i=1}^{a}\left(X_{i j}-\bar{X}_{i}-\bar{X}_{. j}+\bar{X}_{. .}\right)^{2}+a \sum_{j=1}^{b}\left(\bar{X}_{. j}-\bar{X}_{. .}\right)^{2} .\)
Fit \(y=a+x\) to the data $$ \begin{array}{l|lll} \mathrm{x} & 0 & 1 & 2 \\ \hline \mathrm{y} & 1 & 3 & 4 \end{array} $$ by the method of least squares.
Fit by the method of least squares the plane \(z=a+b x+c y\) to the five points \((x, y, z):(-1,-2,5),(0,-2,4),(0,0,4),(1,0,2),(2,1,0)\).
The driver of a diesel-powered automobile decided to test the quality of three types of diesel fuel sold in the area based on mpg. Test the null hypothesis that the three means are equal using the following data. Make the usual assumptions and take \(\alpha=0.05\). $$ \begin{array}{llllll} \text { Brand A: } & 38.7 & 39.2 & 40.1 & 38.9 & \\ \text { Brand B: } & 41.9 & 42.3 & 41.3 & & \\ \text { Brand C: } & 40.8 & 41.2 & 39.5 & 38.9 & 40.3 \end{array} $$
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