simplified pre-discussed beer memo regression

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Scatter Plot
12
y = 0.0275x + 0.9526
R² = 0.814
10
Y
8
6
4
2
0
0
50
100
150
200
X
250
300
350
Calories Pct Alcohol
153
4.9
157
5.9
95
4.2
130
5
123
5.5
115
5
110
4.2
116
4.2
145
5
99
4.3
55
2.4
133
4.6
169
5.9
95
4.1
138
4.3
174
6.1
149
4.9
152
5
104
4.2
144
4.7
144
4.7
110
3.9
132
5
149
5.5
103
4.1
157
5.6
145
4.6
166
5.2
155
5
152
4.7
110
4.1
175
4.9
113
4.3
95
4.1
157
5.6
157
5.8
110
4.2
143
4.7
64
2.8
110
4.2
143
4.7
110
4.2
96
4.2
110
4.2
110
4.2
110
4.2
128
144
98
70
146
114
160
202
220
120
160
160
166
195
124
146
190
330
214
157
190
215
231
175
218
194
225
158
153
123
149
113
94
177
163
154
163
153
171
158
146
292
269
314
131
181
150
4.3
5.9
4.5
0.4
4.5
3.8
5.9
7.5
8
4.5
4.9
4.8
5.2
4.7
4.1
4.7
5.9
9.6
6.8
5
5.9
6.7
6.9
5.6
7
5.6
5.8
5
4.4
5
4.6
4.4
4.1
5.2
4.7
4.7
4.7
4.8
5.4
4.7
5.3
10.5
8.7
10.5
4.7
6.5
5.3
158
179
124
148
162
156
148
162
142
103
111
110
170
149
105
163
152
166
165
205
200
200
140
160
155
145
215
146
153
174
179
188
142
222
160
222
135
161
151
147
150
145
135
98
150
135
5.2
6.4
4.6
4.5
5.1
5.9
4.9
5
5.9
4.1
4.4
4.1
4.9
4.9
4.2
4.9
4.7
4.9
4.9
5.6
6.6
7
4.8
5.2
4.8
4.8
7.8
4.7
5
5.3
5.8
6.5
4.6
8.1
6
8.1
4.2
5.1
4.9
4.6
4.8
5
4.4
3.8
4.5
4.4
Calories Pct Alcohol
153
4.9
157
5.9
95
4.2
130
5
123
5.5
115
5
110
4.2
116
4.2
145
5
99
4.3
55
2.4
133
4.6
169
5.9
95
4.1
138
4.3
174
6.1
149
4.9
152
5
104
4.2
144
4.7
144
4.7
110
3.9
132
5
149
5.5
103
4.1
157
5.6
145
4.6
166
5.2
155
5
152
4.7
110
4.1
175
4.9
113
4.3
95
4.1
157
5.6
157
5.8
110
4.2
143
4.7
64
2.8
110
4.2
143
4.7
110
4.2
96
4.2
110
4.2
110
4.2
110
4.2
128
144
98
70
146
114
160
202
220
120
160
160
166
195
124
146
190
330
214
157
190
215
231
175
218
194
225
158
153
123
149
113
94
177
163
154
163
153
171
158
146
292
269
314
131
181
150
4.3
5.9
4.5
0.4
4.5
3.8
5.9
7.5
8
4.5
4.9
4.8
5.2
4.7
4.1
4.7
5.9
9.6
6.8
5
5.9
6.7
6.9
5.6
7
5.6
5.8
5
4.4
5
4.6
4.4
4.1
5.2
4.7
4.7
4.7
4.8
5.4
4.7
5.3
10.5
8.7
10.5
4.7
6.5
5.3
158
179
124
148
162
156
148
162
142
103
111
110
170
149
105
163
152
166
165
205
200
200
140
160
155
145
215
146
153
174
179
188
142
222
160
222
135
161
151
147
150
145
135
98
150
135
5.2
6.4
4.6
4.5
5.1
5.9
4.9
5
5.9
4.1
4.4
4.1
4.9
4.9
4.2
4.9
4.7
4.9
4.9
5.6
6.6
7
4.8
5.2
4.8
4.8
7.8
4.7
5
5.3
5.8
6.5
4.6
8.1
6
8.1
4.2
5.1
4.9
4.6
4.8
5
4.4
3.8
4.5
4.4
Simple Linear Regression Analysis
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.9022
0.8140
0.8126
0.5642
139
ANOVA
df
Regression
Residual
Total
Intercept
Calories
1
137
138
SS
MS
F
Significance F
190.8133 190.8133 599.5098
0.0000
43.6047
0.3183
234.4180
Coefficients Standard Error
0.9526
0.1777
0.0275
0.0011
t Stat
P-value
5.3620
0.0000
24.4849
0.0000
Lower 95%
Upper 95%
0.6013
1.3039
0.0253
0.0297
Calculations
b1, b0 Coefficients
0.0275
0.9526
b1, b0 Standard Error
0.0011
0.1777
R Square, Standard Error
0.8140
0.5642
F , Residual df
599.5098 137.0000
Regression SS , Residual SS
190.8133 43.6047
Confidence level
t Critical Value
Half Width b0
Half Width b1
Lower 95%
0.6013
0.0253
Upper 95%
1.30390
0.02972
95%
1.9774
0.3513
0.0022
Calories
153
157
95
130
123
115
110
116
145
99
55
133
169
95
138
174
149
152
104
144
144
110
132
149
103
157
145
166
155
152
110
175
113
95
157
157
110
143
64
110
143
110
96
110
110
110
128
144
98
70
146
114
160
202
220
120
160
160
166
195
124
146
190
330
214
157
190
215
231
175
218
194
225
158
153
123
149
113
94
177
163
154
163
153
171
158
146
292
269
314
131
181
150
158
179
124
148
162
156
148
162
142
103
111
110
170
149
105
163
152
166
165
205
200
200
140
160
155
145
215
146
153
174
179
188
142
222
160
222
135
161
151
147
150
145
135
98
150
135
Calories
Mean
Median
Mode
Minimum
Maximum
Range
Variance
Standard Deviation
Coeff. of Variation
Skewness
Kurtosis
Count
Standard Error
Calories
152.3165468
150
110
55
330
275
1828.0005
42.7551
28.07%
1.1860
3.3511
139
3.6264
National Regional
4.9
4.1
5.9
5.2
4.2
4.7
5.0
4.7
5.5
4.7
5.0
4.8
4.2
5.4
4.2
4.7
5.0
5.3
4.3
10.5
2.4
8.7
4.6
10.5
5.9
4.7
4.1
6.5
4.3
5.3
6.1
5.2
4.9
6.4
5.0
4.6
4.2
4.5
4.7
5.1
4.7
5.9
3.9
4.9
5.0
5.0
5.5
5.9
4.1
4.1
5.6
4.4
4.6
4.1
5.2
4.9
5.0
4.9
4.7
4.2
4.1
4.9
4.9
4.7
4.3
4.9
4.1
4.9
5.6
5.6
5.8
6.6
4.2
7.0
4.7
4.8
2.8
5.2
4.2
4.8
4.7
4.8
4.2
7.8
4.2
4.7
4.2
5.0
4.2
5.3
4.2
5.8
4.3
5.9
4.5
0.4
4.5
3.8
5.9
7.5
8.0
4.5
4.9
4.8
5.2
4.7
4.1
4.7
5.9
9.6
6.8
5.0
5.9
6.7
6.9
5.6
7.0
5.6
5.8
5.0
4.4
5.0
4.6
4.4
6.5
4.6
8.1
6.0
8.1
4.2
5.1
4.9
4.6
4.8
5.0
4.4
3.8
4.5
4.4
Percent Alcohol and Dist. Type
Mean
Median
Mode
Minimum
Maximum
Range
Variance
Standard Deviation
Coeff. of Variation
Skewness
Kurtosis
Count
Standard Error
National
Regional
4.935897436 5.404918033
4.7
4.9
4.2
4.7
0.4
3.8
9.6
10.5
9.2
6.7
1.4327
1.9428
1.1970
1.3938
24.25%
25.79%
0.4212
2.1506
4.8372
4.8715
78
61
0.1355
0.1785
Pct Alcohol
4.9
5.9
4.2
5.0
5.5
5.0
4.2
4.2
5.0
4.3
2.4
4.6
5.9
4.1
4.3
6.1
4.9
5.0
4.2
4.7
4.7
3.9
5.0
5.5
4.1
5.6
4.6
5.2
5.0
4.7
4.1
4.9
4.3
4.1
5.6
5.8
4.2
4.7
2.8
4.2
4.7
4.2
4.2
4.2
4.2
4.2
4.3
5.9
4.5
0.4
4.5
3.8
5.9
7.5
8.0
4.5
4.9
4.8
5.2
4.7
4.1
4.7
5.9
9.6
6.8
5.0
5.9
6.7
6.9
5.6
7.0
5.6
5.8
5.0
4.4
5.0
4.6
4.4
4.1
5.2
4.7
4.7
4.7
4.8
5.4
4.7
5.3
10.5
8.7
10.5
4.7
6.5
5.3
5.2
6.4
4.6
4.5
5.1
5.9
4.9
5.0
5.9
4.1
4.4
4.1
4.9
4.9
4.2
4.9
4.7
4.9
4.9
5.6
6.6
7.0
4.8
5.2
4.8
4.8
7.8
4.7
5.0
5.3
5.8
6.5
4.6
8.1
6.0
8.1
4.2
5.1
4.9
4.6
4.8
5.0
4.4
3.8
4.5
4.4
Percent Alcohol
Mean
Median
Mode
Minimum
Maximum
Range
Variance
Standard Deviation
Coeff. of Variation
Skewness
Kurtosis
Count
Standard Error
Pct Alcohol
5.141726619
4.9
4.7
0.4
10.5
10.1
1.6987
1.3033
25.35%
1.3832
5.3178
139
0.1105
Percent Alcohol
Five-Number Summary
Minimum
0.4
First Quartile
4.4
Median
4.9
Third Quartile
5.6
Maximum
10.5
Percent Alcohol
Pct Alcohol
0
2
4
6
8
10
12
Percent Alcohol and Dist. Type
Five-Number Summary
National Regional
Minimum
0.4
3.8
First Quartile
4.2
4.7
Median
4.7
4.9
Third Quartile
5.6
5.7
Maximum
9.6
10.5
Percent Alcohol and Dist. Type
Regional
National
0
2
4
6
8
10
12
Calories
Five-Number Summary
Minimum
55
First Quartile
124
Median
150
Third Quartile
166
Maximum
330
Calories
Calories
50
100
150
200
250
300
350
DistType
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
National
DistType
Regional
Dist. Type
0
10
20
30
40
50
60
70
80
90
Dist. Type
Count of DistType
DistType
National
Regional
Grand Total
Total
78
61
139
Regional
Dist. Type and Light
Yes
National
No
0
10
20
30
40
50
60
Dist. Type and Light
Count of DistType
DistType
National
Regional
Grand Total
Light
No
Yes
53
55
108
25
6
31
Grand Total
78
61
139
Brand
Anchor Steam
Anheuser Busch Natural Ice
Anheuser Busch Natural Light
Bud Dry
Bud Ice
Bud Ice Light
Bud Light
Bud Light Lime
Budweiser
Budweiser Select
Budweiser Select 55
Busch Beer
Busch Ice
Busch Light
Carling Black Label
Colt 45 Malt Liquor
Coors
Coors Extra Gold Lager
Coors Light
Hamm’s Beer
Hamm’s Golden Draft
Hamm’s Special Light
Icehouse
Icehouse 5
Icehouse Light
Magnum Malt Liquor
Michael Shea’s
Michelob Amber Boch
Michelob Beer
Michelob Golden Draft
Michelob Golden Draft Light
Michelob Honey Lager
Michelob Light
Michelob Ultra
Mickey’s Fine Malt Liquor
Mickey’s Ice
Miller Chill
Miller Genuine Draft
Miller Genuine Draft 64
Miller Genuine Draft Light
Miller High Life
Miller High Life Light
Miller Lite
Miller Lite Brwer’s Collection Amber
Miller Lite Brwer’s Collection Blonde
Miller Lite Brwer’s Collection Wheat
Pct Alcohol Calories Carbohydrates
DistTypeCODE
4.9
153
16.0
1
5.9
157
8.9
1
4.2
95
3.2
1
5.0
130
7.8
1
5.5
123
8.9
1
5.0
115
7.5
1
4.2
110
6.6
1
4.2
116
8.0
1
5.0
145
10.6
1
4.3
99
3.1
1
2.4
55
1.9
1
4.6
133
10.2
1
5.9
169
12.5
1
4.1
95
3.2
1
4.3
138
12.5
1
6.1
174
11.1
1
4.9
149
12.2
1
5.0
152
12.5
1
4.2
104
5.3
1
4.7
144
12.1
1
4.7
144
12.1
1
3.9
110
8.3
1
5.0
132
8.7
1
5.5
149
9.8
1
4.1
103
5.5
1
5.6
157
11.2
1
4.6
145
13.0
1
5.2
166
15.0
1
5.0
155
13.3
1
4.7
152
14.1
1
4.1
110
7.0
1
4.9
175
17.9
1
4.3
113
6.7
1
4.1
95
2.6
1
5.6
157
11.2
1
5.8
157
11.8
1
4.2
110
6.5
1
4.7
143
13.1
1
2.8
64
2.4
1
4.2
110
7.0
1
4.7
143
13.1
1
4.2
110
7.0
1
4.2
96
3.2
1
4.2
110
6.2
1
4.2
110
6.2
1
4.2
110
6.2
1
Milwaukee’s Best
Milwaukee’s Best Ice
Milwaukee’s Best Light
O’Doul’s
Old Milwaukee Beer
Old Milwaukee Light
Olde English 800
Olde English 800 7.5
Olde English High Gravity 800
Rolling Rock Premium Beer
Sam Adams Boston Ale
Sam Adams Boston Lager
Sam Adams Cherry Wheat
Sam Adams Cream Stout
Sam Adams Light
Schlitz
Sierra Nevada Anniversary Ale
Sierra Nevada Bigfoot
Sierra Nevada Celebration Ale
Sierra Nevada Draft Ale
Sierra Nevada Early Spring Beer
Sierra Nevada Harvest Ale
Sierra Nevada India Pale Ale
Sierra Nevada Pale Ale
Sierra Nevada Pale Bock
Sierra Nevada Porter
Sierra Nevada Stout
Sierra Nevada Summerfest Beer
Sierra Nevada Wheat Beer
Southpaw Light
Stroh’s Beer
Stroh’s Light
Aspen Edge
Big Sky Moose Drool Brown Ale
Big Sky Scape Goat Pale Ale
Big Sky Summer Honey Ale
Big Sky Trout Slayer Ale
Blatz Beer
Blue Moon
Flying Dog Doggie Style
Flying Dog Dogtober Fest
Flying Dog Double Dog Pale Ale
Flying Dog Gonzo
Flying Dog Horn Dog
Flying Dog In Heat Wheat
Flying Dog K-9 Cruiser
Flying Dog Old Scratch
4.3
5.9
4.5
0.4
4.5
3.8
5.9
7.5
8.0
4.5
4.9
4.8
5.2
4.7
4.1
4.7
5.9
9.6
6.8
5.0
5.9
6.7
6.9
5.6
7.0
5.6
5.8
5.0
4.4
5.0
4.6
4.4
4.1
5.2
4.7
4.7
4.7
4.8
5.4
4.7
5.3
10.5
8.7
10.5
4.7
6.5
5.3
128
144
98
70
146
114
160
202
220
120
160
160
166
195
124
146
190
330
214
157
190
215
231
175
218
194
225
158
153
123
149
113
94
177
163
154
163
153
171
158
146
292
269
314
131
181
150
11.4
7.3
3.5
13.3
12.9
8.3
10.5
13.4
14.6
10.0
19.9
18.0
16.9
23.9
9.7
12.1
17.3
32.1
19.4
13.4
16.7
19.3
20.0
14.1
19.7
18.4
22.3
13.7
13.1
6.6
12.0
7.0
2.6
15.6
13.9
11.6
13.9
11.6
13.7
11.4
11.4
15.0
18.6
18.9
8.3
10.6
9.6
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Flying Dog Road Dog
Flying Dog Snake Dog
Flying Dog Tire Bite
Genesee Beer
Genesee Cream Ale
Genesee Ice
Genesee Red
George Killian’s Irish Red
Keystone Ice
Keystone Light
Keystone Premium
Leinenkugel Amber Light
Leinenkugel Creamy Dark
Leinenkugel Honey Weiss
Leinenkugel Light
Leinenkugel Northwoods Lager
Leinenkugel Original
Leinenkugel Red
Leinenkugel Sunset Wheat
New Belgium 1554
New Belgium 2 Below
New Belgium Abbey
New Belgium Blue Paddle
New Belgium Fat Tire
New Belgium Mothership Wit
New Belgium Sunshine Wheat
New Belgium Trippel
Olympia Premium Lager
Pabst Blue Ribbon
Pete’s Wicked Ale
Red Hook ESB
Red Hook IPA
Schaefer
Steel Reserve
Steel Reserve Six
Steel Reserve Triple Export
Weinhard’s Amber Light
Weinhard’s Blonde Lager
Weinhard’s Hefweizen
Weinhard’s Pale Ale
Weinhard’s Private Reserve
Yuengling Ale
Yuengling Lager
Yuengling Light
Yuengling Porter
Yuengling Premium Beer
5.2
6.4
4.6
4.5
5.1
5.9
4.9
5.0
5.9
4.1
4.4
4.1
4.9
4.9
4.2
4.9
4.7
4.9
4.9
5.6
6.6
7.0
4.8
5.2
4.8
4.8
7.8
4.7
5.0
5.3
5.8
6.5
4.6
8.1
6.0
8.1
4.2
5.1
4.9
4.6
4.8
5.0
4.4
3.8
4.5
4.4
158
179
124
148
162
156
148
162
142
103
111
110
170
149
105
163
152
166
165
205
200
200
140
160
155
145
215
146
153
174
179
188
142
222
160
222
135
161
151
147
150
145
135
98
150
135
12.0
10.6
7.1
13.5
15.0
14.5
14.0
14.8
5.9
5.0
5.8
7.4
16.8
12.0
5.7
15.3
13.9
16.2
16.0
25.0
17.0
18.0
14.0
15.0
15.0
13.0
20.0
11.9
12.0
17.7
14.2
12.7
12.1
16.0
11.0
16.0
11.5
14.0
12.2
13.0
9.9
10.0
12.0
6.6
14.0
12.0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
DistType
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
Light
No
No
Yes
No
No
Yes
Yes
Yes
No
Yes
Yes
No
No
Yes
No
No
No
No
Yes
No
No
Yes
No
No
Yes
No
No
No
No
No
Yes
No
Yes
Yes
No
No
No
No
Yes
Yes
No
Yes
Yes
Yes
Yes
Yes
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
National
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
No
No
Yes
No
No
Yes
No
No
No
No
No
No
No
No
Yes
No
No
No
No
No
No
No
No
No
No
No
No
No
No
Yes
No
Yes
Yes
No
No
No
No
No
No
No
No
No
No
No
No
No
No
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
Regional
No
No
No
No
No
No
No
No
No
Yes
No
Yes
No
No
Yes
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
Yes
No
No
No
No
No
No
Yes
No
No
Frequency Distribution for Pct Alcohol
bins
Midpts. Frequency Percentage
-0.01 -0
0.0%
0.99
0.5
1
0.7%
1.99
1.5
0
0.0%
2.99
2.5
2
1.4%
3.99
3.5
3
2.2%
4.99
4.5
72
51.8%
5.99
5.5
41
29.5%
6.99
6.5
9
6.5%
7.99
7.5
4
2.9%
8.99
8.5
4
2.9%
9.99
9.5
1
0.7%
10.99
10.5
2
1.4%
Total
139
100.0%
7-Beer Study
(A random sample of 139 beers.)
Scenario
You are employed as a research assistant at the Alcohol and Tobacco Tax and Trade Bureau (U. S.
Department of the Treasury) and your supervisor, R. Tyler Paterson, asks you to study the characteristics
of beers sold throughout the United States. For this purpose you take a sample of 139 beers.
For each beer you collect data on several variables (see the Variable INFO tab) but for the reports you will
be preparing you decide to focus on one key numerical variable, “Pct Alcohol” (i.e., the alcoholic content
in percentage), and one key categorical variable, “Light” (i.e., whether or not the beer product is
considered to be light). You also decide to use a grouping categorical variable “Distribution Type” since
each beer is distributed nationally or regionally. This will enable you to make comparisons of both the
percent alcohol of the beer and whether or not the beer is considered to be light based on its distribution
– national or regional. In addition, you have also selected the numerical variable “Calories” in the beer to
develop a simple linear regression model to predict the “Pct Alcohol.”
Introduction to Simple Linear
Regression Modeling
(Prepared by Mark L. Berenson)
This chapter is an introduction to regression analysis modeling techniques that enable you
to use a numerical independent variable to predict the values of a numerical dependent
variable of interest. For example, the placement officer at your university can predict the
expected starting salary (in thousands of dollars) of a graduating business student by
developing a simple linear regression model that uses cumulative grade point average as
the numerical independent variable.
Regression analysis is fundamental to business decision-making because it involves
prediction/estimation/forecasting – three words used here synonymously. Below are
some examples of practical uses of regression analysis:
•
•
•
•
•
An investment analyst can estimate your credit score rating based on current
salary.
A family doctor can forecast your relative’s survivability from surgery based on
hours in surgery.
A financial analyst can predict your company’s sustainability based on revenues
generated through the year.
A real estate agent can estimate the value of your house (in dollars) based on its
size in square feet.
The admissions director of an MBA program can forecast your chances of success
by estimating your graduate grade point average based on your GMAT score.
In a regression analysis, the dependent variable, given by the symbol Y, is the numerical
variable of interest that you want to predict. The dependent variable is often referred to as
the response variable. The independent variable, given by the symbol X, is the numerical
variable used to make the prediction. The independent variable is often referred to as the
predictor or explanatory variable.
1 Developing the Simple Linear Regression Model
A regression analysis begins by visually observing the relationship between the two
numerical variables in a scatter plot. Therefore, to develop a simple linear regression
model you need two numerical measurements on each item in your sample, such as the
expected starting salary of a graduating student and his/her cumulative grade point
average, or the selling price of a house and its size in square feet. Each pair of
measurements is plotted such that the dependent variable of interest is on the vertical or Y
axis, and the independent or predictor variable is on the horizontal or X axis.
Figure 1 shows scatter plots demonstrating both strong and weak linear relationships.
1
FIGURE 1 Scatter plots of strong and weak linear relationships
In the top two panels of Figure 1 you observe positive relationships between X and Y; in
the bottom two panels you see negative relationships between X and Y.
Sometimes the cloud of points will appear to follow a curved pattern instead of a straight
line pattern. In such circumstances be sure to consult with a professional statistician –
curvilinear regression analysis is outside the scope of this course, which focuses on
simple linear regression analysis.
Figure 2 depicts scatter plots with linear and curvilinear relationships.
FIGURE 2 Scatter plots of linear and curvilinear relationships
2
To demonstrate the development of a simple linear regression model, Figure 3 displays
part of an Excel worksheet of a data file constructed at Bergen University (“BU”). Figure
4 is the scatter plot representing the test scores achieved by the sample of 93 business
students based on the number of hours the students claimed to have studied for their
comprehensive (i.e., all topics covered in the semester) final exam in their core-required
operations management course. You can see that the cloud of points plot upward and
toward the right. Any straight line drawn through these points would therefore indicate a
positive relationship between X and Y – a positive correlation and a positive slope.
FIGURE 3 Excel worksheet of the Bergen University data file containing
93 students and displaying student ID number,
test scores, and hours studied
ID Number
ID0001
ID0002
ID0003
ID0004
ID0005
ID0006
ID0007
ID0008
:
ID0086
ID0087

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