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Chapter 4

Regression Analysis

Data analysts use regressions to determine if relationships exists between variables…...

In a simple linear regression we test whether measured values of the dependent variable (on the Y axis) vary with provided independent variable (on the X axis)

Does an increase in advertising (X) coincide to an increase in sales (Y)….?

Can more training hours (X) leads to decreased scrap (Y)….?

Will more spending on HR benefits (X) prompt an increase in employee retention (Y)…?

We use sample data (x,y) and sample y intercept

X is the independent variable (we are given)

Y is the dependent variable (we measure)

is the slope (change in Y for a change in X)

is the value of y when x=0

is the random error

When performing regressions there are 3 rules we must follow:

(Rule 1) Do not predict values far beyond the data we are working with

In the example below we see a linear relationship between X and Y.

What is the predicted Y value at X=12?

In this case the relationship changed (from linear to curvilinear) when x exceeded 6.

Conclusion: we can only apply extrapolate values near the test range

When performing regressions there are 3 rules we must follow:

(Rule 2) Data deviations from the predicted line are assumed to be random

Sales

The data points ( ) are randomly scattered around the regression line. Meaning there is not an underlying influence on Y values other than the X values we are considering

When performing regressions there are 3 rules we must follow:

(Rule 3) Variables X and Y are normally distributed

Regression line

How do we determine if our data is normally distributed?

To test data for skewsness we use the formula =SKEW(). If SKEW value is between -1 (negative skew) and +1 (positive skew) we can say the data is normal in X

To test data for kurtosis we use the formula =KURT(). If the KURT values are between -1 (flat) and +1 (peaked) we can say the data is normal in Y

In this example the data X and Y are normally distributed because SKEW and KURT values are all between +1 and -1.

	X	Y
	8	200
	2	230
	7	220
	3	210
	7	240
	6	200
	4	210
	9	230
	6	216

SKEW	-0.41576	0.268996
KURTOSIS	-0.86776	-0.99992

Now that we know the rules of regression lets try one…

We start by enabling Excel Add-ins

In Excel 2010 and later go to File > Options

1. Click this

2. Click this

3. Check these

4. Click this.

5. Click “Data”. Now you should be able to see these.

1. On Data tab

2. Select Data Analysis

3. Select Regression

4. Click OK

5. Click to select D3:D10

6. Click to select C3:C10

7. Click as 1st row of X & Y are labels

8. Click to make plot

What does all this mean???

Start by looking at Significance F. If F is < .05, there is < 5% chance of incorrectly accepting a regression exists. In other words, there is >95% chance of a regression existing. At F < .05 we accept the regression.

Next we look at R square (i.e. r2)

The coefficient of determination () tell us the % variation in y (“in our example electrical demand”) explained by x (“time period”)

How does r2 do this?

r2 is a ratio of variation explained by the model to total variation.

In our example = 0.8, so 80% of variation in electrical demand can be explained by variation in time period.

= 56.70 + 10.54x

The F <5% means a regression exists and r2 = 0.8 that it is strong; we can now look to coefficients to find x slope and y intercept of the regression line

Are the regression coefficients significant?

The P values of y intercept (.0029) and slope (.006) are less than .05. So….

There is < 5% chance of incorrectly accepting these coefficients. In other words, there is >95% chance of a regression existing with these coefficients.

Let’s try another…

Determine if a relationship exists between how much Triple A Construction Co. sells and how much it pays in payroll.

The null hypothesis () at 95% confidence (is no relationship between sales and payroll

The X and Y data are normally distributed so we can test for a regression

1. On Data tab

2. Select Data Analysis

3. Select regression

4. Click OK

5. Click to select D8:D14

6. Click to select E8:E14

7. Click as 1st row of X & Y are labels

We look at the Significance F

From our Significance F, there is only a 3.9% chance of incorrectly rejecting the null hypothesis () that no relationship exists between sales and payroll

Since our null hypothesis () is tested at 95% confidence (a 3.9% chance of error is acceptable. We reject no relationship between sales and payroll

With a correlation coefficient (r) of .69, the regression is moderate.

With an intercept of 2 and slope coefficient of 1.25 our estimated linear regression equation

With an intercept p value of .3, we cannot accept this value at 95% confidence. We need to consider standard error.

What does standard error mean?

The Standard Errors are errors associated with regression coefficients. Think of it standard deviation of coefficients.

At a 95% confidence interval (i.e. 2 standard deviations) payroll and y intercept coefficients could vary from:

Coefficient

Lowest value of predicted sales () using payroll (x) is:

Highest value of predicted sales () using payroll (x) is:

Is it possible when we collected sales and payroll numbers, there were external factors we didn’t control that affected results (such as years service, or employee performance ratings, or economy strength)?

From the “residual plots” we can see

Residual error is on the vertical axis. The independent variable on the horizontal axis.

Since the points in this example are randomly scattered around the horizontal axis (sum approximately to 0), we can reject external factors and accept a single variable linear regression.

Payroll (X) Residual Plot

3 4 6 4 2 5 0.25 1 -0.5 -2 0 1.25

Payroll (X)

Residuals

A multiple regression model allows us to predict an output value Y using multiple independent variables X1, X2 ….

Lets look at an example….

Can square footage of a house () or age () or both be used to predict the selling price (Y) of a house?

The null hypothesis () at 95% confidence (is no relationship between sales price and square footage or age

1. On Data tab

2. Select Data Analysis

3. Select regression

4. Click OK

5. Click to select B4:B18

6. Click to select C4:D18

7. Click as 1st row of X1, X2 & Y are labels

We look at Significance F

From our Significance F (.0021), there is only a 0.22% chance of incorrectly rejecting the null hypothesis () that no relationship exists between Y, X1 and X2

Since our null hypothesis () is tested at 95% confidence (a 0.22% chance of error is acceptable. We reject that no relationship exists.

The r2=0.67 tells us the linear regression explains 67% of the variance in the dependent variable (i.e. house selling price). So, we have a moderately strong model.

Since the p-values for square feet (.0013) and age (.0039) are both below .05, square feet and age can both be used to predict price

A non-significant P value (>.05) would have told us the variable does not have predictive capability in the presence of the other; so we would have removed it and refit the model without it.

P values shouldn’t be used to eliminate more than one variable at a time

Why? Because a variable that doesn’t have predictive capability in the presence of other variables may have predictive capability when some of those variables are removed from the model.

With an intercept of 146,630 and slope coefficients of 43.8 & -2,898 our estimated linear regression equation is

At higher values of square feet () and lower values of age () home sale prices are larger

Lowest value of predicted home sales price () using square feet () and age () is:

Highest value of predicted home sales price () using square feet () and age () is:

What do t values tell us?

In multiple linear regression, the absolute size of the coefficient for each independent variable gives you the size of the effect that variable is having on your dependent variable, and the sign on the coefficient (positive or negative) gives you the direction of the effect.

In our case square feet (t=4.26) has a bigger effect on house price than age (t=3.64)

What is the adjusted R2

As additional variables are added to a multiple regression equation, R² increases even when the new variables have no real predictive capability.

When variables are added and adjusted R² doesn't increase the new variables do not improve predictive capability.

Is it possible when we collected house price, house age and square footage, there were external factors we didn’t control that affected price (such as school district, builder, or taxes)?

From the “residual plots” we can see

The points are randomly dispersed around the horizontal axis for both square feet and age; we can reject external factors are impacting our age and square feet multiple regression with house price

Square feet Residual Plot

1926 2069 1720 1396 1706 1847 1950 2323 2285 3752 2300 2525 3800 1740 -49066.406014619977 -2345.7129556209984 -10239.616517664399 -29322.435475499602 14171.227106612627 27584.813911759818 5185.8704494942504 3537.3054888200131 10607.696428610972 -26587.137991958123 4760.9159575959784 3002.8722506209742 17802.96365005814 30907.643711790413

Square feet

Residuals

Age Residual Plot

30 40 30 15 32 38 27 30 26 35 18 17 40 12 -49066.406014619977 -2345.7129556209984 -10239.616517664399 -29322.435475499602 14171.2271 06612627 27584.813911759818 5185.8704494942504 3537.3054888200131 10607.696428610972 -26587.137991958123 4760.9159575959784 3002.8722506209742 17802.96365005814 30907.643711790413

Age

Residuals

When we do linear regressions, there are certain assumptions we make….

Sample sizes are large enough (>30) the t distributions approximates normal distributions

Correlation does not equal causality

An action or occurrence can cause another (such as smoking causes lung cancer), or it can correlate with another (such as smoking is correlated with high alcohol consumption). If one action causes another, then they are most certainly correlated. But just because two things occur together does not mean that one caused the other, even if it seems to make sense.

SUMMARY OUTPUT

Regression Statistics

Multiple R0.894909611

R Square0.800863211

Adjusted R Square0.761035854

Standard Error12.43238858

Observations7

ANOVA

dfSSMSFSignificance F

Regression13108.0357143108.03571420.108370.006493257

Residual5772.8214286154.5642857

Total63880.857143

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

Intercept56.7142857110.50728615.3976150610.00294829.7044469283.7241245129.7044469283.72412451

Time Period10.535714292.3495005984.4842356260.0064934.49613072516.575297854.49613072516.57529785

Sheet1

	Time Period	Electrical Demand
2001	1	74
2002	2	79
2003	3	80
2004	4	90
2005	5	105
2005	6	142
2007	7	122
			SUMMARY OUTPUT
			Regression Statistics
			Multiple R	0.8949096107
			R Square	0.8008632114
			Adjusted R Square	0.7610358536
			Standard Error	12.4323885764
			Observations	7
			ANOVA
				df	SS	MS	F	Significance F
			Regression	1	3108.0357142857	3108.0357142857	20.1083691483	0.0064932569
			Residual	5	772.8214285714	154.5642857143
			Total	6	3880.8571428571
				Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
			Intercept	56.7142857143	10.5072861018	5.3976150611	0.0029479517	29.7044469192	83.7241245093	29.7044469192	83.7241245093
			Time Period	10.5357142857	2.3495005983	4.4842356259	0.0064932569	4.496130725	16.5752978464	4.496130725	16.5752978464
			RESIDUAL OUTPUT
			Observation	Predicted Electrical Demand	Residuals
			1	67.25	6.75
			2	77.7857142857	1.2142857143
			3	88.3214285714	-8.3214285714
			4	98.8571428571	-8.8571428571
			5	109.3928571429	-4.3928571429
			6	119.9285714286	22.0714285714
			7	130.4642857143	-8.4642857143

Time Period Line Fit Plot

Electrical Demand 1 2 3 4 5 6 7 74 79 80 90 105 142 122 Predicted Electrical Demand 1 2 3 4 5 6 7 67.25 77.785714285714292 88.321428571428584 98.857142857142861 109.39285714285714 119.92857142857143 130.46428571428572

Time Period

Electrical Demand

Sheet2

Sheet3

Sheet1

	Time Period	Electrical Demand
2001	1	74
2002	2	79
2003	3	80
2004	4	90
2005	5	105
2005	6	142
2007	7	122
			SUMMARY OUTPUT
			Regression Statistics
			Multiple R	0.8949096107
			R Square	0.8008632114
			Adjusted R Square	0.7610358536
			Standard Error	12.4323885764
			Observations	7
			ANOVA
				df	SS	MS	F	Significance F
			Regression	1	3108.0357142857	3108.0357142857	20.1083691483	0.0064932569
			Residual	5	772.8214285714	154.5642857143
			Total	6	3880.8571428571
				Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
			Intercept	56.7142857143	10.5072861018	5.3976150611	0.0029479517	29.7044469192	83.7241245093	29.7044469192	83.7241245093
			Time Period	10.5357142857	2.3495005983	4.4842356259	0.0064932569	4.496130725	16.5752978464	4.496130725	16.5752978464
			RESIDUAL OUTPUT
			Observation	Predicted Electrical Demand	Residuals
			1	67.25	6.75
			2	77.7857142857	1.2142857143
			3	88.3214285714	-8.3214285714
			4	98.8571428571	-8.8571428571
			5	109.3928571429	-4.3928571429
			6	119.9285714286	22.0714285714
			7	130.4642857143	-8.4642857143

Time Period Line Fit Plot

Time Period

Electrical Demand

Sheet2

Sheet3

Sheet1

	Time Period	Electrical Demand
2001	1	74
2002	2	79
2003	3	80
2004	4	90
2005	5	105
2005	6	142
2007	7	122
			SUMMARY OUTPUT
			Regression Statistics
			Multiple R	0.8949096107
			R Square	0.8008632114
			Adjusted R Square	0.7610358536
			Standard Error	12.4323885764
			Observations	7
			ANOVA
				df	SS	MS	F	Significance F
			Regression	1	3108.0357142857	3108.0357142857	20.1083691483	0.0064932569
			Residual	5	772.8214285714	154.5642857143
			Total	6	3880.8571428571
				Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
			Intercept	56.7142857143	10.5072861018	5.3976150611	0.0029479517	29.7044469192	83.7241245093	29.7044469192	83.7241245093
			Time Period	10.5357142857	2.3495005983	4.4842356259	0.0064932569	4.496130725	16.5752978464	4.496130725	16.5752978464
			RESIDUAL OUTPUT
			Observation	Predicted Electrical Demand	Residuals
			1	67.25	6.75
			2	77.7857142857	1.2142857143
			3	88.3214285714	-8.3214285714
			4	98.8571428571	-8.8571428571
			5	109.3928571429	-4.3928571429
			6	119.9285714286	22.0714285714
			7	130.4642857143	-8.4642857143

Time Period Line Fit Plot

Time Period

Electrical Demand

Sheet2

Sheet3

SUMMARY OUTPUT

Regression Statistics

Multiple R0.833333333

R Square0.694444444

Adjusted R Square0.618055556

Standard Error1.31101106

Observations6

ANOVA

dfSSMSFSignificance F

Regression115.62515.6259.0909090.039351852

Residual46.8751.71875

Total522.5

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

Intercept21.7425436391.1477470.31505-2.8380767576.838076757-2.8380767576.838076757

Payroll (X)1.250.4145780993.0151130.0393520.0989466672.4010533330.0989466672.401053333

Sheet1

Triple A Construction Co.
	Sales (Y)	Payroll (X)
	6	3
	8	4
	9	6
	5	4
	4.5	2
	9.5	5
			SUMMARY OUTPUT
			Regression Statistics
			Multiple R	0.8333333333
			R Square	0.6944444444
			Adjusted R Square	0.6180555556
			Standard Error	1.3110110602
			Observations	6
			ANOVA
				df	SS	MS	F	Significance F
			Regression	1	15.625	15.625	9.0909090909	0.0393518519
			Residual	4	6.875	1.71875
			Total	5	22.5
				Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
			Intercept	2	1.7425436389	1.1477474397	0.3150499206	-2.8380767567	6.8380767567	-2.8380767567	6.8380767567
			Payroll (X)	1.25	0.4145780988	3.0151134458	0.0393518519	0.0989466669	2.4010533331	0.0989466669	2.4010533331
			RESIDUAL OUTPUT
			Observation	Predicted Sales (Y)	Residuals
			1	5.75	0.25
			2	7	1
			3	9.5	-0.5
			4	7	-2
			5	4.5	0
			6	8.25	1.25

Payroll (X) Residual Plot

3 4 6 4 2 5 0.25 1 -0.5 -2 0 1.25

Payroll (X)

Residuals

Payroll (X) Line Fit Plot

Sales (Y) 3 4 6 4 2 5 6 8 9 5 4.5 9.5 Predicted Sales (Y) 3 4 6 4 2 5 5.75 7 9.5 7 4.5 8.25

Payroll (X)

Sales (Y)

Sheet2

Sheet3

Sheet1

Triple A Construction Co.
	Sales (Y)	Payroll (X)
	6	3
	8	4
	9	6
	5	4
	4.5	2
	9.5	5
			SUMMARY OUTPUT
			Regression Statistics
			Multiple R	0.8333333333
			R Square	0.6944444444
			Adjusted R Square	0.6180555556
			Standard Error	1.3110110602
			Observations	6
			ANOVA
				df	SS	MS	F	Significance F
			Regression	1	15.625	15.625	9.0909090909	0.0393518519
			Residual	4	6.875	1.71875
			Total	5	22.5
				Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
			Intercept	2	1.7425436389	1.1477474397	0.3150499206	-2.8380767567	6.8380767567	-2.8380767567	6.8380767567
			Payroll (X)	1.25	0.4145780988	3.0151134458	0.0393518519	0.0989466669	2.4010533331	0.0989466669	2.4010533331
			RESIDUAL OUTPUT
			Observation	Predicted Sales (Y)	Residuals
			1	5.75	0.25
			2	7	1
			3	9.5	-0.5
			4	7	-2
			5	4.5	0
			6	8.25	1.25

Payroll (X) Residual Plot

3 4 6 4 2 5 0.25 1 -0.5 -2 0 1.25

Payroll (X)

Residuals

Payroll (X) Line Fit Plot

Sales (Y) 3 4 6 4 2 5 6 8 9 5 4.5 9.5 Predicted Sales (Y) 3 4 6 4 2 5 5.75 7 9.5 7 4.5 8.25

Payroll (X)

Sales (Y)

Sheet2

Sheet3

Sheet1

Triple A Construction Co.
	Sales (Y)	Payroll (X)
	6	3
	8	4
	9	6
	5	4
	4.5	2
	9.5	5
			SUMMARY OUTPUT
			Regression Statistics
			Multiple R	0.8333333333
			R Square	0.6944444444
			Adjusted R Square	0.6180555556
			Standard Error	1.3110110602
			Observations	6
			ANOVA
				df	SS	MS	F	Significance F
			Regression	1	15.625	15.625	9.0909090909	0.0393518519
			Residual	4	6.875	1.71875
			Total	5	22.5
				Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
			Intercept	2	1.7425436389	1.1477474397	0.3150499206	-2.8380767567	6.8380767567	-2.8380767567	6.8380767567
			Payroll (X)	1.25	0.4145780988	3.0151134458	0.0393518519	0.0989466669	2.4010533331	0.0989466669	2.4010533331
			RESIDUAL OUTPUT
			Observation	Predicted Sales (Y)	Residuals
			1	5.75	0.25
			2	7	1
			3	9.5	-0.5
			4	7	-2
			5	4.5	0
			6	8.25	1.25

Payroll (X) Residual Plot

3 4 6 4 2 5 0.25 1 -0.5 -2 0 1.25

Payroll (X)

Residuals

Payroll (X) Line Fit Plot

Sales (Y) 3 4 6 4 2 5 6 8 9 5 4.5 9.5 Predicted Sales (Y) 3 4 6 4 2 5 5.75 7 9.5 7 4.5 8.25

Payroll (X)

Sales (Y)

Sheet2

Sheet3

Sheet1

Triple A Construction Co.
	Sales (Y)	Payroll (X)
	6	3
	8	4
	9	6
	5	4
	4.5	2
	9.5	5
			SUMMARY OUTPUT
			Regression Statistics
			Multiple R	0.8333333333
			R Square	0.6944444444
			Adjusted R Square	0.6180555556
			Standard Error	1.3110110602
			Observations	6
			ANOVA
				df	SS	MS	F	Significance F
			Regression	1	15.625	15.625	9.0909090909	0.0393518519
			Residual	4	6.875	1.71875
			Total	5	22.5
				Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
			Intercept	2	1.7425436389	1.1477474397	0.3150499206	-2.8380767567	6.8380767567	-2.8380767567	6.8380767567
			Payroll (X)	1.25	0.4145780988	3.0151134458	0.0393518519	0.0989466669	2.4010533331	0.0989466669	2.4010533331
			RESIDUAL OUTPUT
			Observation	Predicted Sales (Y)	Residuals
			1	5.75	0.25
			2	7	1
			3	9.5	-0.5
			4	7	-2
			5	4.5	0
			6	8.25	1.25

Payroll (X) Residual Plot

3 4 6 4 2 5 0.25 1 -0.5 -2 0 1.25

Payroll (X)

Residuals

Payroll (X) Line Fit Plot

Sales (Y) 3 4 6 4 2 5 6 8 9 5 4.5 9.5 Predicted Sales (Y) 3 4 6 4 2 5 5.75 7 9.5 7 4.5 8.25

Payroll (X)

Sales (Y)

Sheet2

Sheet3

Sheet1

Triple A Construction Co.
	Sales (Y)	Payroll (X)
	6	3
	8	4
	9	6
	5	4
	4.5	2
	9.5	5
			SUMMARY OUTPUT
			Regression Statistics
			Multiple R	0.8333333333
			R Square	0.6944444444
			Adjusted R Square	0.6180555556
			Standard Error	1.3110110602
			Observations	6
			ANOVA
				df	SS	MS	F	Significance F
			Regression	1	15.625	15.625	9.0909090909	0.0393518519
			Residual	4	6.875	1.71875
			Total	5	22.5
				Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
			Intercept	2	1.7425436389	1.1477474397	0.3150499206	-2.8380767567	6.8380767567	-2.8380767567	6.8380767567
			Payroll (X)	1.25	0.4145780988	3.0151134458	0.0393518519	0.0989466669	2.4010533331	0.0989466669	2.4010533331
			RESIDUAL OUTPUT
			Observation	Predicted Sales (Y)	Residuals
			1	5.75	0.25
			2	7	1
			3	9.5	-0.5
			4	7	-2
			5	4.5	0
			6	8.25	1.25

Payroll (X) Residual Plot

3 4 6 4 2 5 0.25 1 -0.5 -2 0 1.25

Payroll (X)

Residuals

Payroll (X) Line Fit Plot

Sales (Y) 3 4 6 4 2 5 6 8 9 5 4.5 9.5 Predicted Sales (Y) 3 4 6 4 2 5 5.75 7 9.5 7 4.5 8.25

Payroll (X)

Sales (Y)

Sheet2

Sheet3

Sheet1

Triple A Construction Co.
	Sales (Y)	Payroll (X)
	6	3
	8	4
	9	6
	5	4
	4.5	2
	9.5	5
			SUMMARY OUTPUT
			Regression Statistics
			Multiple R	0.8333333333
			R Square	0.6944444444
			Adjusted R Square	0.6180555556
			Standard Error	1.3110110602
			Observations	6
			ANOVA
				df	SS	MS	F	Significance F
			Regression	1	15.625	15.625	9.0909090909	0.0393518519
			Residual	4	6.875	1.71875
			Total	5	22.5
				Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
			Intercept	2	1.7425436389	1.1477474397	0.3150499206	-2.8380767567	6.8380767567	-2.8380767567	6.8380767567
			Payroll (X)	1.25	0.4145780988	3.0151134458	0.0393518519	0.0989466669	2.4010533331	0.0989466669	2.4010533331
			RESIDUAL OUTPUT
			Observation	Predicted Sales (Y)	Residuals
			1	5.75	0.25
			2	7	1
			3	9.5	-0.5
			4	7	-2
			5	4.5	0
			6	8.25	1.25

Payroll (X) Residual Plot

3 4 6 4 2 5 0.25 1 -0.5 -2 0 1.25

Payroll (X)

Residuals

Payroll (X) Line Fit Plot

Sales (Y) 3 4 6 4 2 5 6 8 9 5 4.5 9.5 Predicted Sales (Y) 3 4 6 4 2 5 5.75 7 9.5 7 4.5 8.25

Payroll (X)

Sales (Y)

Sheet2

Sheet3

Sheet1

Triple A Construction Co.
	Sales (Y)	Payroll (X)
	6	3
	8	4
	9	6
	5	4
	4.5	2
	9.5	5
			SUMMARY OUTPUT
			Regression Statistics
			Multiple R	0.8333333333
			R Square	0.6944444444
			Adjusted R Square	0.6180555556
			Standard Error	1.3110110602
			Observations	6
			ANOVA
				df	SS	MS	F	Significance F
			Regression	1	15.625	15.625	9.0909090909	0.0393518519
			Residual	4	6.875	1.71875
			Total	5	22.5
				Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
			Intercept	2	1.7425436389	1.1477474397	0.3150499206	-2.8380767567	6.8380767567	-2.8380767567	6.8380767567
			Payroll (X)	1.25	0.4145780988	3.0151134458	0.0393518519	0.0989466669	2.4010533331	0.0989466669	2.4010533331
			RESIDUAL OUTPUT
			Observation	Predicted Sales (Y)	Residuals
			1	5.75	0.25
			2	7	1
			3	9.5	-0.5
			4	7	-2
			5	4.5	0
			6	8.25	1.25

Payroll (X) Residual Plot

3 4 6 4 2 5 0.25 1 -0.5 -2 0 1.25

Payroll (X)

Residuals

Payroll (X) Line Fit Plot

Sales (Y) 3 4 6 4 2 5 6 8 9 5 4.5 9.5 Predicted Sales (Y) 3 4 6 4 2 5 5.75 7 9.5 7 4.5 8.25

Payroll (X)

Sales (Y)

Sheet2

Sheet3

Sheet1

Triple A Construction Co.
	Sales (Y)	Payroll (X)
	6	3
	8	4
	9	6
	5	4
	4.5	2
	9.5	5
			SUMMARY OUTPUT
			Regression Statistics
			Multiple R	0.8333333333
			R Square	0.6944444444
			Adjusted R Square	0.6180555556
			Standard Error	1.3110110602
			Observations	6
			ANOVA
				df	SS	MS	F	Significance F
			Regression	1	15.625	15.625	9.0909090909	0.0393518519
			Residual	4	6.875	1.71875
			Total	5	22.5
				Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
			Intercept	2	1.7425436389	1.1477474397	0.3150499206	-2.8380767567	6.8380767567	-2.8380767567	6.8380767567
			Payroll (X)	1.25	0.4145780988	3.0151134458	0.0393518519	0.0989466669	2.4010533331	0.0989466669	2.4010533331
			RESIDUAL OUTPUT
			Observation	Predicted Sales (Y)	Residuals
			1	5.75	0.25
			2	7	1
			3	9.5	-0.5
			4	7	-2
			5	4.5	0
			6	8.25	1.25

Payroll (X) Residual Plot

3 4 6 4 2 5 0.25 1 -0.5 -2 0 1.25

Payroll (X)

Residuals

Payroll (X) Line Fit Plot

Sales (Y) 3 4 6 4 2 5 6 8 9 5 4.5 9.5 Predicted Sales (Y) 3 4 6 4 2 5 5.75 7 9.5 7 4.5 8.25

Payroll (X)

Sales (Y)

Sheet2

Sheet3

SUMMARY OUTPUT

Regression Statistics

Multiple R0.819680305

R Square0.671875802

Adjusted R Square0.612216857

Standard Error24312.60729

Observations14

ANOVA

dfSSMSFSignificance F

Regression2133139369686.66E+0911.261950.002178765

Residual1165021316035.91E+08

Total1319816068571

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

Intercept146630.893625482.082875.7542740.00012890545.20731202716.579890545.20731202716.5798

Square feet43.8193664910.280965074.2621840.00133821.1911149466.4476180421.1911149466.44761804

Age-2898.686247796.5649421-3.638980.003895-4651.913863-1145.45863-4651.913863-1145.45863

Sheet1

Selling Price	Square feet	Age
95000	1926	30
119000	2069	40
124800	1720	30
135000	1396	15
142800	1706	32
145000	1847	38
159000	1950	27
165000	2323	30
182000	2285	26
183000	3752	35
200000	2300	18
211000	2525	17
215000	3800	40
219000	1740	12
			SUMMARY OUTPUT
			Regression Statistics
			Multiple R	0.8196803049
			R Square	0.6718758022
			Adjusted R Square	0.6122168572
			Standard Error	24312.6072850603
			Observations	14
			ANOVA
				df	SS	MS	F	Significance F
			Regression	2	13313936968.4553	6656968484.22766	11.261945743	0.0021787652
			Residual	11	6502131602.97325	591102872.997569
			Total	13	19816068571.4286
				Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
			Intercept	146630.893555974	25482.0828687578	5.7542742605	0.0001275664	90545.2073136126	202716.579798335	90545.2073136126	202716.579798335
			Square feet	43.8193664901	10.2809650702	4.2621841618	0.0013380948	21.1911149391	66.447618041	21.1911149391	66.447618041
			Age	-2898.686246708	796.5649420672	-3.638982955	0.0038949963	-4651.9138632471	-1145.4586301689	-4651.9138632471	-1145.4586301689
			RESIDUAL OUTPUT
			Observation	Predicted Selling Price	Residuals
			1	144066.40601462	-49066.40601462
			2	121345.712955621	-2345.712955621
			3	135039.616517664	-10239.6165176644
			4	164322.4354755	-29322.4354754996
			5	128628.772893387	14171.2271066126
			6	117415.18608824	27584.8139117598
			7	153814.129550506	5185.8704494943
			8	161462.69451118	3537.30548882
			9	171392.303571389	10607.696428611
			10	209587.137991958	-26587.1379919581
			11	195239.084042404	4760.915957596
			12	207997.127749379	3002.872250621
			13	197197.036349942	17802.9636500581
			14	188092.35628821	30907.6437117904

Square feet Residual Plot

Square feet

Residuals

Age Residual Plot

30 40 30 15 32 38 27 30 26 35 18 17 40 12 -49066.406014619977 -2345.7129556209984 -10239.616517664399 -29322.435475499602 14171.227106612627 27584.813911759818 5185.8704494942504 3537.3054888200131 10607.696428610972 -26587.137991958123 4760.9159575959784 3002.8722506209742 17802.96365005814 30907.643711790413

Age

Residuals

Square feet Line Fit Plot

Selling Price 1926 2069 1720 1396 1706 1847 1950 2323 2285 3752 2300 2525 3800 1740 95000 119000 124800 135000 142800 145000 159000 165000 182000 183000 200000 211000 215000 219000 Predicted Selling Price 1926 2069 1720 1396 1706 1847 1950 2323 2285 3752 2300 2525 3800 1740 144066.40601461998 121345.712955621 135039.6165176644 164322.4354754996 128628.77289338737 117415.18608824018 153814.12955050575 161462.69451117999 171392.30357138903 209587.13799195812 195239.08404240402 207997.12774937903 197197.03634994186 188092.35628820959

Square feet

Selling Price

Age Line Fit Plot

Selling Price 30 40 30 15 32 38 27 30 26 35 18 17 40 12 95000 119000 124800 135000 142800 145000 159000 165000 182000 183000 200000 211000 215000 219000 Predicted Selling Price 30 40 30 15 32 38 27 30 26 35 18 17 40 12 144066.40601461998 121345.712955621 135039.6165176644 164322.4354754996 128628.77289338737 117415.18608824018 153814.12955050575 161462.69451117999 171392.30357138903 209587.13799195812 195239.08404240402 207997.12774937903 197197.03634994186 188092.35628820959

Age

Selling Price

Sheet2

Sheet3

Sheet1

Selling Price	Square feet	Age
95000	1926	30
119000	2069	40
124800	1720	30
135000	1396	15
142800	1706	32
145000	1847	38
159000	1950	27
165000	2323	30
182000	2285	26
183000	3752	35
200000	2300	18
211000	2525	17
215000	3800	40
219000	1740	12
			SUMMARY OUTPUT
			Regression Statistics
			Multiple R	0.8196803049
			R Square	0.6718758022
			Adjusted R Square	0.6122168572
			Standard Error	24312.6072850603
			Observations	14
			ANOVA
				df	SS	MS	F	Significance F
			Regression	2	13313936968.4553	6656968484.22766	11.261945743	0.0021787652
			Residual	11	6502131602.97325	591102872.997569
			Total	13	19816068571.4286
				Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
			Intercept	146630.893555974	25482.0828687578	5.7542742605	0.0001275664	90545.2073136126	202716.579798335	90545.2073136126	202716.579798335
			Square feet	43.8193664901	10.2809650702	4.2621841618	0.0013380948	21.1911149391	66.447618041	21.1911149391	66.447618041
			Age	-2898.686246708	796.5649420672	-3.638982955	0.0038949963	-4651.9138632471	-1145.4586301689	-4651.9138632471	-1145.4586301689
			RESIDUAL OUTPUT
			Observation	Predicted Selling Price	Residuals
			1	144066.40601462	-49066.40601462
			2	121345.712955621	-2345.712955621
			3	135039.616517664	-10239.6165176644
			4	164322.4354755	-29322.4354754996
			5	128628.772893387	14171.2271066126
			6	117415.18608824	27584.8139117598
			7	153814.129550506	5185.8704494943
			8	161462.69451118	3537.30548882
			9	171392.303571389	10607.696428611
			10	209587.137991958	-26587.1379919581
			11	195239.084042404	4760.915957596
			12	207997.127749379	3002.872250621
			13	197197.036349942	17802.9636500581
			14	188092.35628821	30907.6437117904

Square feet Residual Plot

Square feet

Residuals

Age Residual Plot

30 40 30 15 32 38 27 30 26 35 18 17 40 12 -49066.406014619977 -2345.7129556209984 -10239.616517664399 -29322.435475499602 14171.227106612627 27584.813911759818 5185.8704494942504 3537.3054888200131 10607.696428610972 -26587.137991958123 4760.9159575959784 3002.8722506209742 17802.96365005814 30907.643711790413

Age

Residuals

Square feet Line Fit Plot

Square feet

Selling Price

Age Line Fit Plot

Age

Selling Price

Sheet2

Sheet3

Sheet1

Selling Price	Square feet	Age
95000	1926	30
119000	2069	40
124800	1720	30
135000	1396	15
142800	1706	32
145000	1847	38
159000	1950	27
165000	2323	30
182000	2285	26
183000	3752	35
200000	2300	18
211000	2525	17
215000	3800	40
219000	1740	12
			SUMMARY OUTPUT
			Regression Statistics
			Multiple R	0.8196803049
			R Square	0.6718758022
			Adjusted R Square	0.6122168572
			Standard Error	24312.6072850603
			Observations	14
			ANOVA
				df	SS	MS	F	Significance F
			Regression	2	13313936968.4553	6656968484.22766	11.261945743	0.0021787652
			Residual	11	6502131602.97325	591102872.997569
			Total	13	19816068571.4286
				Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
			Intercept	146630.893555974	25482.0828687578	5.7542742605	0.0001275664	90545.2073136126	202716.579798335	90545.2073136126	202716.579798335
			Square feet	43.8193664901	10.2809650702	4.2621841618	0.0013380948	21.1911149391	66.447618041	21.1911149391	66.447618041
			Age	-2898.686246708	796.5649420672	-3.638982955	0.0038949963	-4651.9138632471	-1145.4586301689	-4651.9138632471	-1145.4586301689
			RESIDUAL OUTPUT
			Observation	Predicted Selling Price	Residuals
			1	144066.40601462	-49066.40601462
			2	121345.712955621	-2345.712955621
			3	135039.616517664	-10239.6165176644
			4	164322.4354755	-29322.4354754996
			5	128628.772893387	14171.2271066126
			6	117415.18608824	27584.8139117598
			7	153814.129550506	5185.8704494943
			8	161462.69451118	3537.30548882
			9	171392.303571389	10607.696428611
			10	209587.137991958	-26587.1379919581
			11	195239.084042404	4760.915957596
			12	207997.127749379	3002.872250621
			13	197197.036349942	17802.9636500581
			14	188092.35628821	30907.6437117904

Square feet Residual Plot

Square feet

Residuals

Age Residual Plot

30 40 30 15 32 38 27 30 26 35 18 17 40 12 -49066.406014619977 -2345.7129556209984 -10239.616517664399 -29322.435475499602 14171.227106612627 27584.813911759818 5185.8704494942504 3537.3054888200131 10607.696428610972 -26587.137991958123 4760.9159575959784 3002.8722506209742 17802.96365005814 30907.643711790413

Age

Residuals

Square feet Line Fit Plot

Square feet

Selling Price

Age Line Fit Plot

Age

Selling Price

Sheet2

Sheet3

Sheet1

Selling Price	Square feet	Age
95000	1926	30
119000	2069	40
124800	1720	30
135000	1396	15
142800	1706	32
145000	1847	38
159000	1950	27
165000	2323	30
182000	2285	26
183000	3752	35
200000	2300	18
211000	2525	17
215000	3800	40
219000	1740	12
			SUMMARY OUTPUT
			Regression Statistics
			Multiple R	0.8196803049
			R Square	0.6718758022
			Adjusted R Square	0.6122168572
			Standard Error	24312.6072850603
			Observations	14
			ANOVA
				df	SS	MS	F	Significance F
			Regression	2	13313936968.4553	6656968484.22766	11.261945743	0.0021787652
			Residual	11	6502131602.97325	591102872.997569
			Total	13	19816068571.4286
				Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
			Intercept	146630.893555974	25482.0828687578	5.7542742605	0.0001275664	90545.2073136126	202716.579798335	90545.2073136126	202716.579798335
			Square feet	43.8193664901	10.2809650702	4.2621841618	0.0013380948	21.1911149391	66.447618041	21.1911149391	66.447618041
			Age	-2898.686246708	796.5649420672	-3.638982955	0.0038949963	-4651.9138632471	-1145.4586301689	-4651.9138632471	-1145.4586301689
			RESIDUAL OUTPUT
			Observation	Predicted Selling Price	Residuals
			1	144066.40601462	-49066.40601462
			2	121345.712955621	-2345.712955621
			3	135039.616517664	-10239.6165176644
			4	164322.4354755	-29322.4354754996
			5	128628.772893387	14171.2271066126
			6	117415.18608824	27584.8139117598
			7	153814.129550506	5185.8704494943
			8	161462.69451118	3537.30548882
			9	171392.303571389	10607.696428611
			10	209587.137991958	-26587.1379919581
			11	195239.084042404	4760.915957596
			12	207997.127749379	3002.872250621
			13	197197.036349942	17802.9636500581
			14	188092.35628821	30907.6437117904

Square feet Residual Plot

Square feet

Residuals

Age Residual Plot

30 40 30 15 32 38 27 30 26 35 18 17 40 12 -49066.406014619977 -2345.7129556209984 -10239.616517664399 -29322.435475499602 14171.227106612627 27584.813911759818 5185.8704494942504 3537.3054888200131 10607.696428610972 -26587.137991958123 4760.9159575959784 3002.8722506209742 17802.96365005814 30907.643711790413

Age

Residuals

Square feet Line Fit Plot

Square feet

Selling Price

Age Line Fit Plot

Age

Selling Price

Sheet2

Sheet3

Sheet1

Selling Price	Square feet	Age
95000	1926	30
119000	2069	40
124800	1720	30
135000	1396	15
142800	1706	32
145000	1847	38
159000	1950	27
165000	2323	30
182000	2285	26
183000	3752	35
200000	2300	18
211000	2525	17
215000	3800	40
219000	1740	12
			SUMMARY OUTPUT
			Regression Statistics
			Multiple R	0.8196803049
			R Square	0.6718758022
			Adjusted R Square	0.6122168572
			Standard Error	24312.6072850603
			Observations	14
			ANOVA
				df	SS	MS	F	Significance F
			Regression	2	13313936968.4553	6656968484.22766	11.261945743	0.0021787652
			Residual	11	6502131602.97325	591102872.997569
			Total	13	19816068571.4286
				Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
			Intercept	146630.893555974	25482.0828687578	5.7542742605	0.0001275664	90545.2073136126	202716.579798335	90545.2073136126	202716.579798335
			Square feet	43.8193664901	10.2809650702	4.2621841618	0.0013380948	21.1911149391	66.447618041	21.1911149391	66.447618041
			Age	-2898.686246708	796.5649420672	-3.638982955	0.0038949963	-4651.9138632471	-1145.4586301689	-4651.9138632471	-1145.4586301689
			RESIDUAL OUTPUT
			Observation	Predicted Selling Price	Residuals
			1	144066.40601462	-49066.40601462
			2	121345.712955621	-2345.712955621
			3	135039.616517664	-10239.6165176644
			4	164322.4354755	-29322.4354754996
			5	128628.772893387	14171.2271066126
			6	117415.18608824	27584.8139117598
			7	153814.129550506	5185.8704494943
			8	161462.69451118	3537.30548882
			9	171392.303571389	10607.696428611
			10	209587.137991958	-26587.1379919581
			11	195239.084042404	4760.915957596
			12	207997.127749379	3002.872250621
			13	197197.036349942	17802.9636500581
			14	188092.35628821	30907.6437117904

Square feet Residual Plot

Square feet

Residuals

Age Residual Plot

30 40 30 15 32 38 27 30 26 35 18 17 40 12 -49066.406014619977 -2345.7129556209984 -10239.616517664399 -29322.435475499602 14171.227106612627 27584.813911759818 5185.8704494942504 3537.3054888200131 10607.696428610972 -26587.137991958123 4760.9159575959784 3002.8722506209742 17802.96365005814 30907.643711790413

Age

Residuals

Square feet Line Fit Plot

Square feet

Selling Price

Age Line Fit Plot

Age

Selling Price

Sheet2

Sheet3

Sheet1

Selling Price	Square feet	Age
95000	1926	30
119000	2069	40
124800	1720	30
135000	1396	15
142800	1706	32
145000	1847	38
159000	1950	27
165000	2323	30
182000	2285	26
183000	3752	35
200000	2300	18
211000	2525	17
215000	3800	40
219000	1740	12
			SUMMARY OUTPUT
			Regression Statistics
			Multiple R	0.8196803049
			R Square	0.6718758022
			Adjusted R Square	0.6122168572
			Standard Error	24312.6072850603
			Observations	14
			ANOVA
				df	SS	MS	F	Significance F
			Regression	2	13313936968.4553	6656968484.22766	11.261945743	0.0021787652
			Residual	11	6502131602.97325	591102872.997569
			Total	13	19816068571.4286
				Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
			Intercept	146630.893555974	25482.0828687578	5.7542742605	0.0001275664	90545.2073136126	202716.579798335	90545.2073136126	202716.579798335
			Square feet	43.8193664901	10.2809650702	4.2621841618	0.0013380948	21.1911149391	66.447618041	21.1911149391	66.447618041
			Age	-2898.686246708	796.5649420672	-3.638982955	0.0038949963	-4651.9138632471	-1145.4586301689	-4651.9138632471	-1145.4586301689
			RESIDUAL OUTPUT
			Observation	Predicted Selling Price	Residuals
			1	144066.40601462	-49066.40601462
			2	121345.712955621	-2345.712955621
			3	135039.616517664	-10239.6165176644
			4	164322.4354755	-29322.4354754996
			5	128628.772893387	14171.2271066126
			6	117415.18608824	27584.8139117598
			7	153814.129550506	5185.8704494943
			8	161462.69451118	3537.30548882
			9	171392.303571389	10607.696428611
			10	209587.137991958	-26587.1379919581
			11	195239.084042404	4760.915957596
			12	207997.127749379	3002.872250621
			13	197197.036349942	17802.9636500581
			14	188092.35628821	30907.6437117904

Square feet Residual Plot

Square feet

Residuals

Age Residual Plot

30 40 30 15 32 38 27 30 26 35 18 17 40 12 -49066.406014619977 -2345.7129556209984 -10239.616517664399 -29322.435475499602 14171.227106612627 27584.813911759818 5185.8704494942504 3537.3054888200131 10607.696428610972 -26587.137991958123 4760.9159575959784 3002.8722506209742 17802.96365005814 30907.643711790413

Age

Residuals

Square feet Line Fit Plot

Square feet

Selling Price

Age Line Fit Plot

Age

Selling Price

Sheet2

Sheet3

Sheet1

Selling Price	Square feet	Age
95000	1926	30
119000	2069	40
124800	1720	30
135000	1396	15
142800	1706	32
145000	1847	38
159000	1950	27
165000	2323	30
182000	2285	26
183000	3752	35
200000	2300	18
211000	2525	17
215000	3800	40
219000	1740	12
			SUMMARY OUTPUT
			Regression Statistics
			Multiple R	0.8196803049
			R Square	0.6718758022
			Adjusted R Square	0.6122168572
			Standard Error	24312.6072850603
			Observations	14
			ANOVA
				df	SS	MS	F	Significance F
			Regression	2	13313936968.4553	6656968484.22766	11.261945743	0.0021787652
			Residual	11	6502131602.97325	591102872.997569
			Total	13	19816068571.4286
				Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
			Intercept	146630.893555974	25482.0828687578	5.7542742605	0.0001275664	90545.2073136126	202716.579798335	90545.2073136126	202716.579798335
			Square feet	43.8193664901	10.2809650702	4.2621841618	0.0013380948	21.1911149391	66.447618041	21.1911149391	66.447618041
			Age	-2898.686246708	796.5649420672	-3.638982955	0.0038949963	-4651.9138632471	-1145.4586301689	-4651.9138632471	-1145.4586301689
			RESIDUAL OUTPUT
			Observation	Predicted Selling Price	Residuals
			1	144066.40601462	-49066.40601462
			2	121345.712955621	-2345.712955621
			3	135039.616517664	-10239.6165176644
			4	164322.4354755	-29322.4354754996
			5	128628.772893387	14171.2271066126
			6	117415.18608824	27584.8139117598
			7	153814.129550506	5185.8704494943
			8	161462.69451118	3537.30548882
			9	171392.303571389	10607.696428611
			10	209587.137991958	-26587.1379919581
			11	195239.084042404	4760.915957596
			12	207997.127749379	3002.872250621
			13	197197.036349942	17802.9636500581
			14	188092.35628821	30907.6437117904

Square feet Residual Plot

Square feet

Residuals

Age Residual Plot

30 40 30 15 32 38 27 30 26 35 18 17 40 12 -49066.406014619977 -2345.7129556209984 -10239.616517664399 -29322.435475499602 14171.227106612627 27584.813911759818 5185.8704494942504 3537.3054888200131 10607.696428610972 -26587.137991958123 4760.9159575959784 3002.8722506209742 17802.96365005814 30907.643711790413

Age

Residuals

Square feet Line Fit Plot

Square feet

Selling Price

Age Line Fit Plot

Age

Selling Price