Exam 8: Tests of Hypotheses Based on a Single Sample

In hypothesis-testing analysis, a type I error occurs only if

Suppose that when data from an experiment was analyzed, the P-value for testing $H _ { o } : \mu = 50 \text { ver sus } H _ {\pm} : \mu > 50$ was calculated as .0244. Which of the following statements are true?

Suppose that a z test of $H _ { 0 } : \mu = \mu _ { 0 } \text { ver sus } H _ {o } : \mu < \mu _ { 0 }$ is conducted. The critical value $z _ { \boldsymbol { \alpha } }$ is determined by specifying __________ and using the fact that $Z = \left( \bar { X } - \mu _ { 0 } \right)/ ( \sigma / \sqrt { n } )$ has a __________ distribution when $H _ { o }$ is true.

Suppose that a test procedure about the population proportion p is performed, and that the sample proportion $\hat { p }$ is approximately normally distributed. If the alternative hypothesis is $H _ { \pm } ^ { \prime } p < p _ { 0 }$ then the rejection region for a level $\alpha$ test is __________.

For an upper-tailed z test, the level of significance $\alpha$ is just the area under the z curve to the __________ of the critical value __________.

Which of the following statements are necessary to construct an appropriate test procedure?

The smallest level of significance $\alpha$ for which the null hypothesis $H _ { o }$ would be rejected is the tail area captured by the computed value of the test statistic. This smallest $\alpha$ is referred to as the __________.

If the null hypothesis is $H _ { o } : \mu = 100 , \bar { x } = 105 \text {, and } \sigma _ { \bar { x } } = \sigma / \sqrt { n } = 2.5$ then the test statistic value is z = __________.

Suppose a test procedure about the population mean $\mu$ is performed, when the population is normal with known standard deviation $\sigma ,$ then if the alternative hypothesis is $H _ { \pm } : \mu < \mu _ { o }$ the rejection region for a level $\alpha$ test is __________.

Which of the following statements are true if the value of the test statistic for a two-tailed z test is z = -1.56?

Which of the following statements are true?

Suppose that a t test of $H _ { o } : \mu = 25 \text { versus } H _ { o } : \mu > 25$ is based on 10 degrees of freedom. If the calculated value of the test statistic is t = 2.4, then the P-value for this test is __________.

Which of the following statements are true?

If the P-value is smaller than or equal to the level of significance $\alpha$ then the researcher should __________ at level $\alpha .$

The rejection region is called __________ if it consists only of small values of the test statistic.

One must be especially careful in interpreting evidence when the sample size is __________, since any small departure from $H _ { o }$ will almost surely be detected by a test, yet such a departure may have little practical significance.

Light bulbs of a certain type are advertised as having an average lifetime of 800 hours. The price of these bulbs is very favorable, so a potential customer has decided to go ahead with a purchase arrangement unless it can be conclusively demonstrated that the true average lifetime is smaller than what is advertised. A random sample of 50 bulbs was selected, the lifetime of each bulb determined, and the appropriate hypotheses were tested using MINITAB, resulting in the accompanying output. What conclusion would be appropriate for a significance level of .05? A significance level of .01?

A university library ordinarily has a complete shelf inventory done once every year. Because of new shelving rules instituted the previous year, the head librarian believes it may be possible to save money by postponing the inventory. The librarian decides to select at random 1000 books from the library's collection and have them searched in a preliminary manner. If evidence indicates strongly that the true proportion of misshelved or unlocatable books is less than .02, then the inventory will be postponed. a. Among the 1000 books searched, 15 were misshelved or unlocatable. Test the relevant hypotheses and advise the librarian what to do (use $\alpha = .05$ ). b. If the true proportion of misshelved and lost books is actually .01, what is the probability that the inventory will be (unnecessarily) taken? c. If the true proportion is .05, what is the probability that the inventory will be postponed?

Suppose a test procedure about the population mean $\mu$ is performed, when the population is normal and the sample size n is small, then if the alternative hypothesis is $H _ { \pm } : \mu \neq \mu _ { 0 } ,$ the rejection region for a level $\alpha$ test is either __________ or __________.

State DMV records indicate that of all vehicles undergoing emissions testing during the previous year, 70% passed on the first try. A random sample of 200 cars tested in a particular county during the current year yields 160 that passed on the initial test. Does this suggest that the true proportion for this county during the current year differs from the previous statewide proportion? Test the relevant hypotheses using $\alpha = .05$