Let X1,X2, . . . ,Xn be a random sample from the normal

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 \(\sum_{i=1}^{10} X_{i}\). (b) What is a uniformly most powerful critical region of size \(\alpha=0.068\) ? Recall that \(\sum_{i=1}^{10} X_{i}\) has a Poisson distribution with mean \(10 \mu\). (c) Sketch the power function of this test. Equation Transcription: Text Transcription: X_1,X_2,,X_10 H_0:=0.5 H_1:>0.5 sum_i=1^10 X_i alpha =0.068 sum_i=1^10 X_i 10 mu

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.

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 \(H_{0}: \theta=3\) against \(H_{1}: \theta=5\) can be based on the statistic \(\sum_{i=1}^{n} X_{i}\) (b) If \(n=12\), use the fact that \((2 / \theta) \sum_{i=1}^{12} X_{i}\) is \(\chi^{2}(24)\) to find a best critical region of size \(\alpha=0.10\). (c) If \(n=12\), find a best critical region of size \(\alpha=0.10\) for testing \(H_{0}: \theta=3\) against \(H_{1}: \theta=7\). (d) If \(H_{1}: \theta>3\), is the common region found in parts (b) and (c) a uniformly most powerful critical region of size \(\alpha=0.10\)? Equation Transcription: , . Text Transcription: X Theta; f(x;theta)=(1/theta)e^-x/theta, 0<x< infinity X_1,X_2,…,X_n H_0:=3 H_1:=5 sun_i=1^n X_i n=12 (2/theta) sum_i=1^12 X_i chi^2(24) alpha=0.10 . H_1:theta=7 . H_1:theta >3 alpha=0.10

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 \(n=15\), find the value of \(c\) so that \(\alpha=0.05\). Hint: Recall that \(\sum_{i=1}^{n} X_{i}^{2} / \sigma^{2}\) is \(\chi^{2}(n)\). (c) If \(n=15\) and \(c\) is the value found in part (b), find the approximate value of \(\beta=P\left(\sum_{i=1}^{n} X_{i}^{2}<c\right.\); \(\left.\sigma^{2}=16\right)\). Equation Transcription: ; Text Transcription: X_1,X_2,…,X_n N(0, sigma^2) C={(x_1,x_2,…,x_n): sum_i=1^n x_i^2 > or = c} H_0:sigma^2=4 H_1: sigma^2=16 n=15 c alpha=0.05 sum_i=1^n X_i^2/sigma^2 chi^2(n) beta=P(sum_i=1^n X_i^2<c; sigma^2=16)

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 \(C=\left\{\left(x_{1}, x_{2}, \ldots, x_{n}\right): \sum_{i=1}^{n} x_{i} \leq n(0.85)\right\}\) and \(Y=\sum_{i=1}^{n} X_{i}\), find the value of \(n\) such that \(\alpha=\) \(P[Y \leq n(0.85) ; p=0.9] \approx 0.10\). Hint: Use the normal approximation for the binomial distribution. (c) What is the approximate value of \(\beta=P[Y>\) \(n(0.85) ; p=0.8]\) for the test given in part (b)? (d) Is the test of part (b) a uniformly most powerful test when the alternative hypothesis is \(H_{1}: p<0.9\) ? Equation Transcription: . Text Transcription: X_1,X_2,,X_n b(1,p) . H_0:p=0.9 H_1:p=0.8 Y=sum_i=1^n X_i b(n,p) C={(x_1,x_2,,x_n):sum_i=1^n x_i < or = n(0.85)} n alpha=P[Y < or = n(0.85);p=0.9] approx 0.10. beta=P[Y> n(0.85);p=0.8] H_1:p<0.9

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 Example 8.6-4, argue that a uniformly most powerful test for testing \(H_{0}: \mu=50\) against \(H_{1}: \mu \neq 50\) does not exist. Equation Transcription: Text Transcription: X_1,X_2,…,X_n N(mu,36) H_0:=50 H_1:<50 C_2={ bar x: bar x < or = c} H_0:mu =50 H_1:mu not = 50

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: \(C_{1}=\left\{\bar{x}: \bar{x} \geq c_{1}\right\}, C_{2}=\left\{\bar{x}: \bar{x} \leq c_{2}\right\}\), and \(C_{3}=\left\{\bar{x}:|\bar{x}-80| \geq c_{3}\right\}\). (a) If \(n=16\), find the values of \(c_{1}, c_{2}, c_{3}\) such that the size of each critical region is \(0.05\). That is, find \(c_{1}, c_{2}\), \(c_{3}\) such that \(\begin{aligned}0.05 &=P\left(\bar{X} \in C_{1} ; \mu=80\right)=P\left(\bar{X} \in C_{2} ; \mu=80\right) \\&=P\left(\bar{X} \in C_{3} ; \mu=80\right)\end{aligned}\) (b) On the same graph paper, sketch the power functions for these three critical regions. Equation Transcription: Text Transcription: X_1,X_2,…,X_n N(mu,9). H_0: mu=80 H_1:munot = 80 C_1={bar x: bar x > or = c_1,C_2={bar x: bar x < or = c_2} C_3={ bar x:| bar x -80| > or = c_3}. n=16 c_1,c_2,c_3 0.05. 0.05 =P( bar X in C_1;mu=80)=P(bar X in C_2; mu=80) =P( bar X in C_3;=80)

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.

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 the value found in part (b), find the approximate value of = P( n i = 1 X2 i < c; 2 = 16).

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 = 12, use the fact that (2/) 12 i = 1 Xi is 2(24) to find a best critical region of size = 0.10. (c) If n = 12, find a best critical region of size = 0.10 for testing H0: = 3 against H1: = 7. (d) If H1: > 3, is the common region found in parts (b) and (c) a uniformly most powerful critical region of size = 0.10?

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 value of n such that = P[ Y n(0.85); p = 0.9 ] 0.10. Hint: Use the normal approximation for the binomial distribution. (c) What is the approximate value of = P[ Y > n(0.85); p = 0.8 ] for the test given in part (b)?(d) Is the test of part (b) a uniformly most powerful test when the alternative hypothesis is H1: p < 0.9?

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 testing H0: = 50 against H1: = 50 does not exist.

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 size of each critical region is 0.05. That is, find c1, c2, c3 such that 0.05 = P(X C1; = 80) = P(X C2; = 80) = P(X C3; = 80). (b) On the same graph paper, sketch the power functions for these three critical regions.

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 \(\sum_{i=1}^{10}X_i\). (b) What is a uniformly most powerful critical region of size \(\alpha = 0.068\)? Recall that \(\sum^{10}_{ i = 1} X_i\) has a Poisson distribution with mean \(10\mu\). (c) Sketch the power function of this test.

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.

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.