Exam 8: Tests of Hypotheses Based on a Single Sample

Suppose that a t test of $H _ { 0 } : \mu _ { 1 } - \mu _ { 2 } = 0 \text { versus } H _ { o } : \mu _ { 1 } - \mu _ { 2 } \neq 0$ is based on 18 degrees of freedom. If the calculated value of the test statistic is t = - 1.80, then the P-value for this test is __________.

Which of the following statements are not true if a test procedure about the population mean $\mu$ is performed when the population is normal with known standard deviation $\sigma$ ?

Which of the following statements are true about the P-value, where z is the calculated value of the test statistic and $\Phi ( z )$ is the corresponding cumulative area under the standard normal curve?

A __________ error consists of rejecting the null hypothesis $H _ { o } \text { when } H _ { o }$ is true.

Suppose that a z test of $H _ { o } : \mu = \mu _ { o } \text { versus } H _ { o} : \mu > \mu _ { o}$ is conducted. Intuition then suggests rejecting $H _ { o }$ when the value of test statistic z is __________.

If $X _ { 1 } , X _ { 2 } , \ldots \ldots , X _ { n }$ is a random sample from a normal distribution, and the sample size n is small, then the standardized variable $T = ( \bar { X } - \mu ) / ( S / \sqrt { n } )$ has a __________ distribution with __________ degrees of freedom.

Which of the following statements are not true for any significance level?

A plan for an executive traveler's club has been developed by an airline on the premise that 5% of its current customers would qualify for membership. A random sample of 500 customers yielded 40 who would qualify. a. Using this data, test at level .01 the null hypothesis that the company's premise is correct against the alternative that it is not correct. b. What is the probability that when the test of part (a) is used, the company's premise will be judged correct when in fact 10% of all current customers qualify?

Which of the following P-values will lead us to reject the null hypothesis at the .05 level?

In many situations, the alternative hypothesis is referred to as the __________ hypothesis, since it is the statement the researcher would really like to validate.

Which of the following statements are not true if a test procedure about the population proportion p is performed provided that $np _ { { o } } \geq 10 \text { and } n \left( 1 - p _ { 0 } \right) \geq 10 ?$

Let p denote the proportion of individuals in a population who possess a specified property, and let X denote the number of individuals in the sample who possess the same property. Provided that the sample size n is small relative to the population size, X has approximately

Let p denote the proportion of individuals in a population who possess a specified property, and let X denote the number of individuals in the sample who possess the same property. Provided that the sample size n is large, the estimator $\hat { p } = \bar{X} / n$ has approximately

Which of the following statements are true?

A sample of 12 radon detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows: 104.3 89.6 89.9 95.6 95.2 90.0 98.8 103.7 98.3 106.4 102.0 91.1 a. Does this data suggest that the population mean reading under these conditions differs from 100? State and test the appropriate hypotheses using $\alpha$ =.05 b. Suppose that prior to the experiment, a value of $\sigma$ =7.5 had been assumed. How many determinations would then have been appropriate to obtain $\beta = .10$ for the alternative $\mu = 95 ?$

Before agreeing to purchase a large order of polyethylene sheaths for a particular type of high pressure oil-filled submarine power cable, a company wants to see conclusive evidence that the true standard deviation of sheath thickness is less than .05mm. What hypotheses should be tested, and why? In this context, what are the type I and type II errors?

Let the test statistic T have a t distribution when $H _ { 0 }$ is true. Give the significance level for each of the following situations: a. $H _ {\pm } : \mu > \mu _ { 0 } , d f = 15 , \text { rejection regiont } \geq 3.733$ b. $H _ { \pm } : \mu < \mu _ { 0 } , n = 24 , \text { rejection region } t \leq - 2.500$ c. $H _ { \pm } : \mu \neq \mu _ { 1 } , n = 31 \text {, rejection region } t \geq 31.697 \text { or } t \leq - 1.697$

Two different companies have applied to provide cable television service in a certain region. Let p denote the proportion of all potential subscribers who favor the first company over the second. Consider testing $H _ { 0 } : p = .5 \text { versus } H _ { \pm} : p \neq .5$ based on a random sample of 25 individuals. Let X denote the number in the sample who favor the first company and x represent the observed value of X. a. Which of the following rejection regions is most appropriate and why? $R _ { 1 } = \{ x : x \leq 7 \text { or } x \geq 18 \} , R _ { 2 } = \{ x : x \leq 8 \} \text {, }$ or $R _ { 3 } = \{ : x \geq 17 \}$ b. In the context of this problem situation, describe what type I and type II errors are. c. What is the probability distribution of the test statistic X when $H _ { 0 }$ is true? Use it to compute the probability of a type I error.. d. Compute the probability of a type II error for the selected region when p = .6 and p = .7. e. Using the selected region, what would you conclude if 6 of the 25 queried favored company 1?

Which of the following statements are not true?

For any significance level $\alpha$ the two-tailed rejection region $z \geq z _ { \alpha / 2 } \text { or } z \leq - z _ { \alpha / 2 }$ has type I error probability equal to __________.