Regression Analysis Question with Solution

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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 least-squares 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:

y= 1.0416 + 0.0099x

r ² = .5629; the regression equation has accounted for 56.29% of the total sum of squares

rxy = 0.75

t = 2.54 > 2.015 (df = 5); p-value is between .05 and 0.1; (Excel’s results: p-value 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; p-value is between 0.1 and 0.2 (Excel’s result: p-value = .052); do not reject H_o; conclude x and y are not related
b.	t = 2.5376 < 4.032; p-value is between 0.1 and 0.2. (Excel’s result: p-value = .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 least-squares 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:

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.

r ² = .8556; the regression equation has accounted for 85.56% of the total sum of squares

rxy = -0.92

t = -5.44 < -4.032 (df = 5); p-value < .01; (Excel’s result: p-value = .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; p-value < .01; (Excel’s result: p-value = .0028); reject H_o, x and y are related
b.	t = -5.4428 < -4.032; p-value < .01; (Excel’s result: p-value = .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*	*P-value*
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*	*P-value*
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*	*P-value*
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*	*p-value*
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 p-value approach and determine whether or not Y and X are related. Let a = 0.05.
b.	Using the p-value 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*	*P-value*
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*	*P-value*
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*	*P-value*
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*	*P-value*
Intercept	80.390	3.102	25.916	0.000
X	-2.137	0.248	-8.610	0.000

c.	R² = 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*	*P-value*
Intercept	54.076	2.358	22.938	0.000
X	0.029	0.021	1.402	0.169

d.	R² = 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*	*P-value*
Intercept	2.864	0.698	4.104	0.055
X	0.636	0.098	6.481	0.023

a.	= 2.864 + 0.636x
b.	p-value < .05; reject H_o
c.	p-value < .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*	*P-value*
Intercept	6.000	0.490	12.247	0.001
X	-0.800	0.115	-6.928	0.006

a.	= 6 - 0.8 x
b.	p-value < .05; reject H_o
c.	p-value < .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; p-value is between .05 and .1 (critical t = 2.447); do not reject H_o
c.	F = 4.348; p-value 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; p-value (actual p-value using Excel = 0.0059) < .05; reject H_o
c.	F = 17.39; p-value (actual p-value 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; p-value (actual p-value using Excel = 0.0012) < .05; reject H_o; it is significant (critical F = 5.99)
g.	t = 5.74; p-value (actual p-value 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; p-value (actual p-value using Excel = 0.0001) < .05; reject H_o (critical t = 3.18)
c.	F = 225; p-value (actual p-value 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; p-value (actual p-value using Excel = 0.0000) < .05; reject H_o (critical t = 2.179)
c.	F = 183.04; p-value (actual p-value 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; p-value (almost zero) < a = .05; reject H_o (critical t = 2.093)
c.	F = 128.982; p-value (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.	R² = 0.8687
b.	H₀: b₁ = 0
	H_a: b₁ 0
c.	test statistic t = -3.64; p-value is between .05 and .10 (critical t = 9.925); do not reject H₀
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.	H₀: b₁ = 0
	H_a: b₁ 0
	t = -6.218; p-value (actual p-value 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.	H₀: b₁ = 0
	H_a: b₁ 0
	t = -12.018; p-value (actual p-value using Excel = 0.0002) < a = .05; reject H₀ (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; p-value (actual p-value using Excel = 0.0001) < a = .05; reject H_o (critical t = 2.571)
c.	F = 27; p-value (actual p-value 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) = b₀ + b₁X, where Y is profit and X is advertising cost
b.	= 210.0676 + 0.80811X
c.	$371.69
d.	t = 6.496; p-value (actual p-value 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.	R² = 0.0079
b.	H₀: b₁ = 0
	H_a: b₁ 0
	t = 0.154; p-value (actual p-value using Excel = 0.8871) > a = .05; do not reject H₀; (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; p-value (actual p-value using Excel = 0.0008) < a = .05; reject H_o; significant (critical t = 3.182
e.	F = 46.091; p-value (actual p-value 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.	R² = 0.568
c.	t = 1.621; p-value (actual p-value using Excel = 0.2464) > a = .05; do not reject H_o; not significant (critical t = 4.303)
d.	F = 2.628; p-value (actual p-value 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; p-value (actual p-value 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 30-year-old 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; p-value (actual p-value using Excel = 0.2803) > a = .05; not significant (critical t = 2.571)
e.	F = 1.46; p-value (actual p-value 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; p-value (actual p-value using Excel = 0.0002) < a =.05; it is significant (critical t = 2.306)
d.	F = 39.07; p-value (actual p-value 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; p-value is between 0.01 and 0.02; they are related (critical t = 2.101)
c.	R² = 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 least-squares 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.	R² = 0.3703
c.	F = 7.65; p-value 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.	p-value < .05; reject H_o
c.	p-value < .05; reject H_o
d.	0.582

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