Traditional management theory suggests that salespeople who are paid on a commis

Question

Traditional management theory suggests that salespeople who are paid on a commission basis (i.e., where wages are a percentage of total sales) outperform salespeople who are on a fixed monthly salary. An alternative theory is that in some industries, salespeople on fixed salaries perform better because they put less pressure on customers.

A researcher wishes to test whether there is any difference in mean sales between salespeople on a fixed salary (Group 1) and salespeople working for a commission (Group 2). Two random samples were taken, one of 77 salespeople on fixed salaries and another of 90 people on commission, from the employee records of a chain of retail stores operating in Texas. Total sales amounts (in dollars) for each sales person were recorded for one month.

The computer output of a two-sample t test from Excel is shown below. The test was done at the 0.05 level of significance.

Find a 95% confidence interval for the true difference in the mean sales amount between employees on a fixed salary and employees working on a commission. Interpret the interval in the context of the problem. Read carefully!

We are 95% confident, based on the method used to calculate the interval, that the true difference in mean sales between salespeople on a fixed salary and salespeople working on a commission is between -5881.87 and 481.67. We cannot conclude which type of payment leads to higher mean sales.

We are 95% confident, based on the method used to calculate the interval, that the true difference in the proportion of sales between salespeople on a fixed salary and salespeople working on a commission is between -5881.87 and 481.67. We cannot conclude which type of payment leads to a higher proportion of sales.

We are 99% confident, based on the method used to calculate the interval, that the true difference in the mean sales between salespeople on a fixed salary and salespeople working on a commission is between -5881.87 and 481.67. Having employees work on a fixed salary leads to higher mean sales.

We are 99% confident, based on the method used to calculate the interval, that the true difference in the mean sales between salespeople on a fixed salary and salespeople working on a commission is between -5881.87 and 481.67. Having employees work on a commission leads to higher mean sales.

We are 90% confident, based on the method used to calculate the interval, that the true difference in the mean sales between salespeople on a fixed salary and salespeople working on a commission is between -5364.73 and -35.47. Having employees work on a commission leads to higher mean sales.

We are 95% confident, based on the method used to calculate the interval, that the true difference in the mean sales between salespeople on a fixed salary and salespeople working on a commission is between -6848.48 and 1448.28. We cannot conclude which type of payment leads to higher mean sales.

t-Test: Two-Sample Assuming Equal Variances Fixed Salary (Group 1) Commission (Group 2) Mean 60862.8 63562.9 Variance 98360720.95 115677015.6 Observations (n) 77 90 Pooled Variance* 107701025.3 Hypothesized Mean Difference 0 df 165 t Stat -1.676 P(T<=t) one-tail 0.0478 t Critical one-tail 1.654 P(T<=t) two-tail 0.0956 t Critical two-tail 1.974

Explanation / Answer

Note that this is

1. A confidence interval for differences of MEANS, not proportions
2. It is inconclusive, because the interval contains 0.

Calculating the means of each group,              
              
X1 =    60862.8          
X2 =    63562.9          
              
Calculating the standard deviations of each group,              
              
s1 =    9917.697361          
s2 =    10755.32499          
              
Thus, the pooled standard deviation is given by              
              
S = sqrt[((n1 - 1)s1^2 + (n2 - 1)(s2^2))/(n1 + n2 - 2)]               
              
As n1 =    77   , n2 =    90  
              
Then              
              
S =    10377.91045          
              
Thus, the standard error of the difference is              
              
Sd = S sqrt (1/n1 + 1/n2) =    1611.022245          
              
df = 165      
              
For the   0.95   confidence level, then      
              
alpha/2 = (1 - confidence level)/2 =    0.025          
t(alpha/2) =    1.97444563          
              
lower bound = [X1 - X2] - t(alpha/2) * Sd =    -6849.818307          
upper bound = [X1 - X2] + t(alpha/2) * Sd =    1449.618307          
              
Thus, the confidence interval is              
              
(   -6849.818307   ,   1449.618307   )

Hence, it is

OPTION F: We are 95% confident, based on the method used to calculate the interval, that the true difference in the mean sales between salespeople on a fixed salary and salespeople working on a commission is between -6848.48 and 1448.28. We cannot conclude which type of payment leads to higher mean sales. [ANSWER, LAST OPTION]

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