It is pretty common across most schools to find the grades at the MBA level divided between A’s and B’s. As such, you expect the mean GPA to be around 3.50. Using the sample of 200 MBA students, conduct a one-sample hypothesis test to determine if the mean GPA is different from 3.50. Use a .05 significance level.
Assume you read in the Whatsamatta U website that the average age of their MBA students is 45. Is this really true or have they failed to update this correctly? You think it is far less because there have been a lot more students going straight from their Bachelors to their Masters since the economy is so bad. You took a sample of 200 students (in the data file). Conduct a one-sample hypothesis test to determine if the mean age is less than 45. Use a .05 significance level.
You have heard from idle chatter that most students don’t declare a major in their MBA programs. You took a sample of 200 students (in the data file). Conduct a one-sample hypothesis test to determine if the proportion without a major is greater than 50%. Use a .05 significance level.
Hyp Test Prop – One Sample
One Sample Hypothesis Test for the Proportion
Null Hypothesis
Level of Significance
Number of Successes
Sample Size
P =
0,5
0,05
40
100
This is an lower-tailed test since we are testing if the proportion is less than 50%.
The p-value of .0228 is less than .05, and so we reject the null hypothesis.
Thus we conclude that the proportion of homes made of brick is less than 50%.
Sample Proportion
0,40
(computed from Successes / Sample Size)
Z Test Statistic (Computed)
-2,00
Direction of Test
Lower Crit Value
Upper Crit Value
p -Value
Decision
Two-Tailed Test
H1: P 0.5
-1,9600
1,9600
0,0455
Reject the null hypothesis
Upper-Tail Test
H1: P > 0.5
n/a
1,6449
0,9772
Do not reject the null hypothesis
Lower-Tail Test
H1: P < 0.5
-1,6449
n/a
0,0228
Reject the null hypothesis
If you REJECT the null hypothesis, conclude that H1 is true.
If you DO NOT REJECT the null hypothesis, there is insufficient evidence to conclude that H1 is true.
NEVER conclude that the null hypothesis is true (i.e., we CANNOT ACCEPT the null).
©2007 DrJimMirabella.com
ID
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
Gender
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
Finance
No Major
No Major
Finance
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
Finance
No Major
No Major
No Major
Finance
No Major
Finance
Finance
No Major
Finance
No Major
Finance
Finance
No Major
Finance
Finance
Employ
Unemployed
Full Time
Part Time
Full Time
Full Time
Unemployed
Full Time
Full Time
Part Time
Full Time
Part Time
Full Time
Full Time
Full Time
Part Time
Full Time
Full Time
Part Time
Full Time
Unemployed
Full Time
Part Time
Full Time
Full Time
Part Time
Full Time
Part Time
Unemployed
Part Time
Full Time
Full Time
Unemployed
Full Time
Full Time
Part Time
Part Time
Full Time
Unemployed
Full Time
Part Time
Full Time
Full Time
Full Time
Unemployed
Full Time
Age
39
55
43
56
38
54
30
37
38
42
52
35
37
53
51
40
33
53
43
35
57
32
59
48
34
53
35
38
37
46
44
31
51
47
56
42
44
54
51
42
45
55
47
43
57
MBA_GPA
2.82
4
3.45
2.61
3.5
4
3
2.5
2.84
3.72
3.21
3.44
3.65
3.02
3.03
3.8
4
3.26
3.53
3.75
3.15
3.66
3.36
3.79
2.85
3.74
3.23
3.52
3.32
2.89
2.83
2.93
3.71
3.47
3.52
2.83
3.64
2.96
3.59
3.33
3.38
3.44
3.31
3.03
3.26
BS GPA
3
4
3.5
4
3.3
3.05
4
3.6
3.05
3.7
3.5
3.55
2.78
3.3
3.25
4
3.5
3.5
3.75
3.9
3.2
3.75
3.45
2.55
3.05
3.9
4
3.7
3.45
3.1
3.05
3.1
3.8
2.6
3.8
4
3.55
3.1
3.9
3.9
3.6
3.35
3.9
3.25
3.4
Hrs_Studying
10
15
3
4
5
5
6
6
6
6
6
6
6
6
6
6
6
7
6
7
6
8
8
8
8
8
2
2
2
2
1
1
1
4
4
4
6
6
6
6
6
6
7
7
7
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
1
1
1
1
0
0
0
0
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
1
1
1
1
1
1
1
Finance
No Major
Finance
Finance
Finance
No Major
Finance
No Major
Finance
Finance
Marketing
Marketing
Marketing
Leadership
Leadership
Marketing
Marketing
Marketing
Marketing
Marketing
No Major
No Major
No Major
No Major
Marketing
Leadership
Leadership
Leadership
Leadership
Leadership
No Major
Leadership
No Major
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
No Major
Marketing
Marketing
No Major
Finance
Finance
Finance
Finance
Full Time
Part Time
Full Time
Full Time
Full Time
Full Time
Full Time
Full Time
Unemployed
Full Time
Part Time
Full Time
Full Time
Part Time
Full Time
Full Time
Full Time
Full Time
Full Time
Full Time
Unemployed
Full Time
Part Time
Full Time
Part Time
Full Time
Part Time
Full Time
Full Time
Full Time
Full Time
Full Time
Full Time
Full Time
Unemployed
Full Time
Full Time
Full Time
Full Time
Full Time
Full Time
Full Time
Part Time
Full Time
Full Time
Full Time
Full Time
36
58
46
53
59
49
34
46
46
33
56
39
51
55
38
33
34
31
37
46
31
47
54
52
43
44
34
59
45
30
32
32
40
48
51
30
31
35
33
35
31
38
46
45
59
58
46
3.04
2.98
2.8
3.75
3.64
3.65
3.18
3.44
3.06
3.51
3.33
2.81
3.64
3.05
2.85
3.56
2.92
3.35
3.46
3.59
3.11
3.65
3.17
2.97
3.77
3.21
3.17
3.65
2.94
3.53
3.65
3.61
3.7
2.91
3.09
3.77
3.79
3.59
3.38
4
2.97
3.44
3.64
3.48
2.76
3.73
2.91
4
3.1
3.05
3.75
3.65
3.8
3.3
4
3.15
3.75
3.4
3.05
3.8
3.4
3.25
3.6
3.1
3.5
3.35
3.75
3.2
3.7
3.5
3.1
3.9
3.2
3.15
3.65
3.1
3.7
3.6
3.7
3.9
3.1
3.25
3.95
3.8
3.6
3.5
3.5
3.1
3.65
3.55
3.4
3.1
3.8
3.05
7
7
7
3
3
3
3
3
3
10
2
2
8
7
3
7
5
7
10
8
6
8
7
5
8
6
6
10
5
8
7
8
8
5
6
9
8
7
8
8
8
8
8
8
8
8
8
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Finance
Finance
Finance
Finance
Finance
Finance
Finance
Finance
No Major
Marketing
Marketing
Leadership
Leadership
No Major
Leadership
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
Leadership
Leadership
Leadership
Leadership
Finance
No Major
No Major
Finance
Finance
Finance
Finance
Finance
Finance
Finance
Finance
Finance
Leadership
Leadership
Leadership
Finance
Finance
Finance
Finance
Full Time
Part Time
Full Time
Full Time
Full Time
Full Time
Full Time
Full Time
Unemployed
Full Time
Part Time
Full Time
Full Time
Full Time
Full Time
Part Time
Full Time
Full Time
Part Time
Full Time
Full Time
Unemployed
Full Time
Full Time
Part Time
Unemployed
Full Time
Part Time
Full Time
Full Time
Part Time
Full Time
Unemployed
Part Time
Full Time
Part Time
Full Time
Unemployed
Part Time
Full Time
Full Time
Full Time
Full Time
Unemployed
Full Time
Part Time
Full Time
35
53
31
50
38
50
48
53
53
30
32
42
56
46
49
32
36
42
37
31
31
42
39
47
28
28
52
35
38
44
38
52
53
53
31
47
51
37
46
48
54
48
36
39
28
45
31
3.78
3.5
3.13
3.14
3.24
3.56
3.16
3.53
3.7
3.3
4
3.5
3.39
3.65
2.78
3.44
3.88
2.84
3.53
3.22
3.56
3.2
3.56
3.41
3.56
3.34
2.56
3.76
3.55
3.88
3.31
3.09
3.82
3.01
3.66
3.64
3.59
3.49
3.13
3.83
3.04
3.91
3.56
3.96
3.46
3.22
3.27
3.95
3.4
3.15
3.25
3.3
3.5
3.25
3.55
3.15
3.35
3.6
3.4
3.4
3.8
3.7
3.6
3.95
3.95
3.6
3.3
3.8
3.25
3.3
3.6
3.7
3.6
3.6
3.8
3.45
3.9
3.45
3.15
4
3.2
3.85
3.7
3.65
3.55
3.2
3.9
3.15
4
3.7
4
3.4
3.15
3.2
9
7
6
6
6
7
6
7
6
6
7
7
7
8
8
7
9
9
7
6
8
6
6
7
8
7
7
8
7
8
7
6
9
6
8
8
7
7
6
8
6
10
8
9
7
6
6
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
0
1
1
1
0
1
1
1
0
1
Finance
Finance
Finance
Finance
Finance
Finance
Finance
Leadership
Leadership
Leadership
Leadership
No Major
No Major
No Major
Marketing
Marketing
Marketing
Marketing
Marketing
Marketing
Marketing
Marketing
Marketing
Marketing
Marketing
Marketing
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Full Time
Part Time
Full Time
Part Time
Unemployed
Full Time
Part Time
Full Time
Unemployed
Part Time
Unemployed
Part Time
Full Time
Unemployed
Full Time
Unemployed
Full Time
Full Time
Unemployed
Unemployed
Part Time
Full Time
Unemployed
Full Time
Part Time
Unemployed
Full Time
Part Time
Full Time
Part Time
Unemployed
Full Time
Part Time
Unemployed
Part Time
Full Time
Full Time
Part Time
Full Time
Unemployed
Full Time
Full Time
Full Time
Full Time
Full Time
Unemployed
Part Time
47
35
52
52
55
52
46
31
33
45
50
33
37
33
46
55
30
51
35
40
29
52
27
51
56
35
46
39
31
52
35
32
44
43
38
54
30
38
45
48
43
34
54
36
45
55
45
3.43
3.85
3.89
3.37
3.32
3.54
3.8
3.74
3.6
2.6
3.8
2.67
3.95
3.56
3.79
3.93
3.79
3.71
3.05
3.22
3.85
3.82
3.23
3.56
3.53
3.62
3.8
3.47
3.64
3.03
3.17
3.22
3.92
3.82
3.26
3.8
3.2
3.46
3.67
4
3.66
3.96
3.75
3.83
3.55
3.36
3.21
3.45
3.95
3.9
3.45
3.3
3.55
3.9
3.85
3.45
3.55
3.3
3.45
4
3.75
3.75
4
3.85
3.85
3.35
3.2
3.95
3.95
3.95
3.65
3.65
4
3.95
3.35
3.65
3.15
3.25
3.2
4
3.95
3.55
3.85
3.2
3.35
3.75
3.4
3.85
4
3.85
3.85
3.2
3.35
3.25
7
9
8
7
6
7
8
8
7
7
6
7
9
8
8
9
8
8
6
6
9
9
9
7
7
9
9
6
7
5
6
6
10
9
7
8
6
6
8
7
8
10
8
8
6
6
6
187
188
189
190
191
192
193
194
195
196
197
198
199
200
1
0
1
1
1
1
1
1
1
1
1
1
1
1
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Part Time
Part Time
Full Time
Full Time
Full Time
Full Time
Unemployed
Full Time
Unemployed
Unemployed
Unemployed
Unemployed
Unemployed
Full Time
34
54
36
24
34
45
33
22
27
33
36
34
55
33
2.97
3.99
3.07
3.65
3.67
3.06
3.98
3.93
3.41
3.43
3.7
3.76
3.9
3.23
3.15
4
3.15
3.65
3.85
3.35
3.7
4
3.3
3.5
3.65
3.75
3.9
3.3
5
10
6
7
8
6
8
10
6
7
7
8
8
6
Works FT
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
0
1
0
0
1
Variable descriptions
Gender = 0 (female), 1 (male)
Major = student's major
Age = age of student in years
MBA_GPA = overall GPA in the MBA program
BS_GPA = overall GPA in the BS program
Hrs_Studying = average hours studied per week
Works FT = 0 (No), 1 (Yes)
0
0
0
1
1
1
0
1
1
0
1
0
1
0
1
1
0
1
1
1
0
1
0
1
1
1
0
0
0
1
1
1
1
1
0
1
1
)
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
0
0
1
1
1
0
1
1
1
0
1
1
1
1
1
0
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
1
1
1
0
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
0
1
1
0
0
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
1
0
1
0
1
Unit 04 Practice Problems
Practice Problem 1
Null Hypothesis
=
205000
Level of Significance
0.05
Sample Size
100
Sample Mean
Standard Deviation
216710.00
43246.60
Test Statistic (Computed)
Direction of Test
2.71
Upper Crit Value
p-Value
Decision
1.6449
0.0034
Reject the null hypothesis
Upper-Tail Test
A One Sample Hypothesis test was conducted to determine if current home prices had risen above the
$205,000 average recorded in the year 2000. Using an upper-tailed test, with the p-value of .0034 being
less than .05 level of significance, we reject the null hypothesis and conclude, based upon our sample,
that the mean sales price for homes is greater than $205,000.
Unit 04 Practice Problems
Practice Problem 2
Null Hypothesis
=
5
Level of Significance
0.05
Sample Size
100
Sample Mean
5.19
Standard Deviation
2.77
Test Statistic (Computed)
0.69
Direction of Test
Lower Crit Value
Upper Crit
Value
p-Value
Two-Tailed Test
Decision
Do not reject the null
-1.9600
1.9600
0.4921
hypothesis
A two-tailedl test was conducted to determine if the average age of homes in the community was really
five years. Based on a sample of 100 homes, with a mean age of 5.19 years, there is insufficient
evidence to conclude that the average age is different from the hypothesized mean (=5). The p-value
of .4921 is greater than .05, and so we retain the null hypothesis that the average home is five years old.
Practice Problem 3
Null Hypothesis
P=
Level of Significance
Number of Brick Homes
Sample Size
0.5
0.05
40
100
Sample Proportion
0.40
Z Test Statistic
(Computed)
Direction of Test
Lower-Tail Test
H1: P < 0.5
-2.00
Lower Crit Value
p-Value
Decision
-1.6449
0.0228
Reject the null hypothesis
A lower-tailed test for proportion was conducted to determine if fewer than half of the homes are made
of brick. Based on a sample of 100 homes, this appears to be true. The p-value of .0228 is less than .05,
and so we reject the null hypothesis and conclude that less than half of the homes are made of brick.
Hyp Mean - One Sample (data)
Observation #
Data
Observation 1
Observation 2
Observation 3
Observation 4
Observation 5
Observation 6
Observation 7
Observation 8
Observation 9
Observation 10
263000
182000
242000
214000
140000
245000
300000
272000
221000
267000
292000
209000
271000
246000
194000
281000
173000
207000
199000
209000
252000
193000
209000
320000
173000
187000
257000
233000
180000
234000
Observation 11
Observation 12
Observation 13
Observation 14
Observation 15
Observation 16
Observation 17
Observation 18
Observation 19
Observation 20
Observation 21
Observation 22
Observation 23
Observation 24
Observation 25
Observation 26
Observation 27
Observation 28
Observation 29
Observation 30
One Sample Hypothesis Test for the Mean
Null Hypothesis
Level of Significance
Sample Size
Sample Mean
Standard Deviation
Test Statistic (Computed)
m=
205000
0.05
100
216710.00
43246.60
This is an upper-tailed test since we are testing if the mean is greater th
The p-value of .0034 is less than .05, and so we reject the null hypothes
Thus we conclude that the mean sales price for homes is greater than $
2.71
Direction of Test
Lower Crit Value
Upper Crit Value
p -Value
Two-Tailed Test
H1: m 205000
-1.9600
1.9600
0.0068
Upper-Tail Test
H1: m > 205000
n/a
1.6449
0.0034
Lower-Tail Test
H1: m < 205000
-1.6449
n/a
0.9966
If you REJECT the null hypothesis, conclude that H1 is true.
If you DO NOT REJECT the null hypothesis, there is insufficient evidence to conclude that H1 is true.
NEVER conclude that the null hypothesis is true (i.e., we CANNOT ACCEPT the null).
©2007 DrJimMirabella.com
Hyp Mean - One Sample (data)
Observation 31
Observation 32
Observation 33
Observation 34
Observation 35
Observation 36
Observation 37
Observation 38
Observation 39
Observation 40
Observation 41
Observation 42
Observation 43
Observation 44
Observation 45
Observation 46
Observation 47
Observation 48
Observation 49
Observation 50
Observation 51
Observation 52
Observation 53
Observation 54
Observation 55
Observation 56
Observation 57
Observation 58
Observation 59
Observation 60
Observation 61
Observation 62
207000
248000
166000
177000
183000
216000
312000
200000
273000
206000
232000
198000
205000
176000
308000
269000
225000
172000
217000
193000
236000
172000
251000
246000
147000
176000
228000
166000
189000
290000
270000
154000
Hyp Mean - One Sample (data)
Observation 63
Observation 64
Observation 65
Observation 66
Observation 67
Observation 68
Observation 69
Observation 70
Observation 71
Observation 72
Observation 73
Observation 74
Observation 75
Observation 76
Observation 77
Observation 78
Observation 79
Observation 80
Observation 81
Observation 82
Observation 83
Observation 84
Observation 85
Observation 86
Observation 87
Observation 88
Observation 89
Observation 90
Observation 91
Observation 92
Observation 93
Observation 94
222000
210000
191000
254000
207000
210000
294000
176000
224000
125000
237000
164000
218000
192000
126000
221000
295000
245000
199000
240000
263000
188000
221000
175000
253000
155000
187000
179000
188000
227000
174000
188000
Hyp Mean - One Sample (data)
Observation 95
Observation 96
Observation 97
Observation 98
Observation 99
Observation 100
Observation 101
Observation 102
Observation 103
Observation 104
Observation 105
Observation 106
Observation 107
Observation 108
Observation 109
Observation 110
Observation 111
Observation 112
Observation 113
Observation 114
Observation 115
Observation 116
Observation 117
Observation 118
Observation 119
Observation 120
Observation 121
Observation 122
Observation 123
Observation 124
Observation 125
Observation 126
294000
179000
188000
227000
174000
188000
Hyp Mean - One Sample (data)
Observation 127
Observation 128
Observation 129
Observation 130
Observation 131
Observation 132
Observation 133
Observation 134
Observation 135
Observation 136
Observation 137
Observation 138
Observation 139
Observation 140
Observation 141
Observation 142
Observation 143
Observation 144
Observation 145
Observation 146
Observation 147
Observation 148
Observation 149
Observation 150
Observation 151
Observation 152
Observation 153
Observation 154
Observation 155
Observation 156
Observation 157
Observation 158
Hyp Mean - One Sample (data)
Observation 159
Observation 160
Observation 161
Observation 162
Observation 163
Observation 164
Observation 165
Observation 166
Observation 167
Observation 168
Observation 169
Observation 170
Observation 171
Observation 172
Observation 173
Observation 174
Observation 175
Observation 176
Observation 177
Observation 178
Observation 179
Observation 180
Observation 181
Observation 182
Observation 183
Observation 184
Observation 185
Observation 186
Observation 187
Observation 188
Observation 189
Observation 190
Hyp Mean - One Sample (data)
Observation 191
Observation 192
Observation 193
Observation 194
Observation 195
Observation 196
Observation 197
Observation 198
Observation 199
Observation 200
Hyp Mean - One Sample (data)
test since we are testing if the mean is greater than $205,000.
less than .05, and so we reject the null hypothesis.
t the mean sales price for homes is greater than $205,000.
Decision
Reject the null hypothesis
Reject the null hypothesis
Do not reject the null hypothesis
de that H1 is true.
t evidence to conclude that H1 is true.
we CANNOT ACCEPT the null).
CHAPTER FOUR
HYPOTHESIS TESTING FOR ONE SAMPLE
C
The Logic of Hypothesis Testing A
How can we prove a coin is perfectly fair? And
L can a doctor find you perfectly healthy? And can
you prove your weight has not changed at all?VThe answer is NO in all cases. We can never prove
equality or lack of change, but we can find evidence
to the contrary. In medical tests, a physician
E
looks for problems and in the absence of any being detected, they tell you that you tested negative
R
and failed to find anything wrong with you. With a coin, if you flip five heads and five tails, you
are not convinced of anything regarding that T
coin; likewise if you flip nine heads and one tail, you
,
are not convinced but are starting to get suspicious. But if you flip 90 heads and 10 tails, you may
rightly believe something is wrong, while 60 heads and 40 tails may not be quite enough to sway
you.
T
If you improved your SAT by 10 points usingEa SAT improvement course, would you be satisfied?
R800 on each section and are in 10-point increments.
I doubt it, considering scores range from 200 to
But if you improved your ACT by 10, you would
R be probably be thrilled (the ACT scale is from 0
to 36). Points are relative. There is a threshold
E for everyone such that one will draw a conclusion
with little chance of error.
N
If you step on and off a scale, you notice theC
weights vary. Why? Do you lose weight merely by
stepping on and off? What if you weighed in
E at 165, stepped off the scale, stepped back on and
registered 164 – would you claim that you lost a pound from step aerobics by stepping off / on the
scale? Or if you exercised for 20 minutes and
1 got on the scale to see that same one pound loss,
would you then feel better at attributing the weight loss to that exercise? At what point do you
8
consider it a weight loss significant enough to declare it was attributed to exercise and not to the
imbalance of the scale? If you lose 5 pounds5in 5 days you might be convinced, but 3 pounds in 5
days may be shifts in water weight combined9with scale imbalance. Where is the fine line?
T
When we have a theory about a parameter (the average is ..., the proportion is ..., etc.), we can test
S
that theory via a hypothesis test. If the results are far enough from the hypothesized number, we
can conclude that with minimal error, the statement must be true. If it isn’t far enough, we state
that we just lack sufficient evidence to draw such a conclusion. Just like with weight loss, which is
a form of a hypothesis test, we look for evidence (often in the form of data) to make our case, and
then make a decision after reviewing the evidence.
Copyright 2011, Savant Learning SystemsTM
Introduction to Statistics by Jim Mirabella
4-1
Chapter Four: Hypothesis Testing for One Sample
One of the best examples of hypothesis testing can be found in the judicial system. Here the null hypothesis
(Ho) is that the defendant is innocent, while the alternate hypothesis (Ha) is that the defendant is guilty.
No one is ever found innocent, because we can never prove anyone innocent (no matter what evidence
you have about who did the crime, you will never know if the defendant was or wasn’t involved somehow,
so short of being in a coma for a year or so, we can never prove his innocence with evidence). Can you
imagine how hard it would be if you had to be proven innocent? All of the examples of honesty cannot
prove exclusive innocence, while one example of dishonesty surely proves guilt. This is why we assume
innocence until proven guilty, but we never actually declare innocence. We either prove guilt, or we lack
enough evidence to conclusively prove guilt; and the choice of verdicts are GUILTY or NOT GUILTY
(meaning that there is insufficient evidence to find the defendant guilty). INNOCENT is not an option for
C
jurors thankfully; they only judge if the evidence is sufficient
or not to judge as GUILTY. All hypothesis
tests are designed to either prove Ha (the alternate hypothesis)
is
true or have insufficient evidence to prove
A
Ha is true. But let’s continue with this example.
L
Anyway, the prosecutor’s job is to gather evidence inVthe form of fingerprints, witnesses, blood, etc. and
try to convince the jury that there is enough evidence E
to convict the defendant. The jury has a significance
level that serves as a cutoff for error (known as reasonable
R doubt). The jury either decides that sufficient
evidence exists beyond reasonable doubt and therefore
T rejects the null that he is innocent and finds the
defendant guilty (meaning the chance of convicting by mistake is very small), or the jury finds that there
,
is not sufficient evidence to convict and therefore finds the defendant not guilty. Note that a “not guilty”
verdict does not mean they find the defendant innocent, because the evidence is still there, but the evidence
wasn’t enough to convince them beyond reasonableTdoubt (i.e., too great a risk of error of sending an
E
innocent person to prison). This is like showing a 2-pound
weight loss on the scale – yes you might have
actually lost weight, but you are not convinced beyond
Rreasonable doubt of that fact; you don’t declare that
you haven’t lost weight at all, but that there is insufficient
R evidence to prove you lost weight.
And how is reasonable doubt assessed? In hypothesisEtesting, we call it a significance level, and the norm
N for error, meaning that you are willing to reject
is to set it to .05 or 5%. This is essentially your tolerance
the null by mistake 5% of the time. The value can beCincreased or decreased at the will of the researcher,
depending on the circumstances. In a jury trial, would
E you want to convict a person knowing that there
is a 5% probability you might be convicting an innocent person? That might be too high. But if you
were screening airline passengers for weapons, would you care about having the metal detector go off by
1
mistake and having to search a suspect? Chances are, you wouldn’t mind having that error occur more
8 no real cost in searching someone who is unarmed
often if it meant saving lives (especially since there is
5 someone who is armed and dangerous. To help
and innocent, but there is a heavy cost for not searching
understand this further, we need to discuss Type 1 and9Type 2 errors.
T
What bothers people most – convicting an innocent man or acquitting a guilty man? Both are detestable,
S When you reject the null and the null is true, you
but our system is predicated on the former being worse.
have committed a Type 1 error; the probability of such an error is the significance level which you choose
for the hypothesis test (note that the judge doesn’t give you the measure of reasonable doubt, as it is hard
to put an exact number on it). So if the significance level is at 2%, then should the defendant be innocent,
there is a 2% probability of his being convicted by mistake (since the significance level = the probability
of a Type 1 error). When you fail to reject the null and it is false, you have committed a Type 2 error (this
Copyright 2011, Savant Learning SystemsTM
Introduction to Statistics by Jim Mirabella
4-2
Chapter Four: Hypothesis Testing for One Sample
is the equivalent of a guilty defendant being acquitted); the probability of such an error is unknown and
cannot be computed, but we do know that as the Type 1 error increases, the Type 2 error decreases by some
unknown amount, and vice versa. Thus, you can indirectly raise or lower the probability of a Type 2 error
by decreasing or increasing the significance level, respectively.
So let’s see if we can fix this judicial system to the satisfaction of all. Since we don’t like to see innocent
men go to jail, we would like to see the probability of a Type 1 error = 0, so we can set the significance level
to 0, BUT that would mean everyone must be acquitted, as it is the only way to guarantee that no innocent
person is ever convicted. As a result, the probability of a Type 2 error grows to 100% because all guilty
people will go free. And what about the problem of acquitting the guilty – for this we need to make sure
the probability of a Type 2 error = 0 so that never happens,
but the only way that is possible is to convict
C
everyone, which means setting the significance level or the probability of a Type 1 error = 100%. So we
A
essentially stop convicting innocent people by setting the significance level to 0% and we stop acquitting
L
the guilty by setting the significance level to 100% -- problem solved! Obviously not. It is impossible to
eliminate both errors, and to eliminate one of the twoVmeans the other occurs 100% of the time. Thus, the
best solution is to allow the possibility of both errors E
(which is what our court system currently does). We
may not like it, but it is truly the best solution. We just
R decided as a nation that convicting the innocent is
worse, so we set our level of reasonable doubt as a low
T one, and while it means a greater chance of a guilty
person going free, it means the prosecutor needs to be better prepared with evidence before going to trial.
,
You do have some input, however, in that you can raise or lower the significance level a bit, depending on
which of the two errors is more serious. In the court system,
the Type 1 error is more serious, so you might
T
set the significance level to .01, making it only a 1% chance
of
convicting an innocent person. If you were
E
in charge of airport security and the null hypothesis is that the passenger is unarmed, a Type 1 error would
R
mean having to frisk a person in error (because the alarm sounded) and a Type 2 error would mean letting
R
an armed man walk through. Of course a Type 2 error is worse here, so the best thing to do is raise the
significance level to maybe .10 (10% probability thatE
an unarmed person is searched), thus making it very
unlikely that an armed person gets by. Likewise on a N
medical test, it is far better that a person tests positive
by mistake and gets further testing than tests negative C
and goes home, only to potentially die unnecessarily;
here the Type 2 error is worse so you would likely increase
the significance level. If neither error is truly
E
a big deal (like if you were testing the preference for peanut butter), the rule of thumb is to just leave the
significance level at 5%.
1
What determines if you reject or don’t reject the null?
8 If you did the test using formulas and paper, you
would rely on the test statistic and tables, but in this high
5 tech world, you will find that all statistical software
produces a p-value in the output for a hypothesis test. A p-value is merely the probability of making a type
9
one error if you decide to reject the null. Going back to the judicial example, it is the probability that the
T
defendant is truly innocent should you decide to convict him. While the significance level is your tolerance
S are wrong, the risk you would be taking. When
for being wrong, the p-value is the likelihood you actually
the p-value is smaller than the significance level, than the risk of error is less than your tolerance for error
and so you reject the null hypothesis; when the p-value is larger than the significance level, the risk is too
great for your tolerance and so you don’t reject the null hypothesis.
The output you see from statistics software typically includes a p-value (also shown as a sig. value on
some). This p-value is the risk of error from the hypothesis test, while the significance level that you set
Copyright 2011, Savant Learning SystemsTM
Introduction to Statistics by Jim Mirabella
4-3
Chapter Four: Hypothesis Testing for One Sample
is the tolerance for error. If the p-value is less than the significance level, then your risk for error is within
your tolerance level and you reject the null; in the judicial example it means you feel very confident of
your decision to convict. If the p-value is greater than the significance level, then your risk is too great and
you fail to reject the null (thus acquitting the defendant due to lack of evidence). The p-value of .009, for
example, means that if you convict the defendant, there is a 0.9% chance you are wrong and he is innocent,
but as this is very small, you go ahead and convict; a p-value of .25 means that if you convict the defendant,
there is a 25% chance you are wrong and he is innocent, but since this is very large, you choose not to
convict. The cutoff is determined by the significance level you set (typically 5% in most studies). Thus, a
low p-value is typically the goal in research so you can actually prove something with evidence.
Hypothesis tests can be done on virtually any population
C measurement, but the most common tests are about
means and proportions. A hypothesis test is essentially the same as comparing a confidence interval against
A
a theorized value. Every election poll shows a percentage of people responding a certain way, and knowing
L
that it takes 50.1% to win the election, if the sample proportion +/- the margin of error is completely above
50%, then you are X% confident that person will winV
(and if it is completely below 50%, then you are X%
confident that person will lose). Hypothesis tests forEproportions work the same way; the null hypothesis
might be that the population proportion = .50, and withRthe sample proportion computed, you would use the
test results to determine if the sample proportion is significantly
different from .50. Significantly different
T
really means that the margin of error is smaller than the distance to .50. If it is significantly different, you
,
reject the null hypothesis and conclude the alternate hypothesis is true. If it is not significantly different,
you fail to reject the null hypothesis and state that there is insufficient evidence to prove the alternate is true.
If the Bush-Kerry poll showed Bush with 52% of the T
vote in Florida, you might fail to reject the null – this
E
doesn’t mean they are tied, but it does mean the results are still inconclusive (the media calls this a swing
state).
R
R
E
N
C
E
Figure 1: Screen display from Hypothesis_Tests_One_Sample.xls
1
8
5
9
T
S
In the example shown here, we are testing if a sample of 30 students at Whatsamatta U. has a mean GMAT
score that differs from the national average of 500. The null hypothesis is that the mean = 500. The
Copyright 2011, Savant Learning SystemsTM
Introduction to Statistics by Jim Mirabella
4-4
Chapter Four: Hypothesis Testing for One Sample
sample mean is 535, and the known standard deviation is 100. There are 3 options of p-values to use,
based on the direction of the test. The hypotheses can be two-tailed (non-directional), upper-tailed or
lower-tailed. A two-tailed test is one in which the null hypothesis has an equals sign and the alternate has
a “not equals” sign; you are testing if the sample mean is different from the hypothesized value. An uppertailed test is one in which the alternate hypothesis tests if the sample mean is greater than the hypothesized
value, and a lower-tailed test is one in which the alternate hypothesis tests if the sample mean is less than
the hypothesized value. By conducting an upper-tailed or lower-tailed test (also called directional tests),
the p-value is cut in half and it is easier to reject the null hypothesis. While this sounds great, it is also
dangerous and irresponsible to set up the test as directional unless you have good reason to do so. For
example, if you are testing a diet pill, it is reasonable to test if the average weight has declined. It would
also be understandable to test if the mean SAT score C
at a top-tier school is above the national average. If
you wanted to study the grade performance of students
A who drink Pepsi, it would not be reasonable to test
if grades are higher or lower than a norm. The best L
decision is to use a non-directional approach unless
you have justification to pick a direction. For this particular problem, though, we are testing if the GMAT
V
is different from 500, so we would use a two-tailed test, and the resulting p-value is .0552, which is greater
E
than the significance level of .05; thus the null hypothesis is not rejected and there is insufficient evidence
R from the national mean of 500.
to conclude that the mean GMAT at Whatsamatta U. differs
T
Suppose instead that we had been testing if Whatsamatta U. has standards which exceed the national
,
average, and so we wanted to test if the mean was greater than 500. In that case we would have used an
upper-tailed test, and with the resulting p-value of .0276 which is less than .05, we would have rejected the
T greater than the national average of 500.
null hypothesis and concluded that the mean GMAT was
E
R
R
E
N
C
E
Figure 2: Screen display from Hypothesis_Tests_One_Sample.xls
1
8
5
9
T
S
Now let’s look at how proportions are testing. Suppose a baseball player made errors 15% of the time
last year, so he went to a training camp to improve his defense. This season he had 100 opportunities to
make a play and committed 9 errors. Test the hypothesis that the error rate is now less than 15%. The
Copyright 2011, Savant Learning SystemsTM
Introduction to Statistics by Jim Mirabella
4-5
Chapter Four: Hypothesis Testing for One Sample
null hypothesis entered here would be that P = .15. The sample size is 100 and the number of successes
is 9 (note that a success is not a good thing or bad thing – it is just whatever is being measured). Leaving
the significance level at .05, we choose to do a lower-tail test to determine if P < .15. The p-value is .0464
which is less than .05, and so we reject the null and conclude that the proportion of errors has decreased
from last season.
Z-tests vs. t-tests
Hypothesis tests for means can use one of two approaches – a z-test or a t-test. To use a z-test, you must
know the population standard deviation (which can be obtained from historical data or prior analysis). You
could also use the z-test when you don’t know the population
standard deviation provided that your sample
C
is large enough (at least 30); the rationale here is thatAwith a large sample, the sample standard deviation
tends to be similar to the population standard deviation.L If you don’t know the population standard deviation
and your sample is not large enough, fear not because you can use a t-test. The t-test helps to deal with the
V
bias from the sample standard deviation. While there are slightly different formulas and different tables in
E
the textbook, those using Excel will find the only difference
between the two tests is negligible.
Closing Notes
R
T
,
Remember that the aim of a study is to prove the alternate hypothesis (often called the research hypothesis).
Researchers spend a lot of time and money trying to prove the alternate is true, as there is no value in failing
T don’t want to go to trial unless they can convict,
to reject the null and being where they began. Prosecutors
E is to reject the null and conclude that the alternate
or else it is a waste of time and money. Since the object
R A directional test will cut the p-value and help the
is true, researchers do what they can to help their odds.
chances of rejection, but it must be appropriate to doRso. Raising the level of significance gives a wider
berth for rejecting the null, but can you really tolerateEthat much error. Lastly, increasing the sample size
will make a small difference more significant (it is always acceptable to take an additional sample if your
N
p-value is close to the rejection mark, but it may not be convenient to do).
C
If a governor had 60% support in a poll, it may not E
be statistically significant (e.g., 3 out of 5 surveyed
support him), but if he had 51%, it may be statistically significant (e.g., 51,000 out of 100,000 support him).
The percents can be misleading; the sample size plays a big factor in determining if the results are truly
1
significant. So don’t jump to conclusions based on a percent sounding significant or sounding close.
8
Hypothesis testing is truly a shift in the way you think,
5 but it makes sense. Once you learn to set up the
hypotheses properly, things get easier. Of course, the9results are completely meaningless if the data is not
valid or reliable. Don’t cut corners just to get the results you want. Would you want to try drugs that were
T
improperly tested by an anxious researcher who just wanted to get published? Even if a researcher proves
S
a drug to cure what ails you, a small percentage of studies
will incur an error of that type, which is why
reputable pharmaceutical companies will re-test, as the aim is to sure. Never let ethics take a backseat to
desired results. Hypothesis testing should open your eyes to how election polling is done, how medical
tests are conducted, how psychological studies are done, etc., and should give you a greater appreciation
for the results.
I suggest you also read Chapter 24 of http://stat-www.berkeley.edu/users/stark/SticiGui/Text/
for an excellent write-up on hypothesis testing.
Copyright 2011, Savant Learning SystemsTM
Introduction to Statistics by Jim Mirabella
4-6
CHAPTER FOUR KNOWLEDGE ASSESSMENT
Hypothesis Testing for One Sample
Discussion Questions
DISCUSSION QUESTION 1
TRUSTING THE RESULTS: If you were to read the results of a study showing that daily use of a
certain exercise machine resulted in an average 10-pound weight loss, what more would you want
to know about the numbers in addition to the average? (Hint: Do you think everyone who used the
machine lost 10 pounds?)
C
A
WHAT’S IN A SAMPLE?: Why is it important that
L a sample be random and representative when
conducting hypothesis testing?
V
E
R
Practice Problems: Real Estate
T
, can check your work.
Solutions are provided to practice problems so you
DISCUSSION QUESTION 2
Use the Real_Estate.xls file which consists of 100Thomes purchased in 2007 and appraised in 2008.
It includes variables regarding the number of bedrooms, number of bathrooms, whether the house
E
has a pool or garage, the age, size and price of the home, what the house is constructed from, how
R
far it is to the city center, and the appraisals from two agents.
R
E
PRACTICE PROBLEM 1:
Assume that in 2000, the average sales price of a N
home was $205,000. Has this increased in seven
years? Using the sample of 100 homes, conductC
a one-sample hypothesis test to determine if the
mean sales price of a home is greater than $205,000.
E Use a .05 significance level.
PRACTICE PROBLEM 2:
1
Assume you read in an advertisement that the average
home in a community is 5 years. Is this
8
really true? You took a sample of 100 homes (in the data file). Conduct a one-sample hypothesis
5
test to determine if the mean age is different from 5 years. Use a .05 significance level.
9
T
PRACTICE PROBLEM 3:
S business that fewer than half of the homes are
You have heard from many folks in the real estate
made of brick, and you decide to put data to the test. You took a sample of 100 homes (in the data
file). Conduct a one-sample hypothesis test to determine if the proportion of homes made of brick
is less than 50%. Use a .05 significance level.
Copyright 2011, Savant Learning SystemsTM
Introduction to Statistics by Jim Mirabella
4-7
CHAPTER FOUR KNOWLEDGE ASSESSMENT
Hypothesis Testing for One Sample
Assigned Problems: Student Data
Use the Student_Data.xls file which consists of 200 MBA at Whatsamattu U. It includes variables
regarding their age, gender, major, GPA, Bachelors GPA, course load, English speaking status,
family, weekly hours spent studying.
ASSIGNED PROBLEM 1:
C grades at the MBA level divided between A’s
It is pretty common across most schools to find the
and B’s. As such, you expect the mean GPA to A
be around 3.50. Using the sample of 200 MBA
students, conduct a one-sample hypothesis test L
to determine if the mean GPA is different from
3.50. Use a .05 significance level.
V
E
ASSIGNED PROBLEM 2:
R
Assume you read in the Whatsamatta U website that the average age of their MBA students is
T
45. Is this really true or have they failed to update this correctly? You think it is far less because
, from their Bachelors to their Masters since the
there have been a lot more students going straight
economy is so bad. You took a sample of 200 students (in the data file). Conduct a one-sample
hypothesis test to determine if the mean age is less
T than 45. Use a .05 significance level.
E
ASSIGNED PROBLEM 3:
R
You have heard from idle chatter that most students
R don’t declare a major in their MBA programs.
You took a sample of 200 students (in the data file). Conduct a one-sample hypothesis test to
E
determine if the proportion without a major is greater than 50%. Use a .05 significance level.
N
C
E
1
8
5
9
T
S
Copyright 2011, Savant Learning SystemsTM
Introduction to Statistics by Jim Mirabella
4-8
