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Statistical Analysis [3-7]

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Paired t-test

A paired t-test can be used to compare the means of two populations where the observations come in pairs.

It is often used when comparing the "before" and "after" results.

Example

Data Inspection; Input ID $ Before After; Datalines; PID9012 167 155 PID9013 182 187 PID9014 197 198 PID9015 114 109 PID9016 112 97 PID9017 137 126 PID9018 181 179 PID9019 174 173 PID9020 179 172 PID9021 136 120 PID9022 137 129 PID9023 166 165 PID9024 220 217 PID9025 153 152 PID9026 199 199 PID9027 108 104 PID9028 182 186 PID9029 151 158 PID9030 200 208 PID9031 218 217 PID9032 131 118 PID9033 192 188 PID9034 186 196 PID9035 167 162 PID9036 216 203 PID9037 237 248 PID9038 126 130 PID9039 137 129 PID9040 184 185 PID9041 158 139 PID9042 126 127 PID9043 194 186 PID9044 129 109 PID9045 124 118 PID9046 157 158 PID9047 198 185 PID9048 145 141 PID9049 170 178 PID9050 143 150 PID9051 171 164 PID9052 122 120 PID9053 134 136 PID9054 146 142 PID9055 160 157 PID9056 120 115 PID9057 190 175 PID9058 203 184 PID9059 184 167 PID9060 129 132 PID9061 115 114 ; Run;

A weight loss product claims to have a significant effect on helping people lose weight.

An independent committee has conducted an inspection and selected 50 healthy participants who have used this particular weight loss product.

Their weights were measured before and after they have used the product and the results are captured in the INSPECTION data set above.

The INSPECTION data set contains the following variables:

• ID: Participants' ID
• Before: Weight before using the product
• After: Weight after using the product

The following hypothesis is being tested:

H0: µd=0
H1: µd≠0

where d = before - after.

Example

​​Proc ttest Data=Inspection;
Paired Before*After;
Run;

The paired t-test can be done by using Proc ttest with the PAIRED statement.

The PAIRED statement should list the two paired variables separated by an asterisk (*).

Run the program above and the following results will be generated:

​1. Basic Statistical Measurements

Some basic statistics are generated for the difference between the before- and after- results.

The average weight loss is 4 lbs after using the weight loss product.

The 95% confidence interval is [1.74, 6.26]. 

We are 95% certain the actual weight loss is between 1.74 to 6.26 lbs.

This rejects the null hypothesis that the difference in weight loss is zero (no effect).

The p-value of 0.0009 also rejects the null hypothesis.

Looks like the weight loss product is indeed effective.

2. Graphs

Some of the related graphs such as the histogram and the Q-Q plot are also plotted.

One of the main assumptions for paired t-test is that the difference should be approximately normally distributed.

The linear pattern from the q-q plot suggests the difference is normally distributed.

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Exercise

Copy and run the GROUPON data set from the yellow box below:

Data Groupon; Input Restaurant N_Bfr Profit_Bfr N_Aftr Profit_Aftr ; Datalines; 1 90 28 328 15 2 139 37 229 7 3 96 18 225 9 4 167 49 346 18 5 90 32 266 8 6 95 16 321 4 7 90 37 342 3 8 161 61 298 14 9 123 39 385 6 10 192 71 355 7 11 90 48 261 12 12 90 69 273 13 13 134 29 252 5 14 137 10 267 8 15 117 37 263 13 16 156 38 242 14 17 158 21 320 11 18 180 80 234 12 19 188 50 295 7 20 90 43 173 10 21 114 44 401 16 22 175 20 239 7 23 90 38 289 10 24 127 15 299 13 25 150 30 336 9 26 142 10 250 9 27 90 76 179 12 28 117 53 285 7 29 111 20 297 9 30 104 45 385 12 31 90 71 277 11 32 143 69 333 9 33 174 37 335 11 34 120 44 267 6 35 104 10 305 15 36 126 42 272 12 37 101 18 340 10 38 161 30 333 7 39 114 17 322 10 40 107 10 278 11 41 124 51 329 12 42 166 62 296 11 43 135 41 322 3 44 133 21 348 10 45 106 26 258 17 46 107 61 303 6 47 113 23 311 8 48 174 24 284 14 49 178 46 220 14 50 123 10 261 15 ; Run;

The GROUPON data set contains a list of restaurants who have launched a groupon marketing campaign.

A study is conducted to find out if the groupon campaign is profitable for the restaurants.

The data set contains 5 variables:

1. Restaurant
The restaurant identification number

2. N_Bfr
The average number of customers the 3 months prior to the groupon campaign

3. Profit_Bft
The average profit made per customer the 3 months prior to the groupon campaign

4. N_Aftr
​The average number of customers the 3 months after to the groupon campaign

5. Profit_Aftr
the average profit made per customer the 3 months after to the groupon campaign


The total profit before and after the groupon campaign can be calculated as:

Total Profit = N x Profit

Where

• N = Average number of customer 
• Profit = Average profit per customer

Perform a paired t-test to compare the total profit before and after the groupon campaign.

Is the campaign profitable for the restaurants?

Next

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Need some help? 

Get Hint

HINT:
First, compute the total profit before and after the campaign. Perform the paired t-test on the total profit after.

Get Solution

SOLUTION:
Data Groupon2;
Set Groupon;
Total_Bfr = N_Bfr*Profit_Bfr;
Total_Aftr = N_Aftr*Profit_Aftr;
Run;

Proc ttest data=groupon2;
paired total_bfr*total_aftr;
Run;

The restaurants, on average, make about $1,900 less after the campaign. The p-value is 0.0001, which demonstrate sufficient evidence that the campaign lowers the total profit for the restaurants.

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