Question

The marketing manager of a firm that produces laundry products decides to test market a new laundry product in each of the firm's two sales regions. He wants to determine whether there will be a difference in mean sales per market per month between the two regions. A random sample of 12 supermarkets from Region 1 had mean sales of 81.4 with a standard deviation of 8.4. A random sample of 17 supermarkets from Region 2 had a mean sales of 89.9 with a standard deviation of 6.4. Does the test marketing reveal a difference in potential mean sales per market in Region 2? Let μ1 be the mean sales per market in Region 1 and μ2μ2 be the mean sales per market in Region 2. Use a significance level of α=0.02 for the test. Assume that the population variances are not equal and that the two populations are normally distributed.

Step 1 of 4: State the null and alternative hypotheses for the test.

Step 2 of 4: Compute the value of the t test statistic. Round your answer to three decimal places.

Step 3 of 4: Determine the decision rule for rejecting the null hypothesis H0H0. Round your answer to three decimal places.

Step 4 of 4: State the test's conclusion.

Answer 1

1)

Ho :   µ1 - µ2 =   0
Ha :   µ1-µ2 ╪   0
      

2)

Sample #1   ---->   1                  
mean of sample 1,    x̅1=   81.400                  
standard deviation of sample 1,   s1 =    8.4000                  
size of sample 1,    n1=   12                  
                          
Sample #2   ---->   2                  
mean of sample 2,    x̅2=   89.900                  
standard deviation of sample 2,   s2 =    6.4000                  
size of sample 2,    n2=   17                  
                          
difference in sample means =    x̅1-x̅2 =    81.4000   -   89.9   =   -8.50  
                          
pooled std dev , Sp=   √([(n1 - 1)s1² + (n2 - 1)s2²]/(n1+n2-2)) =    7.2814                  
std error , SE =    Sp*√(1/n1+1/n2) =    2.7454                  
                          
t-statistic = ((x̅1-x̅2)-µd)/SE = (   -8.5000   -   0   ) /    2.75   =   -3.0961

c)

Degree of freedom, DF=   n1+n2-2 =    27  
t-critical value , t* = ± 2.473   (excel formula =t.inv(α/2,df)

reject Ho if t>2.473 or t <-2.473

d)

reject Ho

There is enough evidence to conclude that  there will be a difference in mean sales per market per month between the two regions.