Mathematics Homework Help

10th edition of Probability and Statistical Inference do Exercises

8.6-1, 8.6-2, 8.6-10, 8.7-1, 8.7-4, 8.7-5.

8.6-1. 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(μ, 9). We shall test the null hypothesis H0: μ = 25 against the alternative hypothesis H1: μ< 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 functionK(μ) of this test? In partic-ular, what is the significance level α = 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)?

8.6-2. 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 ofX is N(μ, 4). To test the null hypothesisH0: μ = 355 against the alternative hypothesis H1: μ< 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(μ) 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 con-clusion 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?

8.6-10. Let X have a Bernoulli distribution with pmf f(x; p) = px(1 − p)1−x,

x = 0, 1, n 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 =

i=1 Xi,where X1,X2,…,Xn is a ran-dom sample of size n from this Bernoulli distribution. Let the critical region be of the form C ={y: y ≥ c}.

(a) Let n = 100. Onthe same set of axes, sketch the graphs of the power functions corresponding to the three crit-ical regions, C1 ={y : y ≥ 40}, C2 ={y : y ≥ 50},and C3 ={y : y ≥ 60}. Use the normal approximation to compute the probabilities.

(b) LetC ={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.

8.7-1. Let X1,X2,…,Xn be a random sample from a normal distribution N(μ, 64).

(a) Show that C ={(x1, x2,…, xn): x ≤ c} is a best critical region for testing H0: μ = 80 against H1: μ = 76.

(b) Find n and c so that α ≈ 0.05 and β ≈ 0.05.

8.7-4. 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 Y = i=1 Xi,which is b(n, p).

(b) If C ={(x1, x2,…, xn): n

n i=1 xi ≤ n(0.85)} and i=1 Xi, find the value of n such that α = 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 β = 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?

8.7-5. Let X1,X2,…,Xn be a random sample from the normal distribution N(μ, 36).

(a) Show that a uniformly most powerful critical region for testing H0: μ = 50 against H1: μ< 50 is given by C2 ={x: x ≤ c}.

(b) With this result and that of Example 8.7-4, argue that a uniformly most powerful test for testing H0: μ = 50 against H1: μ = 50 does not exist.