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a collection of statistics problems

1. A taxi company manager is trying to decide whether the use of radial tires instead of regular belted tires improves fuel economy. Twelve cars were equipped with radial tires and driven over a prescribed test course. Without changing drivers, the same cars were then equipped with regular belted tires and driven once again over the test course. The gasoline consumption, in kilometers per liter, is shown in the table below.

Kilometers per Liter
Car Radial Tires Belted Tires
1 4.2 4.1
2 4.7 4.9
3 6.6 6.2
4 7.0 6.9
5 6.7 6.8
6 4.5 4.4
7 5.7 5.7
8 6.0 5.8
9 7.4 6.9
10 4.9 4.7
11 6.1 6.0
12 5.2 4.9

a. Assuming that distribution of differences in kilometers per liter is approximately normal, can we conclude that cars equipped with radial tires give better fuel economy than those equipped with belted tires? Use a .05 level of significance.

Complete the following:
1. State H0.
2. State H1.
3. State the value of ?.
4. State the value of the test statistic.
5. State the p-value.
6. State the decision in terms of H0 and why.
7. State the decision in terms of the problem.

b. Construct a 95% confidence interval estimate of the difference in kilometers per liter. Do the results of parts (a) and (b) agree? Explain why or why not.

c. Suppose you use a .01 level of significance instead of a .05 level. Without doing the problem again, would the result be different from that in part (a). Explain your answer.

d. Is it important that the same cars were driven with both radial and belted tires, that drivers didn't change when the belted tires replaced the radial tires, and that the same course was used in all tests? Explain why or why not.

2. Most air travelers now use e-tickets. Electronic ticketing allows passengers to not worry about a paper ticket, and it costs the airline companies less to handle than paper ticketing. However, in recent times, the airlines have received complaints from passengers regarding their e-tickets, particularly when connecting flights and a change of airlines were involved. To investigate the problem an independent watchdog agency contacted a random sample of 20 airports and collected information on the number of complaints the airport had with e-tickets during the month of March. The information is shown in the table below.

14 14 16 12 12 14 13 16 15 14
12 15 15 14 13 13 12 13 10 13

a. Assuming that the data are approximately normally distributed, is there sufficient evidence for the watchdog agency to conclude that the mean number of complaints per airport is less than 15 per month? Use a .05 level of significance.
Complete the following:
1. State H0.
2. State H1.
3. State the value of ?.
4. State the value of the test statistic.
5. State the p-value.
6. State the decision in terms of H0 and why.
7. State the decision in terms of the problem.

b. Is the normality assumption in part a necessary? Explain your answer.

c. Using a graphical approach discussed in the course, determine whether or not the assumption of normality appears to be valid. Show your graph and explain your answer.

3. A recent insurance industry report claimed that 40 percent of those persons involved in minor traffic accidents this year have been involved in at least one other traffic accident in the last five years. An advisory group decided to investigate this claim, believing it was too large. A sample of 200 traffic accidents this year showed 74 persons were also involved in another accident within the last five years.

a. At the .01 level of significance is there evidence that the advisory group is correct?

Complete the following:
1. State H0.
2. State H1.
3. State the value of ?.
4. State the value of the test statistic.
5. State the p-value.
6. State the decision in terms of H0 and why.
7. State the decision in terms of the problem.

b. Is it appropriate to use the z-statistic for this test? Explain your answer.

c. Explain the meaning of the p-value in this problem.

4. The Damon family owns a large grape vineyard in western New York along Lake Erie. The grapevines must be sprayed at the beginning of the growing season to protect against various insects and diseases. Two new insecticides have been marketed, Pernod 5 and Action. To test their effectiveness, three long rows were selected and sprayed with Pernod 5, and three others were sprayed with Action. When the grapes ripened, 400 of the vines treated with Pernod 5 were checked for infestation. Likewise, a sample of 400 vines sprayed with Action was checked. The results are as follows:

Insecticide Number of Vines Checked
(Sample Size) Percent of
Infected Vines
Pernod 5 400 6%
Action 400 9%

a. At the .05 level of significance, can we conclude that there is a difference in the proportion of vines infested using Pernod 5 as opposed to Action?

Complete the following:
1. State H0.
2. State H1.
3. State the value of ?.
4. State the value of the test statistic.
5. State the p-value.
6. State the decision in terms of H0 and why.
7. State the decision in terms of the problem.

b. Suppose the sample sizes were both 600 instead of 400 and the percentages of infected plants in the samples remain the same for each type spray. Does that change the conclusion you reached in part (a)? How?

c. Discuss the effect that sample size had on the outcome of this analysis and, in general, on the effect sample size plays in hypothesis-testing.

5. The manufacturer of an MP3 player wanted to know whether a 10 percent reduction in price is enough to increase the sales of the product. To investigate, the owner randomly selected eight outlets and sold the MP3 player at the reduced price. At seven randomly selected outlets, the MP3 player was sold at the regular price. Reported below is the number of units sold last month at the sampled outlets.

Regular Price Reduced Price
138 128
121 134
88 152
115 135
141 114
125 106
96 112
120

a. At the .01 level of significance, can the manufacturer conclude that the price reduction resulted in an increase in sales? Assume that the populations are approximately normally distributed.

Complete the following:
1. State H0.
2. State H1.
3. State the value of ?.
4. State the value of the test statistic.
5. State the p-value.
6. State the decision in terms of H0 and why.
7. State the decision in terms of the problem.

b. Construct separate 99% confidence interval estimates of the mean number of sales at regular price and at the reduced price.

c. Do the results of part (b) agree with the results of part (a)? Explain why or why not.

6. A student team in a business statistics course conducted an experiment to test the download times of the three different types of computers (Mac, iMac, and Dell) available at the university library. The students randomly selected one computer of each type. The students went to the Microsoft game Web site and clicked on the download link for the NBA game. The time (in seconds) between clicking on the link and the completion of the download was recorded. After each download, the file was deleted, and the trash folder was emptied. A total of 30 downloads were completed in random order. The results are shown below. NOTE: You can copy and paste the data into Excel/PHStat.

Mac iMac Dell
156 160 236
166 165 238
148 184 257
160 192 242
139 197 282
151 172 253
158 189 270
167 179 256
142 200 267
219 193 259

a. One assumption of ANOVA is that the data are approximately normally distributed. Do the download times for each type computer appear to be approximately normally distributed? Support your answer with appropriate calculations or graphs.

b. Another assumption of ANOVA is that the variances of the populations are equal. At the .05 level of significance, is there evidence of a difference in the variations in the download times for the three types of computers?

Complete the following:
1. State H0.
2. State H1.
3. State the value of ?.
4. State the value of the test statistic.
5. State the p-value.
6. State the decision in terms of H0 and why.
7. State the decision in terms of the problem.

c. At the .05 level of significance, is there evidence of a difference in the mean download times for the three computers?

Complete the following:
1. State H0.
2. State H1.
3. State the value of ?.
4. State the value of the test statistic.
5. State the p-value.
6. State the decision in terms of H0 and why.
7. State the decision in terms of the problem.

d. If appropriate, use the Tukey-Kramer procedure to determine which download times differ significantly.

e. Based on the above, which computer should be chosen if you are interested in the shortest download time?

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The solution provides detailed explanation to multiple statistics questions.

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