Exam 8: Tests of Hypotheses Based on a Single Sample

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 $H _ { \pm } : \mu < \mu _ { o }$ the rejection region for a level $\alpha$ test is __________.

Which of the following statements are true in testing $H _ { 0 } : \mu = 275 \text { versus } H _ { \pm } : > 275$ based on a sample of size 15 from a normal population with unknown standard deviation $\sigma$ ?

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?

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 $H a : p \neq p _ { o }$ , then the rejection region for a level $\alpha$ test is either __________ or __________.

Which of the following statements are not correctly stated?

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 $H _ { a } : \mu \neq \mu _ { 0 } ,$ the rejection region for a level $\alpha$ test is either __________ or __________.

A __________ error involves not rejecting the null hypothesis $H _ { o } \text { when } H _ { o }$ is false.

Which of the following statements are not generally true?

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

A certain pen has been designed so that "true" average writing lifetime under controlled conditions (involving the use of a writing machine) is at least 12 hours. A random sample of 18 pens is selected, the writing lifetime of each is determined, and a normal probability plot of the resulting data supports the use of a one-sample t test. a. What hypotheses should be tested if the investigator believe a priori that the design specification has been satisfied? b. What conclusion is appropriate if the hypotheses of part (a) are tested, t = -2.5, and $\alpha = .05$ ? c. What conclusion is appropriate if the hypotheses of part (a) are tested, t = -2, and $\alpha = .01$ ? d. What should be concluded if the hypotheses of part (a) are tested and t = -3.25?

Each of a group of 20 intermediate tennis players is given two tennis rackets, one with the two rackets, each player will be asked to state a preference for one of the two types of strings. Let p denote the proportion of all such players who would prefer gut to nylon, and let X be the number of players in the sample who prefer gut. Because gut strings are more expensive, consider the null hypothesis that at most 50% of all such players prefer gut. We simplify this to $H _ { 0 } : p = .5$ planning to reject $H _ { 0 }$ only if sample evidence strongly favors gut strings. a. Which of the rejecting regions { 15, 16, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 5}, or { 0, 1, 2, 3, 17, 18, 19, 20} is most appropriate, and why are the other two not appropriate? b. What is the probability of a type I error for the chosen region of part (a)? Does the region specify a level .05 test? Is it the best level .05 test? c. If 60% of all enthusiasts prefer gut, calculate the probability of a type II error using the appropriate region from part (a). Repeat if 80% of all enthusiasts prefer gut. d. If 13 out of the 20 players prefer gut, should $H _ { 0 }$ be rejected using a significance level of .10?

Which of the following statements are true?

Let the test statistic Z have a standard normal distribution when $H _ { 0 }$ is true. Give the significance level for each of the following situations. a. $H _ { ± } : \mu > \mu _ { 1 } , \text { rejection region } z \geq 2.602$ b. $H _ { \pm } : \mu < \mu _ { 0 } , \text { rejection region } z \leq - 2.069$ c. $H _ { \pm } : \mu = \mu _ { 0 } \text {, rejection region } z \geq 2.042 \text { or } z \leq 2.042$

Let $\mu$ denote the true average radioactivity level (picocuries per liter). The value 5 pCi/L is considered the dividing line between safe and unsafe water. Would you recommend testing $H _ { 0 } : \mu = 5$ versus $H _ { \pm } : \mu > 5 \text { or } H _ { 0 } : \mu < 5 ?$ Explain your reasoning. (Hint: Think about the consequences of a type I and type II error for each possibility.)

Suppose that a t test of $H _ { o } : \mu = 250 \text { versus } H _ { \pm } : \mu \neq 250$ is based on 12 degrees of freedom. If the calculated value of the test statistic is 2.8, then the P-value is

Let p denote the proportion of individuals in a population who possess a specified property, and X denote the number of individuals in the sample who possess the same property. The estimator $\hat { p } = X / n$ is __________ if $E ( \hat { p } ) = p$ and its standard deviation $\sigma _ { \bar {P} }$ = __________.

If the calculated test statistic for two-tailed z test is -1.84, then the P-value is __________.

If the P-value is larger than the level of significance $\alpha$ , then the researcher should __________ at level $\alpha .$

An engineer has suggested a change in the production process in the belief that it will result in a reduced defective rate. Let p denote the true proportion of defective items resulting from the changed process, and that 5% of items produced by a manufacturer during a certain period were defective. Then the research hypothesis is the assertion that __________.