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Chapter 8: Q97SE (page 452)
Consider two events A and B, with ,,and
a.Are A and B mutually exclusive?
b.Are A and B independent?
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Ages of self-employed immigrants. Is self-employment for immigrant workers a faster route to economic advancement in the country? This was one of the questions studied in research published in the International Journal of Manpower (Vol. 32, 2011). One aspect of the study involved comparing the ages of self-employed and wage-earning immigrants. The researcher found that in Sweden, native wage earners tend to be younger than self-employed natives. However, immigrant wage earners tend to be older than self-employed immigrants. This inference was based on the table's summary statistics for male Swedish immigrants.
Self-employed immigrants | Wage-earning immigrants | |
Sample Size | ||
Mean |
Source: Based on L. Andersson, "Occupational Choice and Returns to Self-Employment Among Immigrants," International Journal of Manpower, Vol. 32, No. 8, 2011 (Table I).
a. Based on the information given, why is it impossible to provide a measure of reliability for the inference "Self-employed immigrants are younger, on average, than wage-earning immigrants in Sweden"?
b. What information do you need to measure reliability for the inference, part a?
c. Give a value of the test statistic that would conclude that the true mean age of self-employed immigrants is less than the true mean age of wage-earning immigrants if you are willing to risk a Type I error rate of .01.
d. Assume that s, the standard deviation of the ages is the same for both self-employed and wage-earning immigrants. Give an estimate of s that would lead you to conclude that the true mean age of self-employed immigrants is less than the true mean age of wage-earning immigrants using .
e. Is the true value of s likely to be larger or smaller than the one you calculated in part d?
Homework assistance for accounting students. How much assistance should accounting professors provide students for completing homework? Is too much assistance counterproductive? These were some of the questions of interest in a Journal of Accounting Education (Vol. 25, 2007) article. A total of 75 junior-level accounting majors who were enrolled in Intermediate Financial Accounting participated in an experiment. All students took a pretest on a topic not covered in class; then, each was given a homework problem to solve on the same topic. However, the students were randomly assigned different levels of assistance on the homework. Some (20 students) were given the completed solution, some (25 students) were given check figures at various steps of the solution, and the rest (30 students) were given no help. After finishing the homework, each student was given a posttest on the subject. One of the variables of interest to the researchers was the knowledge gain (or test score improvement), measured as the difference between the posttest and pretest scores. The sample means knowledge gains for the three groups of students are provided in the table.
a. The researchers theorized that as the level of homework assistance increased, the test score improvement from pretest to post test would decrease. Do the sample means reported in the table support this theory?
b. What is the problem with using only the sample means to make inferences about the population mean knowledge gains for the three groups of students?
c. The researchers conducted a statistical test of the Hypothesis to compare the mean knowledge gain of students in the "no solutions" group with the mean knowledge gain of students in the "check figures" group. Based on the theory, part a sets up the null and alternative hypotheses for the test.
d. The observed significance level of the t-test of the part was reported as Using , interpret this result.
e. The researchers conducted a statistical test of the hypothesis to compare the mean knowledge gain of students in the "completed solutions" group with the mean knowledge gain of students in the "check figures" group. Based on the theory, part a sets up the null and alternative hypotheses for the test.
f. The observed significance level of the role="math" localid="1652694732458" -test of part e was reported as 1849. Using , interpret this result.
g. The researchers conducted a statistical test of the Hypothesis to compare the mean knowledge gain of students in the "no solutions" group with the mean knowledge gain of students in the "completed solutions" group. Based on the theory, part a sets up the null and alternative hypotheses for the test.
h. The observed significance level of the -test of the part was reported as Using role="math" localid="1652694677616" , interpret this result.
Drug content assessment. Scientists at GlaxoSmithKlineMedicines Research Center used high-performance liquidchromatography (HPLC) to determine the amountof drug in a tablet produced by the company (Analytical
Chemistry, Dec. 15, 2009). Drug concentrations (measuredas a percentage) for 50 randomly selected tablets are listedin the table below and saved in the accompanying file.
a. Descriptive statistics for the drug concentrations areshown at the top of the XLSTAT printout on the nextpage. Use this information to assess whether the dataare approximately normal.
b. An XLSTAT normal probability plot follows. Use thisinformation to assess whether the data are approximatelynormal.
91.28 92.83 89.35 91.90 82.85 94.83 89.83 89.00 84.62
86.96 88.32 91.17 83.86 89.74 92.24 92.59 84.21 89.36
90.96 92.85 89.39 89.82 89.91 92.16 88.67 89.35 86.51
89.04 91.82 93.02 88.32 88.76 89.26 90.36 87.16 91.74
86.12 92.10 83.33 87.61 88.20 92.78 86.35 93.84 91.20
93.44 86.77 83.77 93.19 81.79
Descriptive statistics(Quantitative data) | |
Statistic | Content |
Nbr.of Observation | 50 |
Minimum | 81.79 |
Maximum | 94.83 |
1st Quartile | 87.2725 |
Median | 89.375 |
3rd Quartile | 91.88 |
Mean | 89.2906 |
Variance(n-1) | 10.1343 |
Standard deviation(n-1) | 3.1834 |
Question: Refer to the Bulletin of Marine Science (April 2010) study of lobster trap placement, Exercise 6.29 (p. 348). Recall that the variable of interest was the average distance separating traps鈥攃alled trap-spacing鈥攄eployed by teams of fishermen. The trap-spacing measurements (in meters) for a sample of seven teams from the Bahia Tortugas (BT) fishing cooperative are repeated in the table. In addition, trap-spacing measurements for eight teams from the Punta Abreojos (PA) fishing cooperative are listed. For this problem, we are interested in comparing the mean trap-spacing measurements of the two fishing cooperatives.
BT Cooperative | 93 | 99 | 105 | 94 | 82 | 70 | 86 | |
PA Cooperative | 118 | 94 | 106 | 72 | 90 | 66 | 98 |
Source: Based on G. G. Chester, 鈥淓xplaining Catch Variation Among Baja California Lobster Fishers Through Spatial Analysis of Trap-Placement Decisions,鈥� Bulletin of Marine Science, Vol. 86, No. 2, April 2010 (Table 1).
a. Identify the target parameter for this study.b. Compute a point estimate of the target parameter.c. What is the problem with using the normal (z) statistic to find a confidence interval for the target parameter?d. Find aconfidence interval for the target parameter.e. Use the interval, part d, to make a statement about the difference in mean trap-spacing measurements of the two fishing cooperatives.f. What conditions must be satisfied for the inference, part e, to be valid?
Working on summer vacation. According to a Harris Interactive (July 2013) poll of U.S. adults, about 60% work during their summer vacation. (See Exercise 3.13, p. 169.) Assume that the true proportion of all U.S. adults who work during summer vacation is p = .6. Now consider a random sample of 500 U.S. adults.
a. What is the probability that between 55% and 65% of the sampled adults work during summer vacation?
b. What is the probability that over 75% of the sampled adults work during summer vacation?
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