Probability and Statistical Inference 9th Edition solutions

Probability and Statistical Inference 9
PROBLEM 1E Let X1,X2, . . . ,Xn be a random sample from a normal distribution N(?, 64). (a) Show that C = {(x1, x2, . . . , xn) : x ? c} is a best critical region for testing H0: ? = 80 against H1: ? = 76. (b) Find n and c so that ? ? 0.05 and ? ? 0.05.

Probability and Statistical Inference 9
Let \(X_{1}, X_{2}, \ldots, X_{n}\)be a random sample from \(N(0, \sigma^{2}\). a) Show that \(C=\left\{\left(x_{1}, x_{2}, \ldots, x_{n}\right): \sum_{i=1}^{n} x_{i}^{2} \geq c\right\}\) is a best critical region for testing \(H_{0}: \sigma^{2}=4\) against \(H_{1}: \sigma^{2}=16\) . (b) If \(n

Probability and Statistical Inference 9
Let \(X\) have an exponential distribution with a mean of \(\theta\); that is, the pdf of \(X\) is \(f(x ; \theta)=(1 / \theta) e^{-x / \theta}\), \(0<x<\infty\) . Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from this distribution. (a) Show that a best critical region for testing \(H_{0}

Probability and Statistical Inference 9
Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample of Bernoulli trials \((b(1, p)\). (a) Show that a best critical region for testing \(H_{0}: p=0.9\) against \(H_{1}: p=0.8\) can be based on the statistic \(Y=\sum_{i=1}^{n} X_{i}\), which is \(b(n, p)\). (b) If \(C=\left\{\left(x_{1}, x_{2},

Probability and Statistical Inference 9
Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from the normal distribution \(N(\mu, 36)\). (a) Show that a uniformly most powerful critical region for testing \(H_{0}: \mu=50\) against \(H_{1}: \mu<50\) is given by \(C_{2}=\{\bar{x}: \bar{x} \leq c\}\). (b) With this result and that of Exa

Probability and Statistical Inference 9
Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from the normal distribution \(N(\mu, 9)\). To test the hypothesis \(H_{0}:\) \(\mu=80\) against \(H_{1}: \mu \neq 80\), consider the following three critical regions: \(C_{1}=\left\{\bar{x}: \bar{x} \geq c_{1}\right\}, C_{2}=\left\{\bar{x}: \bar

Probability and Statistical Inference 9
Let\(X_{1}, X_{2}, \ldots, X_{10}\) be a random sample of size 10 from a Poisson distribution with mean \(\mu\). (a) Show that a uniformly most powerful critical region for testing \(H_{0}: \mu=0.5\) against \(H_{1}: \mu>0.5\) can be defined with the use of the statistic \(\sum_{i=1}^{10} X_{i}\).

Probability and Statistical Inference 9
Let X1, X2, ... , Xn be a random sample from a normal distribution N(, 64). (a) Show that C = {(x1, x2, ... , xn) : x c} is a best critical region for testing H0: = 80 against H1: = 76. (b) Find n and c so that 0.05 and 0.05.

Probability and Statistical Inference 9
Let X1, X2, ... , Xn be a random sample from N(0, 2). (a) Show that C = {(x1, x2, ... , xn) : n i = 1 x2 i c} is a best critical region for testing H0: 2 = 4 against H1: 2 = 16. (b) If n = 15, find the value of c so that = 0.05. Hint: Recall that n i = 1 X2 i /2 is 2(n). (c) If n = 15 and c is th

Probability and Statistical Inference 9
Let X have an exponential distribution with a mean of ; that is, the pdf of X is f(x; ) = (1/)ex/ , 0 < x < . Let X1, X2, ... , Xn be a random sample from this distribution. (a) Show that a best critical region for testing H0: = 3  against H1: = 5 can be based on the statistic n i = 1 Xi. (b) If n

Probability and Statistical Inference 9
Let X1, X2, ... , Xn be a random sample of Bernoulli trials b(1, p). (a) Show that a best critical region for testing H0: p = 0.9 against H1: p = 0.8 can be based on the statistic Y = n i = 1 Xi, which is b(n, p). (b) If C = {(x1, x2, ... , xn) : n i = 1 xi n(0.85)} and Y = n i = 1 Xi, find the v

Probability and Statistical Inference 9
Let X1, X2, ... , Xn be a random sample from the normal distribution N(, 36). (a) Show that a uniformly most powerful critical region for testing H0: = 50 against H1: < 50 is given by C2 = {x: x c}. (b) With this result and that of Example 8.6-4, argue that a uniformly most powerful test for testi

Probability and Statistical Inference 9
Let X1, X2, ... , Xn be a random sample from the normal distribution N(, 9). To test the hypothesis H0: = 80 against H1: = 80, consider the following three critical regions: C1 = {x : x c1}, C2 = {x : x c2}, and C3 = {x: |x 80| c3}. (a) If n = 16, find the values of c1, c2, c3 such that the si

Probability and Statistical Inference 9
Let \(X_1, X_2, \ldots , X_{10}\) be a random sample of size 10 from a Poisson distribution with mean \(\mu\). (a) Show that a uniformly most powerful critical region for testing \(H_0: \mu = 0.5\) against \(H_1: \mu > 0.5\) can be defined with the use of the statistic \(\sum_{i=1}^{10}X_i\).

Probability and Statistical Inference 9
Consider a random sample X1, X2, ... , Xn from a distribution with pdf f(x; ) = (1 x)1, 0 < x < 1, where 0 < . Find the form of the uniformly most powerful test of H0: = 1 against H1: > 1.

Probability and Statistical Inference 9
Let X1, X2, ... , X5 be a random sample from the Bernoulli distribution p(x; ) = x(1 )1x. We reject H0: = 1/2 and accept H1: < 1/2 if Y = 5 i = 1 Xi c. Show that this is a uniformly most powerful test and find the power function K() if c = 1.