Question

A market research firm supplies manufacturers with estimates of the retail sales of their products from samples of retail stores. Marketing managers are prone to look at the estimate and ignore sampling error. An SRS of 29 stores this year shows mean sales of 74 units of a small appliance, with a standard deviation of 12.2 units. During the same point in time last year, an SRS of 26 stores had mean sales of 62.166 units, with standard deviation 15.2 units. An increase from 62.166 to 74 is a rise of about 16%.

1. Construct a 99% confidence interval estimate of the difference μ1−μ2μ1−μ2, where μ1μ1 is the mean of this year's sales and μ2μ2 is the mean of last year's sales.

(a) ________<(μ1−μ2)<_____________

(b) The margin of error is:

2. At a 0.010.01 significance level, is there sufficient evidence to show that sales this year are different from last year?

A. Yes
B. No

Answer 1

1)

Sample #1   ---->   this year
mean of sample 1,    x̅1=   74.00
standard deviation of sample 1,   s1 =    12.2
size of sample 1,    n1=   29
      
Sample #2   ---->   last year
mean of sample 2,    x̅2=   62.166
standard deviation of sample 2,   s2 =    15.20
size of sample 2,    n2=   26

Level of Significance ,    α =    0.01
DF = min(n1-1 , n2-1 )=   25              
t-critical value , t* =    2.7874   (excel formula =t.inv(α/2,df)          
                  
                  
                  
std error , SE =    √(s1²/n1+s2²/n2) =    3.744          
margin of error, E = t*SE =    2.7874   *   3.744   =   10.4365
                  
difference of means = x̅1-x̅2 =    74.0000   -   62.166   =   11.8340
confidence interval is                   
Interval Lower Limit = (x̅1-x̅2) - E =    11.8340   -   10.4365   =   1.3975
Interval Upper Limit = (x̅1-x̅2) + E =    11.8340   -   10.4365   =   22.2705

b)

margin of error, E = t*SE =    2.7874   *   3.744   =   10.4365

2)

YES, there is sufficient evidence to show that sales this year are different from last year because confidence interval does not contain zero.