Regression Analysis Question with Solution Sample Assignment
PROBLEM
1. Assume you have noted the following prices for books and the number of pages that each book contains.
Book 
Pages (x) 
Price (y) 
A 
500 
$7.00 
B 
700 
7.50 
C 
750 
9.00 
D 
590 
6.50 
E 
540 
7.50 
F 
650 
7.00 
G 
480 
4.50 
a. 
Develop a leastsquares estimated regression line. 
b. 
Compute the coefficient of determination and explain its meaning. 
c. 
Compute the correlation coefficient between the price and the number of pages. Test to see if x and y are related. Use a = 0.10. 
ANS:
a. 
y= 1.0416 + 0.0099x 
b. 
r ^{2} = .5629; the regression equation has accounted for 56.29% of the total sum of squares 
c. 
rxy = 0.75 t = 2.54 > 2.015 (df = 5); pvalue is between .05 and 0.1; (Excel’s results: pvalue of 0.052); reject H_{o}, and conclude x and y are related 
2. Assume you have noted the following prices for books and the number of pages that each book contains.
Book 
Pages (x) 
Price (y) 
A 
500 
$7.00 
B 
700 
7.50 
C 
750 
9.00 
D 
590 
6.50 
E 
540 
7.50 
F 
650 
7.00 
G 
480 
4.50 
a. 
Perform an F test and determine if the price and the number of pages of the books are related. Let a = 0.01. 
b. 
Perform a t test and determine if the price and the number of pages of the books are related. Let a = 0.01. 
c. 
Develop a 90% confidence interval for estimating the average price of books that contain 800 pages. 
d. 
Develop a 90% confidence interval to estimate the price of a specific book that has 800 pages. 
ANS:
a. 
F = 6.439 < 16.26; pvalue is between 0.1 and 0.2 (Excel’s result: pvalue = .052); do not reject H_{o}; conclude x and y are not related 
b. 
t = 2.5376 < 4.032; pvalue is between 0.1 and 0.2. (Excel’s result: pvalue = .052); do not reject H_{o}; conclude x and y are not related 
c. 
$7.29 to $10.63 (rounded) 
d. 
$5.62 to $12.31 (rounded) 
3. The following data represent the number of flash drives sold per day at a local computer shop and their prices.
Price (x) 
Units Sold (y) 
$34 
3 
36 
4 
32 
6 
35 
5 
30 
9 
38 
2 
40 
1 
a. 
Develop a leastsquares regression line and explain what the slope of the line indicates. 
b. 
Compute the coefficient of determination and comment on the strength of relationship between x and y. 
c. 
Compute the sample correlation coefficient between the price and the number of flash drives sold. Use a= 0.01 to test the relationship between x and y. 
ANS:
a. 
y= 29.7857  0.7286x The slope indicates that as the price goes up by $1, the number of units sold goes down by 0.7286 units. 
b. 
r ^{2} = .8556; the regression equation has accounted for 85.56% of the total sum of squares 
c. 
rxy = 0.92 t = 5.44 < 4.032 (df = 5); pvalue < .01; (Excel’s result: pvalue = .0028); reject H_{o}, and conclude x and y are related 
4. The following data represent the number of flash drives sold per day at a local computer shop and their prices.
Price (x) 
Units Sold (y) 
$34 
3 
36 
4 
32 
6 
35 
5 
30 
9 
38 
2 
40 
1 
a. 
Perform an F test and determine if the price and the number of flash drives sold are related. Let a = 0.01. 
b. 
Perform a t test and determine if the price and the number of flash drives sold are related. Let a = 0.01. 
ANS:
a. 
F = 29.624 > 16.26; pvalue < .01; (Excel’s result: pvalue = .0028); reject H_{o}, x and y are related 
b. 
t = 5.4428 < 4.032; pvalue < .01; (Excel’s result: pvalue = .0028); reject H_{o}, x and y are related 
5. Shown below is a portion of an Excel output for regression analysis relating Y (dependent variable) and X (independent variable).
ANOVA 

df 
SS 

Regression 
1 
110 
Residual 
8 
74 
Total 
9 
184 
Coefficients 
Standard Error 

Intercept 
39.222 
5.943 
x 
0.5556 
0.1611 
a. 
What has been the sample size for the above? 
b. 
Perform a t test and determine whether or not X and Y are related. Let a = 0.05. 
c. 
Perform an F test and determine whether or not X and Y are related. Let a = 0.05. 
d. 
Compute the coefficient of determination. 
e. 
Interpret the meaning of the value of the coefficient of determination that you found in d. Be very specific. 
ANS:
a through d
Summary Output 

Regression Statistics 

Multiple R 
0.7732 
R Square 
0.5978 
Adjusted R Square 
0.5476 
Standard Error 
3.0414 
Observations 
10 
ANOVA 

df 
SS 
MS 
F 
Significance F 

Regression 
1 
110 
110 
11.892 
0.009 
Residual 
8 
74 
9.25 

Total 
9 
184 
Coefficients 
Standard Error 
t Stat 
Pvalue 

Intercept 
39.222 
5.942 
6.600 
0.000 
x 
0.556 
0.161 
3.448 
0.009 
e. 
59.783% of the variability in Y is explained by the variability in X. 
6. Shown below is a portion of a computer output for regression analysis relating Y (dependent variable) and X (independent variable).
ANOVA 

df 
SS 

Regression 
1 
24.011 
Residual 
8 
67.989 
Coefficients 
Standard Error 

Intercept 
11.065 
2.043 
x 
0.511 
0.304 
a. 
What has been the sample size for the above? 
b. 
Perform a t test and determine whether or not X and Y are related. Let a = 0.05. 
c. 
Perform an F test and determine whether or not X and Y are related. Let a = 0.05. 
d. 
Compute the coefficient of determination. 
e. 
Interpret the meaning of the value of the coefficient of determination that you found in d. Be very specific. 
ANS:
a through d
Summary Output 

Regression Statistics 

Multiple R 
0.511 
R Square 
0.261 
Adjusted R Square 
0.169 
Standard Error 
2.915 
Observations 
10 
ANOVA 

df 
SS 
MS 
F 
Significance F 

Regression 
1 
24.011 
24.011 
2.825 
0.131 
Residual 
8 
67.989 
8.499 

Total 
9 
92 
Coefficients 
Standard Error 
t Stat 
Pvalue 

Intercept 
11.065 
2.043 
5.415 
0.001 
x 
0.511 
0.304 
1.681 
0.131 
e. 
26.1% of the variability in Y is explained by the variability in X. 
7. 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 
? 
? 
0.632 
Residual 
? 
? 
11.45 

Total 
14 
? 
Coefficients 
Standard Error 
t Stat 
Pvalue 

Intercept 
8.6 
2.2197 
? 
0.0019 
x 
0.25 
0.5101 
? 
0.632 
ANS:
Summary Output 

Regression Statistics 

Multiple R 
0.1347 
R Square 
0.0181 
Adjusted R Square 
0.0574 
Standard Error 
3.384 
Observations 
15 
ANOVA 


df 
SS 
MS 
F 
Significance F 
Regression 
1 
2.750 
2.75 
0.2402 
0.6322 
Residual 
13 
148.850 
11.45 

Total 
14 
151.600 
Coefficients 
Standard Error 
t Stat 
pvalue 

Intercept 
8.6 
2.2197 
3.8744 
0.0019 
x 
0.25 
0.5101 
0.4901 
0.6322 
8. Shown below is a portion of a computer output for a regression analysis relating Y (dependent variable) and X (independent variable).
ANOVA 

df 
SS 

Regression 
1 
115.064 
Residual 
13 
82.936 
Total 

Coefficients 
Standard Error 

Intercept 
15.532 
1.457 
x 
1.106 
0.261 
a. 
Perform a t test using the pvalue approach and determine whether or not Y and X are related. Let a = 0.05. 
b. 
Using the pvalue approach, perform an F test and determine whether or not X and Y are related. 
c. 
Compute the coefficient of determination and fully interpret its meaning. Be very specific. 
ANS:
a and b
Summary Output 

Regression Statistics 

Multiple R 
0.7623 
R Square 
0.5811 
Adjusted R Square 
0.5489 
Standard Error 
2.5258 
Observations 
15 
ANOVA 

df 
SS 
MS 
F 
Significance F 

Regression 
1 
115.064 
115.064 
18.036 
0.001 
Residual 
13 
82.936 
6.380 

Total 
14 
198 
Coefficients 
Standard Error 
t Stat 
Pvalue 

Intercept 
15.532 
1.457 
10.662 
0.000 
x 
1.106 
0.261 
4.247 
0.001 
c. 
58.11% of the variability in Y is explained by the variability in X. 
9. 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 
Coefficients 
Standard Error 
t Stat 
Pvalue 

Intercept 
? 
29.4818 
3.7946 
0.0043 
x 
? 
0.7000 
3.0911 
0.0129 
ANS:
Summary Output 

Regression Statistics 

Multiple R 
0.7176 
R Square 
0.5149 
Adjusted R Square 
0.4611 
Standard Error 
7.3413 
Observations 
11 
ANOVA 

df 
SS 
MS 
F 
Significance F 

Regression 
1 
514.9455 
514.9455 
9.5546 
0.0129 
Residual 
9 
485.0545 
53.8949 

Total 
10 
1000.0000 
Coefficients 
Standard Error 
t Stat 
Pvalue 

Intercept 
111.8727 
29.4818 
3.7946 
0.0043 
x 
2.1636 
0.7000 
3.0911 
0.0129 
10. Shown below is a portion of a computer output for a regression analysis relating Y (demand) and X (unit price).
ANOVA 

df 
SS 

Regression 
1 
5048.818 
Residual 
46 
3132.661 
Total 
47 
8181.479 
Coefficients 
Standard Error 

Intercept 
80.390 
3.102 
X 
2.137 
0.248 
a. 
Perform a t test and determine whether or not demand and unit price are related. Let a = 0.05. 
b. 
Perform an F test and determine whether or not demand and unit price are related. Let a = 0.05. 
c. 
Compute the coefficient of determination and fully interpret its meaning. Be very specific. 
d. 
Compute the coefficient of correlation and explain the relationship between demand and unit price. 
ANS:
a and b
Summary Output 

Regression Statistics 

Multiple R 
0.786 
R Square 
0.617 
Adjusted R Square 
0.609 
Standard Error 
8.252 
Observations 
48 
ANOVA 

df 
SS 
MS 
F 
Significance F 

Regression 
1 
5048.818 
5048.818 
74.137 
0.000 
Residual 
46 
3132.661 
68.101 

Total 
47 
8181.479 
Coefficients 
Standard Error 
t Stat 
Pvalue 

Intercept 
80.390 
3.102 
25.916 
0.000 
X 
2.137 
0.248 
8.610 
0.000 
c. 
R^{2} = 0.617; 61.7% of the variability in demand is explained by the variability in price. 
d. 
R = 0.786; since the slope is negative, the coefficient of correlation is also negative, indicating that as unit price increases demand decreases. 
11. Shown below is a portion of a computer output for a regression analysis relating supply (Y in thousands of units) and unit price (X in thousands of dollars).
ANOVA 

df 
SS 

Regression 
1 
354.689 
Residual 
39 
7035.262 
Coefficients 
Standard Error 

Intercept 
54.076 
2.358 
X 
0.029 
0.021 
a. 
What has been the sample size for this problem? 
b. 
Perform a t test and determine whether or not supply and unit price are related. Let a = 0.05. 
c. 
Perform and F test and determine whether or not supply and unit price are related. Let a = 0.05. 
d. 
Compute the coefficient of determination and fully interpret its meaning. Be very specific. 
e. 
Compute the coefficient of correlation and explain the relationship between supply and unit price. 
f. 
Predict the supply (in units) when the unit price is $50,000. 
ANS:
a through c
Regression Statistics 

Multiple R 
0.219 
R Square 
0.048 
Adjusted R Square 
0.024 
Standard Error 
13.431 
Observations 
41 
ANOVA 

df 
SS 
MS 
F 
Significance F 

Regression 
1 
354.689 
354.689 
1.966 
0.169 
Residual 
39 
7035.262 
180.391 

Total 
40 
7389.951 
Coefficients 
Standard Error 
t Stat 
Pvalue 

Intercept 
54.076 
2.358 
22.938 
0.000 
X 
0.029 
0.021 
1.402 
0.169 
d. 
R^{2} = 0.048; 4.8% of the variability in supply is explained by the variability in price. 
e. 
R = 0.219; since the slope is positive, as unit price increases so does supply. 
f. 
supply = 54.076 + .029(50) = 55.526 (55,526 units) 
12. Given below are four observations collected in a regression study on two variables x (independent variable) and y (dependent variable).
x 
y 
2 
4 
6 
7 
9 
8 
9 
9 
a. 
Develop the least squares estimated regression equation. 
b. 
At 95% confidence, perform a t test and determine whether or not the slope is significantly different from zero. 
c. 
Perform an F test to determine whether or not the model is significant. Let a = 0.05. 
d. 
Compute the coefficient of determination. 
ANS:
Regression Statistics 

Multiple R 
0.977 
R Square 
0.955 
Adjusted R Square 
0.932 
Standard Error 
0.564 
Observations 
4 
ANOVA 

df 
SS 
MS 
F 
Significance F 

Regression 
1 
13.364 
13.364 
42.000 
0.023 
Residual 
2 
0.636 
0.318 

Total 
3 
14 
Coefficients 
Standard Error 
t Stat 
Pvalue 

Intercept 
2.864 
0.698 
4.104 
0.055 

X 
0.636 
0.098 
6.481 
0.023 
a. 
= 2.864 + 0.636x 
b. 
pvalue < .05; reject H_{o} 
c. 
pvalue < .05; reject H_{o} 
d. 
0.955 
13. Given below are five observations collected in a regression study on two variables, x (independent variable) and y (dependent variable).
x 
y 
2 
4 
3 
4 
4 
3 
5 
2 
6 
1 
a. 
Develop the least squares estimated regression equation. 
b. 
At 95% confidence, perform a t test and determine whether or not the slope is significantly different from zero. 
c. 
Perform an F test to determine whether or not the model is significant. Let a = 0.05. 
d. 
Compute the coefficient of determination. 
e. 
Compute the coefficient of correlation. 
ANS:
Regression Statistics 

Multiple R 
0.970 
R Square 
0.941 
Adjusted R Square 
0.922 
Standard Error 
0.365 
Observations 
5 
ANOVA 

df 
SS 
MS 
F 
Significance F 

Regression 
1 
6.4 
6.400 
48.000 
0.006 
Residual 
3 
0.4 
0.133 

Total 
4 
6.8 
Coefficients 
Standard Error 
t Stat 
Pvalue 

Intercept 
6.000 
0.490 
12.247 
0.001 

X 
0.800 
0.115 
6.928 
0.006 
a. 
= 6  0.8 x 
b. 
pvalue < .05; reject H_{o} 
c. 
pvalue < .05; reject H_{o} 
d. 
0.941 
e. 
0.970 
14. Below you are given a partial computer output based on a sample of 8 observations, relating an independent variable (x) and a dependent variable (y).
Coefficient 
Standard Error 

Intercept 
13.251 
10.77 
X 
0.803 
0.385 
Analysis of Variance 

SOURCE 
SS 
Regression 

Error (Residual) 
41.674 
Total 
71.875 
a. 
Develop the estimated regression line. 
b. 
At a = 0.05, test for the significance of the slope. 
c. 
At a = 0.05, perform an F test. 
d. 
Determine the coefficient of determination. 
ANS:
a. 
= 13.251 + 0.803x 
b. 
t = 2.086; pvalue is between .05 and .1 (critical t = 2.447); do not reject H_{o} 
c. 
F = 4.348; pvalue is between .05 and .1 (critical F = 5.99); do not reject H_{o} 
d. 
0.42 
15. Below you are given a partial computer output based on a sample of 8 observations, relating an independent variable (x) and a dependent variable (y).
Coefficient 
Standard Error 

Intercept 
9.462 
7.032 
x 
0.769 
0.184 
Analysis of Variance 

SOURCE 
SS 
Regression 
400 
Error (Residual) 
138 
a. 
Develop the estimated regression line. 
b. 
At a = 0.05, test for the significance of the slope. 
c. 
At a = 0.05, perform an F test. 
d. 
Determine the coefficient of determination. 
ANS:
a. 
y= 9.462 + 0.769x 
b. 
t = 4.17; pvalue (actual pvalue using Excel = 0.0059) < .05; reject H_{o} 
c. 
F = 17.39; pvalue (actual pvalue using Excel = 0.0059) < .05; reject H_{o} 
d. 
0.743 
16. The following data represent a company's yearly sales volume and its advertising expenditure over a period of 8 years.
(Y) Sales in Millions of Dollars 
(X) Advertising in ($10,000) 
15 
32 
16 
33 
18 
35 
17 
34 
16 
36 
19 
37 
19 
39 
24 
42 
a. 
Develop a scatter diagram of sales versus advertising and explain what it shows regarding the relationship between sales and advertising. 
b. 
Use the method of least squares to compute an estimated regression line between sales and advertising. 
c. 
If the company's advertising expenditure is $400,000, what are the predicted sales? Give the answer in dollars. 
d. 
What does the slope of the estimated regression line indicate? 
e. 
Compute the coefficient of determination and fully interpret its meaning. 
f. 
Use the F test to determine whether or not the regression model is significant at a = 0.05. 
g. 
Use the t test to determine whether the slope of the regression model is significant at a = 0.05. 
h. 
Develop a 95% confidence interval for predicting the average sales for the years when $400,000 was spent on advertising. 
i. 
Compute the correlation coefficient. 
ANS:
a. 
The scatter diagram shows a positive relation between sales and advertising. 
b. 
= 10.42 + 0.7895X 
c. 
$21,160,000 
d. 
As advertising is increased by $10,000, sales are expected to increase by $789,500. 
e. 
0.8459; 84.59% of variation in sales is explained by variation in advertising 
f. 
F = 32.93; pvalue (actual pvalue using Excel = 0.0012) < .05; reject H_{o}; it is significant (critical F = 5.99) 
g. 
t = 5.74; pvalue (actual pvalue using Excel = 0.0012) < .05; reject H_{o}; significant (critical t = 2.447) 
h. 
$19,460,000 to $22,860,000 
i. 
0.9197 
17. Given below are five observations collected in a regression study on two variables x (independent variable) and y (dependent variable).
x 
y 
10 
7 
20 
5 
30 
4 
40 
2 
50 
1 
a. 
Develop the least squares estimated regression equation 
b. 
At 95% confidence, perform a t test and determine whether or not the slope is significantly different from zero. 
c. 
Perform an F test to determine whether or not the model is significant. Let a = 0.05. 
d. 
Compute the coefficient of determination. 
e. 
Compute the coefficient of correlation. 
ANS:
a. 
y= 8.3  0.15x 
b. 
t = 15; pvalue (actual pvalue using Excel = 0.0001) < .05; reject H_{o} (critical t = 3.18) 
c. 
F = 225; pvalue (actual pvalue using Excel = 0.0001) < .05; reject H_{o} (critical F = 10.13) 
d. 
0.9868 
e. 
0.9934 
18. Below you are given a partial computer output based on a sample of 14 observations, relating an independent variable (x) and a dependent variable (y).
Predictor 
Coefficient 
Standard Error 
Constant 
6.428 
1.202 
X 
0.470 
0.035 
Analysis of Variance 

SOURCE 
SS 
Regression 
958.584 
Error (Residual) 

Total 
1021.429 
a. 
Develop the estimated regression line. 
b. 
At a = 0.05, test for the significance of the slope. 
c. 
At a = 0.05, perform an F test. 
d. 
Determine the coefficient of determination. 
e. 
Determine the coefficient of correlation. 
ANS:
a. 
= 6.428 + 0.47x 
b. 
t = 13.529; pvalue (actual pvalue using Excel = 0.0000) < .05; reject H_{o} (critical t = 2.179) 
c. 
F = 183.04; pvalue (actual pvalue using Excel = 0.0000) < .05; reject H_{o} (critical F = 4.75) 
d. 
0.938 
e. 
0.968 
19. Below you are given a partial computer output based on a sample of 21 observations, relating an independent variable (x) and a dependent variable (y).
Predictor 
Coefficient 
Standard Error 
Constant 
30.139 
1.181 
X 
0.252 
0.022 
Analysis of Variance 

SOURCE 
SS 
Regression 
1,759.481 
Error 
259.186 
a. 
Develop the estimated regression line. 
b. 
At a = 0.05, test for the significance of the slope. 
c. 
At a = 0.05, perform an F test. 
d. 
Determine the coefficient of determination. 
e. 
Determine the coefficient of correlation. 
ANS:
a. 
y= 30.139  0.252X 
b. 
t = 11.357; pvalue (almost zero) < a = .05; reject H_{o} (critical t = 2.093) 
c. 
F = 128.982; pvalue (almost zero) < a = .05; reject H_{o} (critical F = 4.38) 
d. 
0.872 
e. 
0.934 
20. An automobile dealer wants to see if there is a relationship between monthly sales and the interest rate. A random sample of 4 months was taken. The results of the sample are presented below. The estimated least squares regression equation is
y= 75.061  6.254X
Y 
X 
Monthly Sales 
Interest Rate (In Percent) 
22 
9.2 
20 
7.6 
10 
10.4 
45 
5.3 
a. 
Obtain a measure of how well the estimated regression line fits the data. 
b. 
You want to test to see if there is a significant relationship between the interest rate and monthly sales at the 1% level of significance. State the null and alternative hypotheses. 
c. 
At 99% confidence, test the hypotheses. 
d. 
Construct a 99% confidence interval for the average monthly sales for all months with a 10% interest rate. 
e. 
Construct a 99% confidence interval for the monthly sales of one month with a 10% interest rate. 
ANS:
a. 
R^{2} = 0.8687 
b. 
H_{0}: b_{1} = 0 
H_{a}: b_{1} 0 

c. 
test statistic t = 3.64; pvalue is between .05 and .10 (critical t = 9.925); do not reject H_{0} 
d. 
33.151 to 58.199; therefore, 0 to 58.199 
e. 
67.068 to 92.116; therefore, 0 to 92.116 
21. Jason believes that the sales of coffee at his coffee shop depend upon the weather. He has taken a sample of 6 days. Below you are given the results of the sample.
Cups of Coffee Sold 
Temperature 
350 
50 
200 
60 
210 
70 
100 
80 
60 
90 
40 
100 
a. 
Which variable is the dependent variable? 
b. 
Compute the least squares estimated line. 
c. 
Compute the correlation coefficient between temperature and the sales of coffee. 
d. 
Is there a significant relationship between the sales of coffee and temperature? Use a .05 level of significance. Be sure to state the null and alternative hypotheses. 
e. 
Predict sales of a 90 degree day. 
ANS:
a. 
Cups of coffee sold 
b. 
y= 605.714  5.943X 
c. 
0.95197 
d. 
H_{0}: b_{1} = 0 
H_{a}: b_{1} 0 

t = 6.218; pvalue (actual pvalue using Excel = 0.0034) < a = .05; reject H_{o} (critical t = 2.776) 

e. 
70.8 or 71 cups 
22. Researchers have collected data on the hours of television watched in a day and the age of a person. You are given the data below.
Hours of Television 
Age 
1 
45 
3 
30 
4 
22 
3 
25 
6 
5 
a. 
Determine which variable is the dependent variable. 
b. 
Compute the least squares estimated line. 
c. 
Is there a significant relationship between the two variables? Use a .05 level of significance. Be sure to state the null and alternative hypotheses. 
d. 
Compute the coefficient of determination. How would you interpret this value? 
ANS:
a. 
Hours of Television 
b. 
= 6.564  0.1246X 
c. 
H_{0}: b_{1} = 0 
H_{a}: b_{1} 0 

t = 12.018; pvalue (actual pvalue using Excel = 0.0002) < a = .05; reject H_{0} (critical t = 3.18) 

d. 
0.98 (rounded); 98 % of variation in hours of watching television is explained by variation in age. 
23. Given below are seven observations collected in a regression study on two variables, X (independent variable) and Y (dependent variable).
X 
Y 
2 
12 
3 
9 
6 
8 
7 
7 
8 
6 
7 
5 
9 
2 
a. 
Develop the least squares estimated regression equation. 
b. 
At 95% confidence, perform a t test and determine whether or not the slope is significantly different from zero. 
c. 
Perform an F test to determine whether or not the model is significant. Let a = 0.05. 
d. 
Compute the coefficient of determination. 
ANS:
a. 
= 13.75 1.125X 
b. 
t = 5.196; pvalue (actual pvalue using Excel = 0.0001) < a = .05; reject H_{o} (critical t = 2.571) 
c. 
F = 27; pvalue (actual pvalue using Excel = 0.0001) < a = .05; reject H_{o} (critical F = 6.61) 
d. 
0.844 
24. The owner of a retail store randomly selected the following weekly data on profits and advertising cost.
Week 
Advertising Cost ($) 
Profit ($) 
1 
0 
200 
2 
50 
270 
3 
250 
420 
4 
150 
300 
5 
125 
325 
a. 
Write down the appropriate linear relationship between advertising cost and profits. Which is the dependent variable? Which is the independent variable? 
b. 
Calculate the least squares estimated regression line. 
c. 
Predict the profits for a week when $200 is spent on advertising. 
d. 
At 95% confidence, test to determine if the relationship between advertising costs and profits is statistically significant. 
e. 
Calculate the coefficient of determination. 
ANS:
a. 
E(Y) = b_{0} + b_{1}X, where Y is profit and X is advertising cost 
b. 
= 210.0676 + 0.80811X 
c. 
$371.69 
d. 
t = 6.496; pvalue (actual pvalue using Excel = 0.0013) < a = .05; reject H_{o}; relationship is significant (critical t = 3.182) 
e. 
0.9336 
25. The owner of a bakery wants to analyze the relationship between the expenditure of a customer and the customer's income. A sample of 5 customers is taken and the following information was obtained.
Y 
X 
Expenditure 
Income (In Thousands) 
.45 
20 
10.75 
19 
5.40 
22 
7.80 
25 
5.60 
14 
The least squares estimated line is = 4.348 + 0.0826 X.
a. 
Obtain a measure of how well the estimated regression line fits the data. 
b. 
You want to test to see if there is a significant relationship between expenditure and income at the 5% level of significance. Be sure to state the null and alternative hypotheses. 
c. 
Construct a 95% confidence interval estimate for the average expenditure for all customers with an income of $20,000. 
d. 
Construct a 95% confidence interval estimate for the expenditure of one customer whose income is $20,000. 
ANS:
a. 
R^{2} = 0.0079 
b. 
H_{0}: b_{1} = 0 
H_{a}: b_{1} 0 

t = 0.154; pvalue (actual pvalue using Excel = 0.8871) > a = .05; do not reject H_{0}; (critical t = 3.182) 

c. 
0.185 to 12.185 
d. 
9.151 to 21.151 
26. Below you are given information on annual income and years of college education.
Income (In Thousands) 
Years of College 
28 
0 
40 
3 
36 
2 
28 
1 
48 
4 
a. 
Develop the least squares regression equation. 
b. 
Estimate the yearly income of an individual with 6 years of college education. 
c. 
Compute the coefficient of determination. 
d. 
Use a t test to determine whether the slope is significantly different from zero. Let a = 0.05. 
e. 
At 95% confidence, perform an F test and determine whether or not the model is significant. 
ANS:
a. 
y= 25.6 + 5.2X 
b. 
$56,800 
c. 
0.939 
d. 
t = 6.789; pvalue (actual pvalue using Excel = 0.0008) < a = .05; reject H_{o}; significant (critical t = 3.182 
e. 
F = 46.091; pvalue (actual pvalue using Excel = 0.0008) < a = .05; reject H_{o}; significant (critical F = 10.13) 
27. Below you are given information on a woman's age and her annual expenditure on purchase of books.
Age 
Annual Expenditure ($) 
18 
210 
22 
180 
21 
220 
28 
280 
a. 
Develop the least squares regression equation. 
b. 
Compute the coefficient of determination. 
c. 
Use a t test to determine whether the slope is significantly different from zero. Let a = 0.05. 
d. 
At 95% confidence, perform an F test and determine whether or not the model is significant. 
ANS:
a. 
y= 54.834 + 7.536X 
b. 
R^{2} = 0.568 
c. 
t = 1.621; pvalue (actual pvalue using Excel = 0.2464) > a = .05; do not reject H_{o}; not significant (critical t = 4.303) 
d. 
F = 2.628; pvalue (actual pvalue using Excel = 0.2464) > a = .05; do not reject H_{o}; not significant (critical F = 18.51) 
28. The following sample data contains the number of years of college and the current annual salary for a random sample of heavy equipment salespeople.
Years of College 
Annual Income (In Thousands) 
2 
20 
2 
23 
3 
25 
4 
26 
3 
28 
1 
29 
4 
27 
3 
30 
4 
33 
4 
35 
a. 
Which variable is the dependent variable? Which is the independent variable? 
b. 
Determine the least squares estimated regression line. 
c. 
Predict the annual income of a salesperson with one year of college. 
d. 
Test if the relationship between years of college and income is statistically significant at the .05 level of significance. 
e. 
Calculate the coefficient of determination. 
f. 
Calculate the sample correlation coefficient between income and years of college. Interpret the value you obtain. 
ANS:
a. 
Y (dependent variable) is annual income and X (independent variable) is years of college 
b. 
= 21.6 + 2X 
c. 
$23,600 
d. 
The relationship is not statistically significant since t = 1.51; pvalue (actual pvalue using Excel = 0.1696) > a = .05 (critical t = 2.306) 
e. 
0.222 
f. 
0.471; there is a positive correlation between years of college and annual income 
29. The following data shows the yearly income (in $1,000) and age of a sample of seven individuals.
Income (in $1,000) 
Age 
20 
18 
24 
20 
24 
23 
25 
34 
26 
24 
27 
27 
34 
27 
a. 
Develop the least squares regression equation. 
b. 
Estimate the yearly income of a 30yearold individual. 
c. 
Compute the coefficient of determination. 
d. 
Use a t test to determine whether the slope is significantly different from zero. Let a = 0.05. 
e. 
At 95% confidence, perform an F test and determine whether or not the model is significant. 
ANS:
a. 
y= 16.204 + 0.3848X 
b. 
$27,748 
c. 
0.2266 
d. 
t = 1.21; pvalue (actual pvalue using Excel = 0.2803) > a = .05; not significant (critical t = 2.571) 
e. 
F = 1.46; pvalue (actual pvalue using Excel = 0.2803) > a = .05; not significant (critical F = 6.61) 
30. The following data show the results of an aptitude test (Y) and the grade point average of 10 students.
Aptitude Test Score (Y) 
GPA (X) 
26 
1.8 
31 
2.3 
28 
2.6 
30 
2.4 
34 
2.8 
38 
3.0 
41 
3.4 
44 
3.2 
40 
3.6 
43 
3.8 
a. 
Develop a least squares estimated regression line. 
b. 
Compute the coefficient of determination and comment on the strength of the regression relationship. 
c. 
Is the slope significant? Use a t test and let a = 0.05. 
d. 
At 95% confidence, test to determine if the model is significant (i.e., perform an F test). 
ANS:
a. 
= 8.171 + 9.4564X 
b. 
0.83; there is a fairly strong relationship 
c. 
t = 6.25; pvalue (actual pvalue using Excel = 0.0002) < a =.05; it is significant (critical t = 2.306) 
d. 
F = 39.07; pvalue (actual pvalue using Excel = 0.0002) < a =.05; it is significant (critical F = 5.32) 
31. Shown below is a portion of the computer output for a regression analysis relating sales (Y in millions of dollars) and advertising expenditure (X in thousands of dollars).
Predictor 
Coefficient 
Standard Error 
Constant 
4.00 
0.800 
X 
0.12 
0.045 
Analysis of Variance 


SOURCE 
DF 
SS 
Regression 
1 
1,400 
Error 
18 
3,600 
a. 
What has been the sample size for the above? 
b. 
Perform a t test and determine whether or not advertising and sales are related. Let a = 0.05. 
c. 
Compute the coefficient of determination. 
d. 
Interpret the meaning of the value of the coefficient of determination that you found in Part c. Be very specific. 
e. 
Use the estimated regression equation and predict sales for an advertising expenditure of $4,000. Give your answer in dollars. 
ANS:
a. 
20 
b. 
t = 2.66; pvalue is between 0.01 and 0.02; they are related (critical t = 2.101) 
c. 
R^{2} = 0.28 
d. 
28% of variation in sales is explained by variation in advertising expenditure. 
e. 
$4,480,000 
32. A company has recorded data on the daily demand for its product (Y in thousands of units) and the unit price (X in hundreds of dollars). A sample of 15 days demand and associated prices resulted in the following data.
SX = 75 
S (Y )(X ) = 59 
SY = 135 
S (X )^{2} = 94 
S (Y )^{2} = 100 

SSE = 62.9681 
a. 
Using the above information, develop the leastsquares estimated regression line and write the equation. 
b. 
Compute the coefficient of determination. 
c. 
Perform an F test and determine whether or not there is a significant relationship between demand and unit price. Let a = 0.05. 
d. 
Would the demand ever reach zero? If yes, at what price would the demand be zero? 
ANS:
a. 
y= 12.138  0.6277X 
b. 
R^{2} = 0.3703 
c. 
F = 7.65; pvalue is between .01 and .025; reject H_{o} and conclude that demand and unit price are related (critical F = 4.67) 
d. 
Yes, at $1,934 
33. A regression and correlation analysis resulted in the following information regarding an independent variable (x) and a dependent variable (y).
SX = 42 
S (Y  )(X  ) = 37 
SY = 63 
S (X  )^{2} = 84 
n = 7 
S (Y  )^{2} = 28 
a. 
Develop the least squares estimated regression equation. 
b. 
At 95% confidence, perform a t test and determine whether or not the slope is significantly different from zero. 
c. 
Perform an F test to determine whether or not the model is significant. Let a = 0.05. 
d. 
Compute the coefficient of determination. 
ANS:
a. 
= 6.3571 + 0.4405x 
b. 
pvalue < .05; reject H_{o} 
c. 
pvalue < .05; reject H_{o} 
d. 
0.582 
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