BA 275Practice Final Examination

Winter 2007

Practice Final Examination

  1. Re-do all quizzes and two midterm examinations.
  1. Re-do all examples in the lecture slides and all weekly review problems.
  1. Dollar Car Rental Co. was originally named Dollar a Day Car Rental because they charged $1.00 per day to rent a car, plus a charge per mile driven. Many customers complained that the odometers on Dollar’s cars recorded more miles than were actually driven. To evaluate these complaints you take a random sample of 6 Dollar’s cars, drive them on a carefully measured 100-mile course, and record the miles driven as registered by the odometers. The results are 100, 105, 109, 102, 107, and 101, with the sample standard deviation around 3.578.
  1. Using these sample results, construct a 95% confidence interval for the population mean miles recorded by all Dollar cars for a 100-mile trip.
  2. As a legal consultant hired by the group of the customers who complained about the odometers, do you have enough evidence to support your clients’ claim? State your hypotheses (H0 vs. Ha), rejection region and both statistical and substantive conclusions.
  1. A study of class attendance and grades among sophomores at OSU showed that in general students who attended a higher percent of their classes earned higher grades. Class attendance explained 16% of the total variation in grade index among the sophomores studied. What is the numerical value of the sample correlation between percent of classes attended and grade index?
  1. Stock-market analysts are keenly interested in determining what factors influence the price of a stock. After some examination, a statistician hypothesized that a stock price (Y in $) would be affected by its quarterly dividends (X1 in $), its price/earnings ratio (X2), and the interest rate of treasury bills (X3 in %). The values of the relevant variables were observed for a period of 40 quarters. When the data were run on STATGRAPHICS, the accompanying printout was created.

Multiple Regression Analysis

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Standard T

Parameter Estimate Error Statistic P-Value

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Constant 17.3925 5.5254 ? 0.0033

X1 ? 7.5016 5.5054 0.0000

X2 -0.4158 0.5228 ? ?

X3 0.57091 ? 1.3982 0.0621

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Analysis of Variance

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Source Sum of Squares Df Mean Square F-Ratio P-Value

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Model ? ?

Residual 180.095 ?

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Total (Corr.) 697.040 ?

R-squared = ??? percent

R-squared (adjusted for d.f.) = 73.22 percent

Standard Error of Est. = 2.237

  1. What is the estimated value of 1?
  2. What is the value of SSModel? What does SSModel mean?
  3. What percentage of total variation in the stock price has NOT been explained by the regression model?
  4. What is the value of the sample correlation?
  5. Test the hypothesis H0: 1 = 0 vs. Ha: 1 0 at the 5% level of significance. Based on the regression results above, what is the statistical conclusion?
  6. Test the hypothesis H0: 3 = 0 vs. Ha: 3 0. Based on the regression results above, which of the following statement is true?

A)Do not reject H0 at the 5% significance level, but reject H0 at the 1% significance level.

B)Reject H0 at the 5% significance level, but do not reject H0 at the 1% significance level.

C)Reject H0 at the 10% significance level, but do not reject H0 at the 5% significance level.

D)Do not reject H0 at the 1% significance level but reject H0 at the 5% significance level.

E)None of the above.

  1. When the interest rate goes up, do you expect the stock price to go up or down? Why?
  2. What is the estimated stock price given the interest rate of treasury bills is 4 (in %), its price/earnings ratio is1.34, and its quarterly dividends are 2.5 dollars.

Answer Key: Practice Final Examination

  1. , s=3.578, and n=6.
  1. (100.245, 107.755) with degrees of freedom = 6-1=5.
  2. H0:  = 100 vs. Ha:  > 100

It is an upper tailed test with small sample (n=6). We should use the test statistic . The rejection region is: Reject H0 if t > t with degrees of freedom n-1. Since  is not given, we will simply use =5% and, therefore, the rejection region is: Reject H0 if t > t= 2.015

The statistical conclusion is to reject the null hypothesis since t=2.7384 > the upper bound 2.015.

The substantive conclusion is: as a legal consultant representing the group of the customers who complained about the odometers, we now have enough evidence to support the clients' claim because the sample supports the hypothesis that is greater than 100.

  1. The statement "Class attendance explained 16% of the total variation in grade index among the sophomores studied." means r2 = 0.16. The correlation between percent of classes attended and grade index is r = 0.4, or 40%.

5.

Multiple Regression Analysis

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Standard T

Parameter Estimate Error Statistic P-Value

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Constant 17.3925 5.5254 3.1477 0.0033

X1 41.2993 7.5016 5.5054 0.0000

X2 -0.4158 0.5228 -0.7953 ?

X3 0.57091 0.4083 1.3982 0.0621

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Analysis of Variance

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Source Sum of Squares Df Mean Square F-Ratio P-Value

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Model 516.945 3

Residual 180.095 36

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Total (Corr.) 697.040 39

R-squared = 74.163 percent

R-squared (adjusted for d.f.) = 73.22 percent

Standard Error of Est. = 2.237

  1. Estimated 1= (Standard Error) * (T Statistic) = (7.5016) (5.5054) = 41.2993
  2. SSModel=SSTotal – SSResidual = 697.040 – 180.095 = 516.945. It means that the amount of uncertainty (or variation) that has been explained by the model.
  3. , which means that 74.163% of total variation in the stock price has been explained by the regression model. Therefore, we still have about 100%-74.163% = 25.837% of total variation in the stock price that can NOT be explained by the regression model.
  4. The value of sample of correlation is
  1. Test the hypothesis H0: 1 = 0 vs. Ha: 1 0 at the 5% level of significance. Since the p-value is 0.0000 <  = 5%, the statistical conclusion is to reject the null hypothesis. That is, we have sufficient evidence to say that 1 0.
  2. Test the hypothesis H0: 3 = 0 vs. Ha: 3 0. The p-value is 0.0621. Since the p-value is greater than  = 1% and 5%, but is less than  = 10%, the null hypothesis should be rejected at  = 10%, but should not be rejected at  = 1% and 5%. Answer: C.

Additional Question: What is the p-value of the test H0: 3 = 0 vs. Ha: 3 > 0?

Answer: Since , there is evidence to support the alternative and it is necessary to calculate the p-value. The p-value for a one-sided test is half of the p-value from the printout. In other words, 0.0621 / 2 = 0.03105.

  1. The coefficient associated with the independent variable (interest rate) is 3, whose estimate is . Since , it indicates that the stock price should go up when the interest rate goes up.
  2. Given the interest rate of treasury bills is 4% (X3 = 4), its price/earnings ratio is1.34 (X2 = 1.34), and its quarterly dividends are 2.5 dollars (X1=2.5), the estimated stock price is 17.3925 + 41.2993 (2.5) - 0.4158 (1.34) + 0.57091 (4) = $122.37.

Hsieh, P-H1