Polygraphs that are used in criminal investigations are supposed to indicate whether a person is lying or telling the truth. However the procedure

is not infallible, as is illustrated by the following example. An experienced

polygraph examiner was asked to make an overall judgment for each of a

total 280 records, of which 140 were from guilty suspects and 140 from innocent suspects. The results are listed in Table 26.2. We view each judgment

as a problem of hypothesis testing, with the null hypothesis corresponding to

"suspect is innocent" and the alternative hypothesis to "suspect is guilty."

Estimate the probabilities of a type I error and a type II error that apply to

this polygraph method on the basis of Table 26.2.

26.2 Consider the testing problem in Exercise 25.11. Compute the probability

of committing a type II error if the true value of is 1.

26.3 One generates a number x from a uniform distribution on the interval

[0, ?]. One decides to test H0 : ? = 2 against H1 : ? = 2 by rejecting H0 if

x ? 0.1 or x ? 1.9.

a. Compute the probability of committing a type I error.

b. Compute the probability of committing a type II error if the true value

of ? is 2.5

Recall Exercises 23.5 and 24.8 about the 1500 m speed-skating results

in the 2002 Winter Olympic Games. The number of races won by skaters

starting in the outer lane is modeled by a random variable X with a Bin(23, p)

distribution. The question of whether there is an outer lane advantage was

investigated in Exercise 24.8 by means of constructing confidence intervals

using the normal approximation. In this exercise we examine this question by

testing the null hypothesis H0 : p = 1/2 against H1 : p > 1/2 using X as the

test statistic. The distribution of X under H0 is given in Table 26.3. Out of

23 completed races, 15 were won by skaters starting in the outer lane.

a. Compute the p-value corresponding to x = 15 and report your conclusion

if we perform the test at level 0.05. Does your conclusion agree with the

confidence interval you found for p in Exercise 24.8b?

b. Determine the critical region corresponding to significance level ? = 0.05.

c. Compute the probability of committing a type I error if we base our

decision rule on the critical region determined in b.

5. (1 pt; 64) Let X1, . .., Xn ~p and p is an unknown infinitely differentiable density function. Let Pr(x) = EK () be the KDE. Assume that we use a kernel function satisfying 0 = ( xK(x)da = ( xK(x)dx = ( xK(x)dx and fx*K(x)da # 0. At a given point To, show that the bias bias(Pn(x0)) = Cihl + o(hi ) for some constant C, and the variance Var (Pr (20)) 5 nh for some constant C2. Note: This type of kernel function is called a higher-order kernel function. It leads to a KDE with a smaller bias compared to the KDE from a regular kernel function. However, this kernel function will take negative value, which implies that Pn(x) may be negative at some place!Binomial Regression Kernel mean function = Logistic Response = Fail Terms = {Temp} Trials = n Coefficient Estimates Label Estimate Std. Error EstKSE p-vaiue Constant 5.08492 3.04906 1.668 0.0954 Temp -0.115600 0.0469567 -2.462 0.0133 b) Estimate the probability that an O-ring fails if Temp = 31. :3) Using 1)), estimate the odds that an O-ring fails if Temp = 31. 3.9 The rescaled Epanechnikov kernel 85] is a symmetric density function fe(x) = ( (1 - x2), |ac|