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.
All the tools & learning materials you need for study success - in one app.
Get started for freeGiven the following observations associated with a two-way classification with \(a=3\) and \(b=4\), compute the \(F\) -statistic used to test the equality of the column means \(\left(\beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=0\right)\) and the equality of the row means \(\left(\alpha_{1}=\alpha_{2}=\alpha_{3}=0\right)\), respectively. $$ \begin{array}{ccccc} \hline \text { Row/Column } & 1 & 2 & 3 & 4 \\ \hline 1 & 3.1 & 4.2 & 2.7 & 4.9 \\ 2 & 2.7 & 2.9 & 1.8 & 3.0 \\ 3 & 4.0 & 4.6 & 3.0 & 3.9 \\ \hline \end{array} $$
Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote a random sample of size \(n\) from a distribution which is \(N\left(0, \sigma^{2}\right)\). Prove that \(\sum_{1}^{n} X_{i}^{2}\) and every quadratic form, which is nonidentically zero in \(X_{1}, X_{2}, \ldots, X_{n}\), are dependent.
Fit by the method of least squares the plane \(z=a+b x+c y\) to the five points \((x, y, z):(-1,-2,5),(0,-2,4),(0,0,4),(1,0,2),(2,1,0)\).
Let \(\mu_{1}, \mu_{2}, \mu_{3}\) be, respectively, the means of three normal distributions with a common but unknown variance \(\sigma^{2}\). In order to test, at the \(\alpha=5\) percent significance level, the hypothesis \(H_{0}: \mu_{1}=\mu_{2}=\mu_{3}\) against all possible alternative hypotheses, we take an independent random sample of size 4 from each of these distributions. Determine whether we accept or reject \(H_{0}\) if the observed values from these three distributions are, respectively, $$ \begin{array}{lrrrr} X_{1}: & 5 & 9 & 6 & 8 \\ X_{2}: & 11 & 13 & 10 & 12 \\ X_{3}: & 10 & 6 & 9 & 9 \end{array} $$
Let \(A\) be the real symmetric matrix of a quadratic form \(Q\) in the observations of a random sample of size \(n\) from a distribution which is \(N\left(0, \sigma^{2}\right)\). Given that \(Q\) and the mean \(\bar{X}\) of the sample are independent, what can be said of the elements of each row (column) of \(\boldsymbol{A}\) ? Hint: Are \(Q\) and \(X^{2}\) independent?
What do you think about this solution?
We value your feedback to improve our textbook solutions.