In Exercise 8.5-6, let be the critical region. Find values

A certain size of bag is designed to hold 25 pounds of potatoes. A farmer fills such bags in the field. Assume that the weight \(X\) of potatoes in a bag is \(N(\mu, 9)\). We shall test the null hypothesis \(H_{0}: \mu=25\) against the alternative hypothesis \(H_{1}: \mu<25\) . Let \(X_{1}, X_{2}, X_{3}, X_{4}\) be a random sample of size 4 from this distribution, and let the critical region \(C\) for this test be defined by \(\bar{x} \leq 22.5\), where \(\bar{x}\) is the observed value of \(\bar{X}\). (a) What is the power function \(K(\mu)\) of this test? In particular, what is the significance level \(\alpha=K(25)\) for your test? (b) If the random sample of four bags of potatoes yielded the values \(x_{1}=21.24, x_{2}=24.81, x_{3}=23.62\), and \(x_{4}=26.82\), would your test lead you to accept or reject \(H_{0}\) ? (c) What is the \(p\)-value associated with \(\bar{x}\) in part (b)? Equation Transcription: Text Transcription: X N(mu,9) H_0:=25 H_1:<25 X_1,X_2,X_3,X_4 Bar x < or = 22.5 X Bar X K(mu) Alpha =K(25) x_1=21.24, x_2=24.81, x_3=23.62 x_4=26.82 H_0 p

Let \(X\) equal the number of milliliters of a liquid in a bottle that has a label volume of \(350 \mathrm{ml}\). Assume that the distribution of \(X\) is \(N(\mu, 4)\). To test the null hypothesis \(H_{0}\) : \(\mu=355\) against the alternative hypothesis \(H_{1}: \mu<355\), let the critical region be defined by \(C=\{\bar{x}: \bar{x} \leq 354.05\}\), where \(\bar{x}\) is the sample mean of the contents of a random sample of \(n=12\) bottles. (a) Find the power function \(K(\mu)\) for this test. (b) What is the (approximate) significance level of the test? (c) Find the values of \(K(354.05)\) and \(K(353.1)\), and sketch the graph of the power function. (d) Use the following 12 observations to state your conclusion from this test: (e) What is the approximate \(p\)-value of the test? Equation Transcription: : Text Transcription: X N(mu,4) H_0 : =355 H_1:<355 C={bar x: bar x < or = 354.05} H_1:<715 C={bar x:bar x < or = 668.94} Bar x n=25 n=12 K(mu) K(354.05) K(353.1) p

Assume that SAT mathematics scores of students who attend small liberal arts colleges are \(N(\mu, 8100)\). We shall test \(H_{0}: \mu=530\) against the alternative hypothesis \(H_{1}: \mu<530\). Given a random sample of size \(n=36\) SAT mathematics scores, let the critical region be defined by \(C=\{\bar{x}: \bar{x} \leq 510.77\}\), where \(\bar{x}\) is the observed mean of the sample. (a) Find the power function, \(K(\mu)\), for this test. (b) What is the value of the significance level of the test? (c) What is the value of \(K(510.77)\) ? (d) Sketch the graph of the power function. (e) What is the \(p\)-value associated with (i) \\bar{x}=507.35\); (ii) \(\bar{x}=497.45\)? Equation Transcription: Text Transcription: N(mu,8100) H_0:=530 H_1:<530 n=36 C={bar x: bar x?510.77} Bar x K(mu) K(510.77) p Bar x=507.35 Bar x=497.45

Let \(X\) be \(N(\mu, 100)\). To test \(H_{0}: \mu=80\) against \(H_{1}: \mu>80\), let the critical region be defined by \(C=\) \(\left\{\left(x_{1}, x_{2}, \ldots, x_{25}\right): \bar{x} \geq 83\right\}\), where \(\bar{x}\) is the sample mean of a random sample of size \(n=25\) from this distribution. (a) What is the power function \(K(\mu)\) for this test? (b) What is the significance level of the test? (c) What are the values of \(K(80), K(83)\), and \(K(86)\) ? (d) Sketch the graph of the power function. (e) What is the \(p\)-value corresponding to \(\bar{x}=83.41\)? Equation Transcription: , ? Text Transcription: X N(mu,100) H_0:=80 H_1:>80 C= {x_1,x_2,…,x_25): bar x > or = 83} Bar x n=25 K(mu) K(80),K(83) K(86) p Bar x=83.41

Let \(X\) equal the yield of alfalfa in tons per acre per year. Assume that \(X\) is \(N(1.5,0.09)\). It is hoped that a new fertilizer will increase the average yield. We shall test the null hypothesis \(H_{0}: \mu=1.5\) against the alternative hypothesis \(H_{1}: \mu>1.5\). Assume that the variance continues to equal \(\sigma^{2}=0.09\) with the new fertilizer. Using \(\bar{X}\), the mean of a random sample of size \(n\), as the test statistic, reject \(H_{0}\) if \(\bar{x} \geq c\). Find \(n\) and \(c\) so that the power function \(K(\mu)=P(\bar{X} \geq c: \mu)\) is such that \(\alpha=K(1.5)=0.05\) and \(K(1.7)=0.95\). Equation Transcription: Text Transcription: X N(1.5,0.09) H_0:=1.5 H_1:>1.5. sigma^2=0.09 Bar X n H_0 bar x > or = c c K(mu)=P(bar X > or = c:mu) alpha=K(1.5)=0.05 K(1.7)=0.95

Let \(X\) equal the butterfat production (in pounds) of a Holstein cow during the 305 -day milking period following the birth of a calf. Assume that the distribution of \(X\) is \(N\left(\mu, 140^{2}\right)\). To test the null hypothesis \(H_{0}: \mu=715\) against the alternative hypothesis \(H_{1}: \mu<715\), let the critical region be defined by \(C=\{\bar{x}: \bar{x} \leq 668.94\}\), where \(\bar{x}\) is the sample mean of \(n=25\) butterfat weights from 25 cows selected at random. (a) Find the power function \(K(\mu)\) for this test. (b) What is the significance level of the test? (c) What are the values of \(K(668.94)\) and \(K(622.88)\) ? (d) Sketch a graph of the power function. (e) What conclusion do you draw from the following 25 observations of \(X\) ? (f) What is the approximate \(p\)-value of the test? Equation Transcription: Text Transcription: X N(mu,140^2) H_0:=715 H_1:<715 C={ bar x: bar x < or =668.94} Bar x n=25 K(mu) K(668.94) K(622.88) p

In Exercise 8.5-6, let \(C=\{\bar{x}: \bar{x} \leq c\}\) be the critical region. Find values for \(n\) and \(c\) so that the significance level of this test is \(\alpha=0.05\) and the power at \(\mu=650\) is \(0.90\). Equation Transcription: Text Transcription: C={bar x: bar xc} n c alpha=0.05 mu=650 0.90

Let \(X\) have a Bernoulli distribution with pmf \(f(x ; p)=p^{x}(1-p)^{1-x}, \quad x=0,1, \quad 0 \leq p \leq 1\). We would like to test the null hypothesis \(H_{0}: p \leq 0.4\) against the alternative hypothesis \(H_{1}: p>0.4\). For the test statistic, use \(Y=\sum_{i=1}^{n} X_{i}\), where \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample of size \(n\) from this Bernoulli distribution. Let the critical region be of the form \(C=\{y: y \geq c\}\). (a) Let \(n=100\). On the same set of axes, sketch the graphs of the power functions corresponding to the three critical regions, \(C_{1}=\{y: y \geq 40\}, C_{2}=\{y:\) \(y \geq 50\}\), and \(C_{3}=\{y: y \geq 60\}\). Use the normal approximation to compute the probabilities. (b) Let \(C=\{y: y \geq 0.45 n\}\). On the same set of axes, sketch the graphs of the power functions corresponding to the three samples of sizes 10,100 , and 1000 . Equation Transcription: . , Text Transcription: X f(x;p)=p^x(1-p)^1-x, x=0,1, 0?p?1 H_0:p < or = 0.4 H_1:p>0.4 Y=sum_i=1^n X_i X_1,X_2,…,X_n n C={y:y > or = c} n=100 C_1={y:y > or = 40},C_2={y: y > or = 50} C_3={y:y > or =60} C={y:y> or = 0.45n}

Let \(X_{1}, X_{2}, X_{3}\) be a random sample of size \(n=3\) from an exponential distribution with mean \(\theta>0\). Reject the simple null hypothesis \(H_{0}: \theta=2\), and accept the composite alternative hypothesis \(H_{1}: \theta<2\), if the observed $\operatorname{sum} \sum_{i=1}^{3} x_{i} \leq 2\) (a) What is the power function \(K(\theta)\), written as an integral? (b) Using integration by parts, define the power function as a summation. (c) With the help of Table III in Appendix B, determine \(\alpha=K(2), K(1), K(1 / 2)\), and \(K(1 / 4)\). Equation Transcription: , , K(1/2) Text Transcription: X_1,X_2, X_3 n=3 Theta > 0 H_0<2 sum_i=1^3 x_ < or =i2 K(theta) alpha=K(2), K(1), K(1/2) K(1/4)

PROBLEM 11E Let p equal the fraction defective of a certain manufactured item. To test H0: p = 1/26 against H1: p > 1/26, we inspect n items selected at random and let Y be the number of defective items in this sample. We reject H0 if the observed y ? c. Find n and c so that ? = K(1/26) ? 0.05 and K(1/10) ? 0.90, where K(p) = P(Y ? c; p). Hint: Use either the normal or Poisson approximation to help solve this exercise.

Let \(X_{1}, X_{2}, \ldots, X_{8}\) be a random sample of size \(n=8\) from a Poisson distribution with mean \(\lambda\). Reject the simple null hypothesis \(H_{0}: \lambda=0.5\), and accept \(H_{1}:\) \(\lambda>0.5\), if the observed \(\operatorname{sum} \sum_{i=1}^{8} x_{i} \geq 8\). (a) Compute the significance level \(\alpha\) of the test. (b) Find the power function \(K(\lambda)\) of the test as a sum of Poisson probabilities. (c) Using Table III in Appendix B, determine \(K(0.75)\), \(K(1)\), and \(K(1.25)\). Equation Transcription: , Text Transcription: X_1,X_2,…,X_8 n=8 H_0:=0.5 H_1: >0.5 sum_i=1^8 x_i > or = 8 K(labda) K(0.75), K(1) K(1.25)

Let \(p\) denote the probability that, for a particular tennis player, the first serve is good. Since \(p=0.40\), this player decided to take lessons in order to increase \(p\). When the lessons are completed, the hypothesis \(H_{0}: p=0.40\) will be tested against $H_{1}: p>0.40$ on the basis of \(n=25\) trials. Let \(y\) equal the number of first serves that are good, and let the critical region be defined by \(C=\{y: y \geq 14\}\). (a) Find the power function \(K(p)\) for this test. (b) What is the value of the significance level, \(\alpha=\) \(K(0.40)\) ? Use Table II in Appendix B. (c) Evaluate \(K(p)\) at \(p=0.45,0.50,0.60,0.70,0.80\), and \(0.90\). Use Table II. (d) Sketch the graph of the power function. (e) If \(y=15\) following the lessons, would \(H_{0}\) be rejected? (f) What is the \(p\)-value associated with \(y=15\)? Equation Transcription: , . Text Transcription: p p=0.40 H_0:p=0.40 H_1:p>0.40 n=25 y C={y:y > or =14} K(p) alpha= K(0.40) p=0.45,0.50,0.60,0.70,0.80 0.90 y=15 H_0

A certain size of bag is designed to hold 25 pounds of potatoes. A farmer fills such bags in the field. Assume that the weight X of potatoes in a bag is N(, 9). We shall test the null hypothesis H0: = 25 against the alternative hypothesis H1: < 25. Let X1, X2, X3, X4 be a random sample of size 4 from this distribution, and let the critical region C for this test be defined by x 22.5, where x is the observed value of X. (a) What is the power function K() of this test? In particular, what is the significance level = K(25) for your test? (b) If the random sample of four bags of potatoes yielded the values x1 = 21.24, x2 = 24.81, x3 = 23.62, and x4 = 26.82, would your test lead you to accept or reject H0? (c) What is the p-value associated with x in part (b)?

Let X equal the number of milliliters of a liquid in a bottle that has a label volume of 350 ml. Assume that the distribution of X is N(, 4). To test the null hypothesis H0: = 355 against the alternative hypothesis H1: < 355, let the critical region be defined by C = {x : x 354.05},where x is the sample mean of the contents of a random sample of n = 12 bottles. (a) Find the power function K() for this test. (b) What is the (approximate) significance level of the test? (c) Find the values of K(354.05) and K(353.1), and sketch the graph of the power function. (d) Use the following 12 observations to state your conclusion from this test: 350 353 354 356 353 352 354 355 357 353 354 355 (e) What is the approximate p-value of the test?

Assume that SAT mathematics scores of students who attend small liberal arts colleges are N(, 8100). We shall test H0: = 530 against the alternative hypothesis H1: < 530. Given a random sample of size n = 36 SAT mathematics scores, let the critical region be defined by C = {x: x 510.77}, where x is the observed mean of the sample. (a) Find the power function, K(), for this test. (b) What is the value of the significance level of the test? (c) What is the value of K(510.77)? (d) Sketch the graph of the power function. (e) What is the p-value associated with (i) x = 507.35; (ii) x = 497.45?

Let X be N(, 100). To test H0: = 80 against H1: > 80, let the critical region be defined by C = {(x1, x2, ... , x25) : x 83}, where x is the sample mean of a random sample of size n = 25 from this distribution. (a) What is the power function K() for this test? (b) What is the significance level of the test? (c) What are the values of K(80), K(83), and K(86)? (d) Sketch the graph of the power function. (e) What is the p-value corresponding to x = 83.41?

Let X equal the yield of alfalfa in tons per acre per year. Assume that X is N(1.5, 0.09). It is hoped that a new fertilizer will increase the average yield. We shall test the null hypothesis H0: = 1.5 against the alternative hypothesis H1: > 1.5. Assume that the variance continues to equal 2 = 0.09 with the new fertilizer. Using X, the mean of a random sample of size n, as the test statistic, reject H0 if x c. Find n and c so that the power function K() = P(X c : ) is such that = K(1.5) = 0.05 and K(1.7) = 0.95.

Let X equal the butterfat production (in pounds) of a Holstein cow during the 305-day milking period following the birth of a calf. Assume that the distribution of X is N(, 1402). To test the null hypothesis H0: = 715 against the alternative hypothesis H1: < 715, let the critical region be defined by C = {x : x 668.94}, where x is the sample mean of n = 25 butterfat weights from 25 cows selected at random. (a) Find the power function K() for this test. (b) What is the significance level of the test? (c) What are the values of K(668.94) and K(622.88)? (d) Sketch a graph of the power function. (e) What conclusion do you draw from the following 25 observations of X? 425 710 661 664 732 714 934 761 744 653 725 657 421 573 535 602 537 405 874 791 721 849 567 468 975 (f) What is the approximate p-value of the test?

In Exercise 8.5-6, let C = {x : x c} be the critical region. Find values for n and c so that the significance level of this test is = 0.05 and the power at = 650 is 0.90.

Let X have a Bernoulli distribution with pmf f(x; p) = px(1 p) 1x, x = 0, 1, 0 p 1. We would like to test the null hypothesis H0: p 0.4 against the alternative hypothesis H1: p > 0.4. For the test statistic, use Y = n i=1 Xi, where X1, X2, ... , Xn is a random sample of size n from this Bernoulli distribution. Let the critical region be of the form C = {y: y c}. (a) Let n = 100. On the same set of axes, sketch the graphs of the power functions corresponding to the three critical regions, C1 = {y : y 40}, C2 = {y : y 50}, and C3 = {y : y 60}. Use the normal approximation to compute the probabilities. (b) Let C = {y : y 0.45n}. On the same set of axes, sketch the graphs of the power functions corresponding to the three samples of sizes 10, 100, and 1000.

Let p denote the probability that, for a particular tennis player, the first serve is good. Since p = 0.40, this player decided to take lessons in order to increase p. When the lessons are completed, the hypothesis \(H_0: p = 0.40\) will be tested against \(H_1: p > 0.40\) on the basis of n = 25 trials. Let y equal the number of first serves that are good, and let the critical region be defined by \(C = {y: y \ge 14}\). (a) Find the power function K(p) for this test. (b) What is the value of the significance level, \(\alpha = K(0.40)\)? Use Table II in Appendix B. (c) Evaluate K(p) at p = 0.45, 0.50, 0.60, 0.70, 0.80, and 0.90. Use Table II. (d) Sketch the graph of the power function. (e) If y = 15 following the lessons, would \(H_0\) be rejected? (f) What is the p-value associated with y = 15?

Let X1, X2, ... , X8 be a random sample of size n = 8 from a Poisson distribution with mean . Reject the simple null hypothesis H0: = 0.5, and accept H1: > 0.5, if the observed sum 8 i=1 xi 8. (a) Compute the significance level of the test. (b) Find the power function K() of the test as a sum of Poisson probabilities. (c) Using Table III in Appendix B, determine K(0.75), K(1), and K(1.25).

Let p equal the fraction defective of a certain manufactured item. To test H0: p = 1/26 against H1: p > 1/26, we inspect n items selected at random and let Y be the number of defective items in this sample. We reject H0 if the observed y c. Find n and c so that = K(1/26) 0.05 and K(1/10) 0.90, where K(p) = P(Y c; p). Hint: Use either the normal or Poisson approximation to help solve this exercise.

Let \(X_1, X_2, X_3\) be a random sample of size n = 3 from an exponential distribution with mean \(\theta > 0\). Reject the simple null hypothesis \(H_0: \theta = 2\), and accept the composite alternative hypothesis \(H_1: \theta < 2\), if the observed sum \(\sum_{i=1}^3 x_i \le 2\). (a) What is the power function \(K(\theta)\), written as an integral? (b) Using integration by parts, define the power function as a summation. (c) With the help of Table III in Appendix B, determine \(\alpha = K(2), K(1), K(1/2),~\text{and } K(1/4)\).