Use the Student_Data.xls file which consists of 200 MBA students at Whatsamattu U. It includes variables regarding their age, gender, major, GPA, Bachelors GPA, course load, English speaking status, family, and weekly hours spent studying.

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