Chapter 7: Q25 E (page 403)
In a test of \({H_0}:\mu = 100\) against \({H_a}:\mu \ne 100\), the sample data yielded the test statistic z = 2.17. Find the p-value for the test.
The p-value for the hypothesis test is 0.030.
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Intrusion detection systems. The Journal of Research of the National Institute of Standards and Technology (November– December 2003) published a study of a computer intrusion detection system (IDS). The IDS is designed to provide an alarm whenever unauthorized access (e.g., an intrusion) to a computer system occurs. The probability of the system giving a false alarm (i.e., providing a warning when no intrusion occurs) is defined by the symbol , while the probability of a missed detection (i.e., no warning given when an intrusion occurs) is defined by the symbol . These symbols are used to represent Type I and Type II error rates, respectively, in a hypothesis-testing scenario
a. What is the null hypothesis, ?
b. What is the alternative hypothesis,?
c. According to actual data collected by the Massachusetts Institute of Technology Lincoln Laboratory, only 1 in 1,000 computer sessions with no intrusions resulted in a false alarm. For the same system, the laboratory found that only 500 of 1,000 intrusions were actually detected. Use this information to estimate the values of and .
Trading skills of institutional investors. Refer to The Journal of Finance (April 2011) analysis of trading skills of institutional investors, Exercise 7.36 (p. 410). Recall that the study focused on “round-trip” trades, i.e., trades in which the same stock was both bought and sold in the same quarter. In a random sample of 200 round-trip trades made by institutional investors, the sample standard deviation of the rates of return was 8.82%. One property of a consistent performance of institutional investors is a small variance in the rates of return of round-trip trades, say, a standard deviation of less than 10%.
a. Specify the null and alternative hypotheses for determining whether the population of institutional investors performs consistently.
b. Find the rejection region for the test using
c. Interpret the value of in the words of the problem.
d. A Minitab printout of the analysis is shown (next column). Locate the test statistic and on the printout.
e. Give the appropriate conclusion in the words of the problem.
f. What assumptions about the data are required for the inference to be valid?
Consider the test \({H_0}:\mu = 70\) versus \({H_a}:\mu \ne 70\) using a large sample of size n = 400. Assume\(\sigma = 20\).
a. Describe the sampling distribution of\(\bar x\).
b. Find the value of the test statistic if\(\bar x = 72.5\).
c. Refer to part b. Find the p-value of the test.
d. Find the rejection region of the test for\(\alpha = 0.01\).
e. Refer to parts c and d. Use the p-value approach to
make the appropriate conclusion.
f. Repeat part e, but use the rejection region approach.
g. Do the conclusions, parts e and f, agree?
For each of the following rejection regions, sketch the sampling distribution for z and indicate the location of the rejection region.
a. \({H_0}:\mu \le {\mu _0}\) and \({H_a}:\mu > {\mu _0};\alpha = 0.1\)
b. \({H_0}:\mu \le {\mu _0}\) and \({H_a}:\mu > {\mu _0};\alpha = 0.05\)
c. \({H_0}:\mu \ge {\mu _0}\) and \({H_a}:\mu < {\mu _0};\alpha = 0.01\)
d. \({H_0}:\mu = {\mu _0}\) and \({H_a}:\mu \ne {\mu _0};\alpha = 0.05\)
e. \({H_0}:\mu = {\mu _0}\) and \({H_a}:\mu \ne {\mu _0};\alpha = 0.1\)
f. \({H_0}:\mu = {\mu _0}\) and \({H_a}:\mu \ne {\mu _0};\alpha = 0.01\)
g. For each rejection region specified in parts a–f, state the probability notation in z and its respective Type I error value.
Refer to Exercise 7.99.
a. Find b for each of the following values of the population mean: 74, 72, 70, 68, and 66.
b. Plot each value of b you obtained in part a against its associated population mean. Show b on the vertical axis and m on the horizontal axis. Draw a curve through the five points on your graph.
c. Use your graph of part b to find the approximate probability that the hypothesis test will lead to a Type II error when m = 73.
d. Convert each of the b values you calculated in part a to the power of the test at the specified value of m. Plot the power on the vertical axis against m on the horizontal axis. Compare the graph of part b with the power curve of this part.
e. Examine the graphs of parts b and d. Explain what they reveal about the relationships among the distance between the true mean m and the null hypothesized mean m0, the value of b, and the power.
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