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Chapter 7: Q7E (page 397)
If you test a hypothesis and reject the null hypothesis in favor of the alternative hypothesis, does your test prove that the alternative hypothesis is correct? Explain.
Saying that a test proves the alternative is correct is an inaccurate statement.
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7.87 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 ofin the words of the problem.
d. A Mini tab printout of the analysis is shown (next column). Locate the test statistic andon 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?
Time required to complete a task. When a person is asked, “How much time will you require to complete this task?” cognitive theory posits that people (e.g., a business consultant) will typically underestimate the time required. Would the opposite theory hold if the question was phrased in terms of how much work could be completed in a given amount of time? This was the question of interest to researchers writing in Applied Cognitive Psychology (Vol. 25, 2011). For one study conducted by the researchers, each in a sample of 40 University of Oslo students was asked how many minutes it would take to read a 32-page report. In a second study, 42 students were asked how many pages of a lengthy report they could read in 48 minutes. (The students in either study did not actually read the report.) Numerical descriptive statistics (based on summary information published in the article) for both studies are provided in the accompanying table.
a. The researchers determined that the actual mean time it takes to read the report in\(\mu = 48\) minutes. Is there evidence to support the theory that the students, on average, overestimated the time it would take to read the report? Test using\(\alpha = 0.10\).
b. The researchers also determined that the actual mean number of pages of the report that is read within the allotted time is\(\mu = 32\)pages. Is there evidence to support the theory that the students, on average, underestimated the number of report pages that could be read? Test using\(\alpha = 0.10\)
c. The researchers noted that the distribution of both estimated time and number of pages is highly skewed (i.e., not normally distributed). Does this fact impact the inferences derived in parts a and b? Explain.
A random sample of n observations is selected from a normal population to test the null hypothesis that µ=10.Specify the rejection region for each of the following combinations of \(Ha,\alpha ,\) and n:
a.\(Ha:\)µ\( \ne 10;\alpha = .05.;n = 14\)
b.\(Ha:\)µ\( > 10;\alpha = .01;n = 24\)\(\)
c.\(Ha:\)µ\( > 10;\alpha = .10;n = 9\)
d.\(Ha:\)µ <\(10:\alpha = .01;n = 12\)
e.\(Ha:\)µ\( \ne 10;\alpha = .10;n = 20\)
f. \(Ha:\)µ<\(10;\alpha = .05;n = 4\)
We reject the null hypothesis when the test statistic falls in the rejection region, but we do not accept the null hypothesis when the test statistic does not fall in the rejection region. Why?
Radon exposure in Egyptian tombs. Refer to the Radiation Protection Dosimetry (December 2010) study of radon exposure in Egyptian tombs, Exercise 6.30 (p. 349). The radon levels—measured in becquerels per cubic meter (\({{Bq} \mathord{\left/ {\vphantom {{Bq} {{m^3}}}} \right. \\} {{m^3}}}\) )—in the inner chambers of a sample of 12 tombs are listed in the table shown below. For the safety of the guards and visitors, the Egypt Tourism Authority (ETA) will temporarily close the tombs if the true mean level of radon exposure in the tombs rises to 6,000\({{Bq} \mathord{\left/ {\vphantom {{Bq} {{m^3}}}} \right. \\} {{m^3}}}\) . Consequently, the ETA wants to conduct a test to determine if the true mean level of radon exposure in the tombs is less than 6,000\({{Bq} \mathord{\left/ {\vphantom {{Bq} {{m^3}}}} \right. \\} {{m^3}}}\) , using a Type I error probability of .10. An SPSS analysis of the data is shown at the bottom of the page. Specify all the elements of the test: \({H_0}\,,{H_a}\) test statistic, p-value,\(\alpha \) , and your conclusion.
50 910 180 580 7800 4000 390 12100 3400 1300 11900 110
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