The following information was entered into Excel as shown and resulted in the regression printout appearing below.

Sales Price Advertising SalesPrice Advertising

410,000 $3.00$2,650 395,000$3.25$2,500

490,000 $2.00$3,000 365,000$3.65$2,300

415,000 $2.95$2,700 400,000$3.10$2,550

440,000 $2.75$2,800 412,000$3.05$2,600

356,000 $3.50$2,150 350,000$3.65$2,300

SUMMARY OUTPUT
Regression Statistics
Multiple R / 0.9937483
R Square / 0.98753569
Adjusted R Square / 0.98397446
Standard Error / 5295.06881
Observations / 10
ANOVA
Df / SS / MS / F / Significance F
Regression / 2 / 15549835724 / 7774917862 / 277.301739 / 2.162E-07
Residual / 7 / 196264275.7 / 28037753.7
Total / 9 / 15746100000
Coefficients / Standard Error / t Stat / P-value / Lower 95% / Upper 95% / Lower 95.0% / Upper 95.0%
Intercept / 420328.327 / 84673.07886 / 4.96413184 / 0.00162996 / 220108.31 / 620548.342 / 220108.31 / 620548.342
Price / -54923.4091 / 10753.10396 / -5.10767955 / 0.00138761 / -80350.46 / -29496.359 / -80350.46 / -29496.3588
Advertising / 59.7592984 / 20.5942121 / 2.90175211 / 0.02292856 / 11.061725 / 108.456872 / 11.061725 / 108.456872

1. Which of the variables is/are independent variables? (Mark all answers that apply.)

  1. price
  2. advertising
  3. sales
  4. More information is needed to answer this question.

2. Assume that  = 0.025. Which variable(s) is (are) significant in this model? (Mark all answers that apply.)

  1. sales
  2. advertising
  3. price
  4. None of the variables are significant in this model.
  5. More information is needed to answer this question.

3. What percentage of the variation in the dependent variable is “explained” by the explanatory variable(s)?

  1. 98.40%
  2. 98.75%
  3. 99.37%
  4. None of the above is correct.
  5. More information is needed to answer this question.

4. What is the sales forecast if the selling price is $3.65 and advertising is $2,300? Round all coefficients to two decimal places and the sales estimate to the nearest unit.

  1. 357
  2. 221,232
  3. 357,306
  4. 357,906
  5. 365,000

5. What is the chance that this model fit the data purely by accident?

  1. 0.0000002162%
  2. 0.00002162%
  3. 2.162%
  4. 97.838%
  5. 98.750%

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SOLUTIONS

1. Which of the variables is/are independent variables? (Mark all answers that apply.)

  1. price
  2. advertising
  3. sales
  4. More information is needed to answer this question.

a and b are correct. The changes in these variables “explain” the changes in sales which is the dependent variable. Notice this example is different from the one in class where there was only one explanatory variable.

2. Assume that  = 0.025. Which variable(s) is (are) significant in this model? (Mark all answers that apply.)

  1. sales
  2. advertising
  3. price
  4. None of the variables are significant in this model.
  5. More information is needed to answer this question.

b and c are correct. The rule is if the p-value is less than alpha, then reject the null hypothesis. The null hypothesis in this case is that there is no relationship between price and sales and advertising and sales. Since the p-values on advertising and price are smaller than alpha, reject the null and conclude that there is a statistically significant relationship between advertising and sales and between price and sales.

3. What percentage of the variation in the dependent variable is “explained” by the explanatory variable(s)?

  1. 98.40%
  2. 98.75%
  3. 99.37%
  4. None of the above is correct.
  5. More information is needed to answer this question.

a is correct. Note that is the ADJUSTED R Square value

4. What is the sales forecast if the selling price is $3.65 and advertising is $2,300? Round all coefficients to two decimal places and the sales estimate to the nearest unit.

  1. 357
  2. 221,232
  3. 357,306
  4. 357,906
  5. 365,000

c is correct. The value is calculated as follows: Sales = 420328.33 – 54923.41(3.65) + 59.76(2300)

5. What is the chance that this model fit the data purely by accident?

  1. 0.0000002162%
  2. 0.00002162%
  3. 2.162%
  4. 97.838%
  5. 98.750%

b is correct. This is the Significance F statistic. Note that the statistic is expressed in scientific notation, so you move the decimal point seven places to the left. But the answer is expressed in percent which then moves the decimal point two places to the right.