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Chapter 9: Q5SE (page 621)

Consider again the conditions of Exercise 4, and suppose that伪(未)is required to be a given value伪0 (0 < 伪0< 1). Determine the test procedure未for which尾 (未)will be a minimum, and calculate this minimum value.

Short Answer

Expert verified

Hence the minimum value for 尾(饾浛)is

Reject the null is X 鈮� 1-鈭毼�0 ; 尾(饾浛) = (1-鈭毼�0)2

Step by step solution

01

Calculate the minimum value for which  β (δ) will be minimum

The optimal such test procedure is obtained by rejecting the null if

\begin{aligned}\frac{f_{1}(x)}{f_{0}(x)}>k\end{aligned}

Where the level condition determines k

Here the ratio on the left turns out to be 1-x/x. Hence this is equivalent to saying that we reject the null when x>c, where the level condition determines c. Hence the level condition yields

伪0 = P(X 鈮� c | 胃 = 2)

= (1- c)2

c =1-鈭毼�0

the type II error thus turns out to be

尾(饾浛) = P(X < c | 胃 = 0)

= c2

= (1-鈭毼�0)2

Reject the null is X 鈮� 1-鈭毼�0 ; 尾(饾浛) = (1-鈭毼�0)2

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Most popular questions from this chapter

Suppose that a sequence of Bernoulli trials is to be carried out with an unknown probability胃of success on each trial, and the following hypotheses are to be tested:

H0 : 胃 = 0.1

H1 : 胃 = 0.2

Let X denote the number of trials required to obtain success, and suppose that H0 is to be rejected if X鈮� 5. Determine the probabilities of errors of type I and type II.

Consider again the conditions of Exercise 2, but suppose now that it is desired to test the following hypotheses:

H0: 渭 鈮� 0,

H1: 渭 > 0.

Suppose also that in the random sample of 10,000 observations, the sample means Xn is 0.03. At what level of significance is this result just significant?

Suppose that \({X_1},....,{X_{10}}\)form a random sample from a normal distribution for which both the mean and the variance are unknown. Construct a statistic that does not depend on any unknown parameters and has the \(F\)distribution with three and five degrees of freedom.

Prove Theorem 9.1.3

Suppose that a random sample of eight observationsX1,鈥�.,Xs is taken from the normal distribution with the unknown mean 碌 and unknown variance 蟽2, and it is desired to test the following hypotheses:

H0 : 碌 = 0

H1 : 碌鈮� 0

Suppose also that sample data are such that \begin{aligned}\sum_{i-1}^{8}X_{i}=-11.2\end{aligned} and \begin{aligned}\sum_{i-1}^{8}X_{i}^{2}=43.7\end{aligned}

if a symmetric t- test is performed at the level of significance 0.10 so that each tail of the critical region has probability 0.05, should the hypothesis H0be rejected or not?
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