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**Statistical Analysis [3-7]**

**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.

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

__Example__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:

The following hypothesis is being tested:

H0: µd=0

H1: µd≠0

where d = before - after.

Proc ttest Data=Inspection;

Paired Before*After;

Run;

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.

**Exercise**

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

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?

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