C22.0103 FINAL EXAM

Name:______

Write your answers to the first five questions on the attached sheets, in the spaces provided. Circle the choice which best answers questions 6-15. Do not write anything else on this page (besides your name and the circles). When you are finished, hand in the entire exam (both question sheets and answer sheets). Please do not remove any pages from the exam paper. There are 15 questions, each worth 5 points. Everyone receives 25 points for free. Good Luck!

1) WRITTEN 11) (A) (B) (C) (D) (E)

2) WRITTEN 12) (A) (B) (C) (D) (E)

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8) (A) (B) (C) (D) (E)

9) (A) (B) (C) (D) (E)

10) (A) (B) (C) (D) (E)

Answer For Question 1: Answer for Question 2: Answer for Question 3: Answer for Question 4: Answer for Question 5: C22.0103 FINAL EXAM

Questions 1) - 5) are based on a data set on the annual Sales of 20 large automobile manufacturers (in billions of US Dollars). The two explanatory variables are Advertising expenditures for the previous year, and Research expenditures for the previous five years, in Millions of US Dollars. Figure 1 gives a fitted line plot for Sales vs. Advertising. This is followed by the corresponding Minitab Simple Linear Regression output.

Fig 1: Sales vs. Advertising For 20 Large Automobile Manufacturers Sales = - 2.664 + 5.439 Advertising

38 S 0.641792 R-Sq 98.1% 36 R-Sq(adj) 98.0% 34

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S 28

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20 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 Advertising Regression Analysis: Sales versus Advertising

The regression equation is Sales = - 2.66 + 5.44 Advertising

Predictor Coef SE Coef T P Constant -2.664 1.070 -2.49 0.023 Advertising 5.4388 0.1770 30.72 0.000

S = 0.641792 R-Sq = 98.1% R-Sq(adj) = 98.0%

Analysis of Variance

Source DF SS MS F P Regression 1 388.73 388.73 943.77 0.000 Residual Error 18 7.41 0.41 Total 19 396.15

1) A) Based on the fitted line plot, do you feel comfortable with the simple linear regression model where X=Advertising, Y=Sales? (1 point).

B) Predict the sales for a company that spent $20 Million on Advertising last year. (2 Points).

C) Give an interpretation of the slope of the fitted regression line, in terms of Sales and Advertising. Be careful to take account of the units for X and Y. (2 points). 2) Based on the simple linear regression output above:

A) Is there evidence that the true intercept is different from zero, at the 1% level of significance? (1 Point).

B) Test the null hypothesis that the true coefficient of Advertising is zero. Should this be a two-tailed, left-tailed or right-tailed test? Why? (4 Points).

3) The table below gives Minitab output for the multiple regression of Sales vs. Advertising and Research. Notice that the estimated coefficient of Advertising is different in the multiple regression than it was in the simple regression. But has it changed by a lot? To answer this, test whether the coefficient in the multiple regression is significantly different from the value you got in the simple regression. Treat the value from the simple regression as a fixed null hypothesis value. Use a significance level of 5%.

Regression Analysis: Sales versus Advertising, Research

The regression equation is Sales = - 3.60 + 5.44 Advertising + 0.152 Research

Predictor Coef SE Coef T P Constant -3.604 1.611 -2.24 0.039 Advertising 5.4432 0.1790 30.40 0.000 Research 0.1525 0.1937 0.79 0.442

S = 0.648683 R-Sq = 98.2% R-Sq(adj) = 98.0%

Analysis of Variance

Source DF SS MS F P Regression 2 388.99 194.50 462.22 0.000 Residual Error 17 7.15 0.42 Total 19 396.15

4) Based on both the simple regression and the multiple regression output, determine whether Research is an important variable for predicting Sales. Justify your answer as completely as possible. 5) Are the results from the F-test in the multiple regression significant at level . 05? At level .01? Do the results imply that both explanatory variables are needed in the regression? Explain.

Questions 6-15 are general and do not pertain to the regression example above.

6) True or False: In hypothesis testing, if the p-value is .02, then the probability that the null hypothesis is true is .02. A) True B) False

7) Consider a game where you toss three dice. If at least one of the dice comes up 5, you win $5. Otherwise, you lose $1. If you play this game 100 times, independently, what is the probability that your total profit is at most $80? (Retain four digits of accuracy in your calculations). A) .9931 B) .0069 C) .4013 D) .5987 E) None of the Above

8) A sample of size 15 yields a sample mean of 2 and a sample standard deviation of 1.5. If you test the null hypothesis that the population mean is 1, using a right-tailed alternative hypothesis, then: A) The null hypothesis cannot be rejected at level .05 or level .01 B) The null hypothesis can be rejected at level .05 but not at level .01 C) The null hypothesis can be rejected at level .01 but not at level .05 D) The null hypothesis can be rejected at level .05 and at level .01.

9) Suppose we wish to test H 0 :   3 versus H A :   3 based on a sample of size 100. If the sample mean is 3.5 and the sample standard deviation is 2, then the p-value is: A) .0062 B) .0031 C) .4013 D) .0124 E) None of the Above

10) In simple linear regression, if the sample size is 30, the total sum of squares is 20, the regression sum of squares is 5, what is the residual mean square, s 2 ? A) .3750 B) .7500 C) .5357 D) .2500 E) None of the Above

11) In a multiple linear regression with two explanatory variables, if the sample size is 21, and the level of significance is 5%, what is the critical value for a left-tailed t-test of the null hypothesis that the true coefficient of the first explanatory variable is zero? A) 1.645 B) 1.729 C) 1.734 D) 1.740 E) None of the Above

12) Suppose that the 95% confidence interval for the mean based on a sample of size 12 is (2.6, 3.1). What is the value of the sample standard deviation? A) .4418 B) .1276 C) .1136 D) .3935 E) None of the Above 13) In multiple regression, if the residual sum of squares is 20 and the total sum of squares is 50, then the coefficient of determination R 2 is: A) .4 B) .6 C) .6325 D) .7746 E) None of the Above.

14) Suppose a coin is tossed 9 times. The number of possible sequences of Heads and Tails (for example, HHHHHTTTT, HTHTHTHTH) is: A) 512 B) 81 C) 18 D) 9 E) None of the Above.

15) If the true standard error of the sample mean is .02 and the sample size is 100, then the population standard deviation is: A) 2 B) .2 C) .02 D) .002 E) .0002