The manager of Discount Furniture designed an incentive plan

The manager of Discount Furniture designed an incentive plan for sales people.To evaluate her plan, 12 sales people were selected at random, and their weekly incomes before and after the plan were recorded.

            Salesperson                Before             After

            Sid                               $320                $340

            Carol                           290                  285

            Tom                             421                  475

            Andy                           510                  540

            Jean                             210                  215

            Jack                             402                  500

            Peg                              625                  631

            Anita                           560                  570

            John                             360                  365

            Carl                             431                  438

            Alan                            506                  525

            Fern                             505                  619

A) Are the assumptions for using the paired t-test satisfied in this example?

B) Assuming the assumptions are satisfied, was there a significant increase in the typical salesperson’s weekly income due to the incentive plan?Use the .05 significance level.

(Please post with steps on how you got to each answer)

Solution

A)

Yes, the assumptions are satisfied because each data point in the first group has a natural pairing with the second group, that is, because they are based on the performance of the same person.

Consider the table:

B)

Let

group 2 = after
group 1 = before

Let ud = u2 - u1.              
Formulating the null and alternative hypotheses,              
              
Ho:   ud   <=   0  
Ha:   ud   >   0  
At level of significance =    0.05          
As we can see, this is a    right   tailed test.      
              
Calculating the standard deviation of the differences (third column):              
              
s =    36.50088062          
              
Thus, the standard error of the difference is sD = s/sqrt(n):              
              
sD =    10.53689662          
              
Calculating the mean of the differences (third column):              
              
XD =    30.25          
              
As t = [XD - uD]/sD, where uD = the hypothesized difference =    0   , then      
              
t =    2.870864267          
              
As df = n - 1 =    11          
              
Then the critical value of t is              
              
tcrit =    +   2.20098516      
              
Thus, comparing t and tcrit, we   WE REJECT THE NULL HYPOTHESIS.          
              
Also, using p values,              
              
p =        0.007608988      
              
Also, comparing this p to the significance level,   WE REJECT THE NULL HYPOTHESIS.          

Thus, there is a significant increase in the typical salesperson’s weekly income due to the incentive plan. [CONCLUSION]