SIMPLE REGRESSION PROBLEMS

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HW Set #4 – SIMPLE REGRESSION PROBLEMS

Note: In problems I, II and III use least squares formula to solve the problem by hand. Do not use the “Regression” feature of Excel. However, you must use Excel for calculations and reporting.

  1. The following table is the ABC company’s data on daily high temperature and number of air-conditioning units sold for eight randomly selected business days during the summer (Use α = 0.05 throughout the problem):

Daily  High                Number of

Day                 Temperature              Units Sold

1                                 83                                4

2                                 88                                5

3                                 73                                1

4                                 76                                0

5                                 92                                5

6                                 79                                3

7                                 81                                2

8                                 77                                2

1)         Fit a least square line to the data.

2)         Interpret the regression coefficients in the context of the problem.

3)         Plot the data and graph the least square line.

4)         Compute SSE and S2.

 

  1. Oil Problem – The price of gas at the pump (in cents) and the price of crude oil (in dollars per barrel) are given in the following table.

1) Plot the Scattergram.

2) Compute the regression coefficients using
Least Squares Method.

3) Write the Least Squares equation.

4) Predict the price of gas at the pump in a year in which

the price of crude oil is $30 per bbl.

5) Graph the Least Squares Line.

6) Interpret the intercept and slope in the context of the

problem.

7) Compute SSE.

8) Compute S2,  the estimator of  σ2, and interpret its

meaning.

9) Compute S, the estimator of s and interpret its

meaning.
10) Is the model useful? Test using (a = 0.05). Draw the

appropriate conclusion in the context of the problem.

11) Develop a 95% confidence interval for the slope of

the population.

12) Compute the coefficient of correlation and interpret

its meaning in the context of the problem.

13) Compute the coefficient of determination and
interpret its meaning in the context of the problem.

III. The sales manager of a company would like to study the effect of the advertising on sales of GameToy.  A sample of 22 cities with approximately equal popula­tion was selected for study during a test period of one month.  Each city was allocated a specific expenditure level ($1000) and sales ($100000) during the test month were determined.  The results are summarized below:

n=22      Σsales = 2695.3         Σsales2 = 355288.93

Σadvertising = 550       Σadvertising2 = 19250                  Σadvertising.sales = 76672

Hint:  First determine which variable is X and which is Y.

1) Use the least square method to find regression coefficients ß0, and ß1.

2) Interpret the meaning of regression coefficients in the context of this problem.

3) Predict the sales for the GameToy for a city in which $20000 has been spent

for advertising.

4) Test whether the straight-line model contributes information for prediction of

the dependent variable? Test using α=0.05.

5) Compute the coefficient of determination and interpret its meaning in the

context of the problem.

6) Compute the sample correlation coefficient and interpret its meaning in the

context of the problem.

7) Compute the variance and define its meaning.

 

  1. Re-solve prob. I using the “Regression” feature of Excel. Compare answers.

 

  1. Re-solve prob. II using the “Regression” feature of Excel. Compare answers.

 

  1. Apply the Excel’s “Regression” feature to the data of the oil problem (problem #2) and obtain the regression output. Answer all the thirteen questions posed in problem 2, based on the Excel’s output.

 

VII.      Part of an Excel output relating X (independent Variable) and Y (dependent variable) is shown below.  Fill in all the blanks marked with “?”.

Summary Output          
           
Regression Statistics          
Multiple R 0.1347        
R Square ?        
Adjusted R Square ?        
Standard Error 3.3838        
Observations ?        
           
ANOVA          
  df SS MS F Significance F
Regression ? 2.7500 ? ? 0.632246859
Residual ? ? 11.45    
Total 14 ?      
           
  Coefficients Standard Error t Stat P-value  
Intercept 8.6 2.2197 ? 0.0019  
x 0.25 0.5101 ? 0.6322  

 

VIII.     Shown below is a portion of a computer output for a regression analysis relating Y (dependent variable) and X (independent variable).

 

Predictor             Coefficient                 Standard Error

Constant                   12.14                                1.27

X                            -0.63                                0.23

ANALYSIS OF VARIANCE

SOURCE                      SS                DF

Regression                37.03                1

Error                           62.97              13

 

  1. Perform a t-test and determine whether or not Y and X are related.  Let a = 0.05.

 

  1. Compute the coefficient of determination and fully interpret its meaning.  Be very specific.

 

 

 

  1. Part of an Excel output relating X (independent Variable) and Y (dependent variable) is shown below. Fill in all the blanks marked with “?”.
 

Summary Output

          
           
Regression Statistics          
Multiple R ?        
R Square 0.5149        
Adjusted R Square ?        
Standard Error 7.3413        
Observations 11        
           
ANOVA          
  df SS MS F Significance F
Regression ? ? ? ? 0.0129
Residual ? ? ?    
Total ? 1000.0000      
           
  Coefficients Standard Error t Stat P-value  
Intercept ? 29.4818 3.7946 0.0043  
x ? 0.7000 -3.0911 0.0129  

 

 

Multiple Choice Questions:

 

Exhibit 1. A sales manager collected the following data on annual sales and years of experience.

Salesperson Experience Sales      
  X Y X_SQ XY Y_SQ
1 1 80 1 80 6400
2 3 97 9 291 9409
3 4 92 16 368 8464
4 4 102 16 408 10404
5 6 103 36 618 10609
6 8 111 64 888 12321
7 10 119 100 1190 14161
8 10 123 100 1230 15129
9 11 117 121 1287 13689
10 13 136 169 1768 18496
  70 1080 632 8128 119082
  1. Refer to Exhibit 1. What is the slope of the least squares line?
  2. 12.86
  3. 4
  4. 12.90
  5. 3
  6. Refer to Exhibit 1. The value of Y-intercept is
  7. 80
  8. 70
  9. 17.9744
  10. none of the above
  11. Refer to Exhibit 1. The Y-intercept in this problem
  12. has a practical meaning
  13. is in cents
  14. has no practical meaning
  15. none of the above
  16. Refer to Exhibit 1. In this problem the slope and intercept parameters are
  17. known
  18. being estimated
  19. not a relevant question
  20. none of the above
  21. Refer to Exhibit 1. The coefficient of determination is
  22. 1.02
  23. – 0.95
  24. 0.96
  25. none of the above
  26. Refer to Exhibit 1. Based on the least squares line estimate the expected sales for a salesperson with 10 years of experience.
  27. $120
  28. $140
  29. $110000
  30. none of the above
  31. Regression Analysis tries to answer:
  32. how a single variable depends on other relevant variables.
  33. if there are differences between distinct populations.
  34. how several variables depend on each other.
  35. none of the above.

 

  1. The error term in simple regression represents:
  2. the difference between the estimated regression line & population line.
  3. the vertical distance from any point to the mean value of Y’s.
  4. the vertical distance from any point to the regression line.
  5. none of the above.

 

  1. 9. The standard error of the estimate () is essentially:
  2. the mean of the residuals
  3. the standard deviation of the residuals
  4. the mean of the explanatory variable
  5. the standard deviation of the explanatory variable
  6. none of the above

 

  1. The variance of regression is a measure of
  2.   variation of the regression coefficients
  3. variation around the fitted line of regression
  4.   variation around the arithmetic mean
  5.  relationship between the dependent and independent variables.

 

  1. What does y represent?
  2. The independent variable.                 b.  The random error component.
  3.   The y‑intercept of the line.             d.   The dependent variable.

 

  1. What value of β1 in the model y = β0 + β1x, would indicate that the variable y is completely unrelated to the variable x.
  2. β1 = 0                                  b.  β1 = 1
  3. β1 = ‑1                                d.  β1 = 0.5

 

  1. In linear regression, we fit the least squares line to a set of values (or points on a scatterplot).  The distance from the line to a point is called the:
  2. fitted value             residual                    c.  correlation
  3. covariance e. none of the above
  4. In a simple linear regression analysis, the following sum of squares are produced:

 

The proportion of the variation in y that is explained by the variation in x is:

  1. 20%   80%            c.  25%            d.  50%            e.  none of the above

 

  1. Given the least squares regression line 8 – 3x,
  2. the relationship between x and y is positive
  3. the relationship between x and y is negative
  4. as x increases, so does y
  5. as x decreases, so does y
  6. there is no relationship between x and y

 

  1. In regression analysis, the variables used to help explain or predict the response variable are called:
  2. the independent variables the dependent variables
  3. the regression variables d. the statistical variables

 

  1. In regression analysis, if there are several explanatory variables, it is called:
  2. simple regression multiple regression
  3. compound regression d. composite regression
  4. none of the above

 

  1. In regression analysis, the variable we are trying to explain or predict is called:
  2. the independent variable the dependent variable
  3. the regression variable                         d. the statistical variable
  4. the residual variable

 

  1. Correlation is a summary measure that indicates:
  2. a curved relationship among the variables
  3. the rate of change in Y for a one unit change in X
  4. the strength of the relationship between pairs of variables
  5. the magnitude of difference between two variables
  6. none of the above

 

  1. A correlation value of zero indicates.
  2. a strong linear relationship a weak linear relationship
  3. no linear relationship d. a perfect linear relationship
  4. none of the above

 

  1. The coefficient of determination () can be interpreted as the fraction (or percent) of variation:
  2. of the explanatory variable explained by the independent variable
  3. of the explanatory variable explained by the regression line
  4. of the response variable explained by the regression line
  5. of the error explained by the regression line
  6. none of the above

 

  1. The correlation value ranges from  to                    .
  2. 0; +1                                –1; +1
  3. –2; +2 d. –¥; + ¥                   e. none of the above

 

 

  1. In a simple regression problem, the sign of the correlation coefficient always depends on
  2. coefficient of determination
  3. the sign of Y_intercept
  4. the sign of slope
  5. none of the above

 

  1. In simple regression, what distribution is used to test the usefulness of the entire model?
  2. standard normal distribution
  3. t distribution
  4. F distribution
  5. none of the above

 

  1. If the coefficient of determination is 0.81, the coefficient of correlation
  2. is 0.6561
  3. must be 0.9
  4. must be positive
  5. none of these answers is correct.

 

  1. If two variables, x and y, have a strong linear relationship, then
  2. there may or may not be any causal relationship between x and y
  3. x causes y to happen
  4. y causes x to happen
  5. none of these answers is correct.

 

  1. In a regression analysis if SSE = 200 and SSR = 300, then the coefficient of determination is
  2. 0.6667
  3. 0.6000
  4. 0.4000
  5. 1.5000

 

  1. A regression analysis between sales (in $1000) and price (in dollars) resulted in the following equation

 

= 50,000 – 8x

 

The above equation implies that an

  1. increase of $1 in price is associated with a decrease of $8 in sales
  2. increase of $8 in price is associated with an increase of $8,000 in sales
  3. increase of $1 in price is associated with a decrease of $42,000 in sales
  4. increase of $1 in price is associated with a decrease of $8000 in sales

 

  1. In a simple regression analysis (where y is a dependent and x an independent variable), if the y intercept is positive, then
  2. there is a positive correlation between x and y
  3. there is a negative correlation between x and y
  4. if x is increased, y must also increase
  5. none of these answers is correct.

 

  1. A procedure used for finding the equation of a straight line that provides the best approximation for the relationship between the independent and dependent variables is
  2. correlation analysis
  3. mean squares method
  4. least squares method
  5. most squares method
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