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Statistics Problems and Solutions for Consulting Firm, Recycled Paper Dealer, Traffic Volume, Batter

## Problem 1: Consulting Firm Health Care Management Proposals

1.A consulting firm often submits health care management proposals for projects in competition with one particular firm of very similar capabilities. A relatively new vice president in charge of the health care division has submitted 16 proposals against the competitor. The firm has won contracts in only 5 of these cases, with the competitor winning the other 11. The president of the firm, noting this fact, concludes that the data prove that the vice president cannot win half the time against the competitor. At 95% level of confidence, do you agree with the president’s conclusion? Please show the necessary steps and explain your results.

2.A dealer in recycled paper places empty trailers at various sites; these are gradually filled by individuals who bring in old newspapers and the like. The trailers are picked up (and replaced by empties) on several schedules. One such schedule involves pickup every second week. This schedule is viable if the average amount of recycled paper is more than 1600 cubic feet per two-week period. The dealer’s records for 22 two-week periods show the following volumes (in cubic feet) at a particular site:

1660    1820    1590    1440    1730    1680    1750    1720    1900    1690    1850    1770
1570    1700    1900    1800    1770    2010    1580    1620    1690    1510    1420    1400

a.Please construct and interpret the 90% and 99% confidence intervals for the population mean amount of recycled paper clearly showing all the necessary steps. What would be the probability that the population mean amount of paper could be greater than the upper limit of the 99% confidence interval? Please explain which confidence interval is wider and why. What do you think will happen to the margins of error and the confidence intervals if the sample size was to be increased to 30 observations? Please justify your answer.

b.Based on the confidence intervals you have constructed do you think the proposed schedule is viable? Please explain how arrived at your answer.

c.Using the given data and at 1% level of significance, please test whether the proposed schedule is viable. Show the necessary steps and explain your conclusion. Is your test result consistent with the conclusion you arrived at based on the 99% confidence interval you constructed under question (a) above? Please explain.

3.The main access road to suburban shopping mall sometimes becomes severely congested. On weekdays, excluding holidays, the average number of vehicles going toward the mall between 9:00 AM and 7:00 PM that pass a counter is 11,260. The highway department tried to improve traffic flow by changing stoplight cycles and improving turn lanes. For the first five non-holiday weekdays after the changes, the volumes were 10,690, 11,452, 12,316, 12, 297, and 12,647. A local politician who reviewed the results said that the data proved there had been significant improvement in traffic volume. At 5% level of significance, is this a reasonable conclusion to draw from the given data? Please show the necessary steps and explain your conclusion.  What will happen to the probability of type II error if you were to redo the test at 1% level of significance?

4.The battery pack of a hand calculator is designed to perform at least 20,000 calculations before needing recharging. A new quality-control manager hired by the manufacturer is concerned that the pack may not be working for as long as the specification states. A sample of 114 packs resulted in an average of 19,695 calculations with estimated population standard deviation of 1,103.The quality-control manager concluded that her concerns are indeed confirmed by these data. At 10% level of significance, do you agree with the manager’s conclusion? Does your conclusion change at 1% level of significance? Please show your steps and explain your conclusion in each case. Also explain the difference between the two tests.

5.A simple linear regression model relating a bank lending interest rate and investment in physical capital by companies is stated as:

a.Which variable (lending interest rate or investment in physical capital) do you think should be the dependent variable in this regression model? Please justify your answer.

b.What sign would you expect for the slope of this regression model for interest rate and investment in physical capita? Please justify your answer.

c.What is the role of the error term in the simple linear regression model like the one stated above? Please state at least three examples of the factors that you think belong in the error term of the simple linear regression model stated above briefly justifying each example.

d.Please explain how you would estimate the relationship between interest rate and investment specified above. [note: you do not need to write any formulas as an answer to this question. Just explain what you will need and the reasoning behind the method you would use to estimate the relationship].

e.For what purpose could you use the estimated simple linear regression model for lending interest rate and investment in physical capital? Why could the simple linear regression model stated above be inadequate for the purpose?

f.Using an example, please explain why a statistically significant relationship in a regression model for two variables does not necessarily imply “cause” and “effect” relationship between the two variables.