91Ó°ÊÓ

State the decision rule that would be used to test the following hypotheses. Evaluate the appropriate test statistic and state your conclusion. (a) \(H_{0}: \mu=120\) versus \(H_{1}: \mu<120 ; \bar{y}=114.2, n=25\), \(\sigma=18, \alpha=0.08\) (b) \(H_{0}: \mu=42.9\) versus \(H_{1}: \mu \neq 42.9 ; \bar{y}=45.1, n=16\), \(\sigma=3.2, \alpha=0.01\) (c) \(H_{0}: \mu=14.2\) versus \(H_{1}: \mu>14.2 ; \bar{y}=15.8, n=9, \sigma=\) \(4.1, \alpha=0.13\)

An herbalist is experimenting with juices extracted from berries and roots that may have the ability to affect the Stanford-Binet IQ scores of students afflicted with mild cases of attention deficit disorder (ADD). A random sample of twenty-two children diagnosed with the condition have been drinking Berry Smart daily for two months. Past experience suggests that children with ADD score an average of 95 on the IQ test with a standard deviation of 15 . If the data are to be analyzed using the \(\alpha=0.06\) level of significance, what values of \(\bar{y}\) would cause \(H_{0}\) to be rejected? Assume that \(H_{1}\) is two-sided.

(a) Suppose \(H_{0}: \mu=\mu_{0}\) is rejected in favor of \(H_{1}: \mu \neq \mu_{0}\) at the \(\alpha=0.05\) level of significance. Would \(H_{0}\) necessarily be rejected at the \(\alpha=0.01\) level of significance? (b) Suppose \(H_{0}: \mu=\mu_{0}\) is rejected in favor of \(H_{1}: \mu \neq\) \(\mu_{0}\) at the \(\alpha=0.01\) level of significance. Would \(H_{0}\) necessarily be rejected at the \(\alpha=0.05\) level of significance?

If \(H_{0}: \mu=\mu_{0}\) is rejected in favor of \(H_{1}: \mu>\mu_{0}\), will it necessarily be rejected in favor of \(H_{1}: \mu \neq \mu_{0}\) ? Assume that \(\alpha\) remains the same.

A random sample of size 16 is drawn from a normal distribution having \(\sigma=6.0\) for the purpose of testing \(H_{0}\) : \(\mu=30\) versus \(H_{1}: \mu \neq 30\). The experimenter chooses to define the critical region \(C\) to be the set of sample means lying in the interval \((29.9,30.1)\). What level of significance does the test have? Why is \((29.9,30.1)\) a poor choice for the critical region? What range of \(\bar{y}\) values should comprise \(C\), assuming the same \(\alpha\) is to be used?

Defeated in his most recent attempt to win a congressional seat because of a sizeable gender gap, a politician has spent the last two years speaking out in favor of women's rights issues. A newly released poll claims to have contacted a random sample of one hundred twenty of the politician's current supporters and found that seventy-two were men. In the election that he lost, exit polls indicated that \(65 \%\) of those who voted for him were men. Using an \(\alpha=0.05\) level of significance, test the null hypothesis that the proportion of his male supporters has remained the same. Make the alternative hypothesis one-sided.

Suppose \(H_{0}: p=0.45\) is to be tested against \(H_{1}: p>\) \(0.45\) at the \(\alpha=0.14\) level of significance, where \(p=P(i\) th trial ends in success). If the sample size is two hundred, what is the smallest number of successes that will cause \(H_{0}\) to be rejected?

Construct a power curve for the \(\alpha=0.05\) test of \(H_{0}: \mu=60\) versus \(H_{1}: \mu \neq 60\) if the data consist of a random sample of size 16 from a normal distribution having \(\sigma=4\).

Suppose a sample of size 1 is taken from the pdf \(f_{Y}(y)=(1 / \lambda) e^{-y / \lambda}, y>0\), for the purpose of testing $$ \begin{gathered} H_{0}: \lambda=1 \\ \text { versus } \\ H_{1}: \lambda>1 \end{gathered} $$ The null hypothesis will be rejected if \(y \geq 3.20\). (a) Calculate the probability of committing a Type I error. (b) Calculate the probability of committing a Type II error when \(\lambda=\frac{4}{3}\). (c) Draw a diagram that shows the \(\alpha\) and \(\beta\) calculated in parts (a) and (b) as areas.

An urn contains ten chips. An unknown number of the chips are white; the others are red. We wish to test \(H_{0}\) : exactly half the chips are white versus \(H_{1}\) : more than half the chips are white We will draw, without replacement, three chips and reject \(H_{0}\) if two or more are white. Find \(\alpha\). Also, find \(\beta\) when the urn is (a) \(60 \%\) white and (b) \(70 \%\) white.