Question

 

A consumer product testing organization uses a survey of readers to obtain customer satisfaction ratings for the nation's largest supermarkets. Each survey respondent is asked to rate a specified supermarket based on a variety of factors such as: quality of products, selection, value, checkout efficiency, service, and store layout. An overall satisfaction score summarizes the rating for each respondent with 100 meaning the respondent is completely satisfied in terms of all factors. Suppose sample data representative of independent samples of two supermarkets' customers are shown below.

Supermarket 1	Supermarket 2
n1 = 270	n2 = 300
x1 = 83	x2 = 82

(a)

Formulate the null and alternative hypotheses to test whether there is a difference between the population mean customer satisfaction scores for the two retailers. (Let μ₁ = the population mean satisfaction score for Supermarket 1's customers, and let μ₂ = the population mean satisfaction score for Supermarket 2's customers. Enter != for ≠ as needed.)

H₀:

H_a:

(b)

Assume that experience with the satisfaction rating scale indicates that a population standard deviation of 16 is a reasonable assumption for both retailers. Conduct the hypothesis test.

Calculate the test statistic. (Use

μ₁ − μ₂.

Round your answer to two decimal places.)

Report the p-value. (Round your answer to four decimal places.)

p-value =

At a 0.05 level of significance what is your conclusion?

Do not reject H₀. There is not sufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers.Reject H₀. There is sufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers.    Do not reject H₀. There is sufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers.Reject H₀. There is not sufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers.

(c)

Which retailer, if either, appears to have the greater customer satisfaction?

Supermarket 1Supermarket 2    neither

Provide a 95% confidence interval for the difference between the population mean customer satisfaction scores for the two retailers. (Use

x₁ − x₂.

Round your answers to two decimal places.)

to

Answer 1

a)

Ho:   p1 - p2 =   0
Ha:   p1 - p2 ╪   0

b)

sample #1   ----->              
first sample size,     n1=   270          
number of successes, sample 1 =     x1=   83          
proportion success of sample 1 , p̂1=   x1/n1=   0.3074          
                  
sample #2   ----->              
second sample size,     n2 =    300          
number of successes, sample 2 =     x2 =    82          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.2733          
                  
difference in sample proportions, p̂1 - p̂2 =     0.3074   -   0.2733   =   0.0341
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.2895          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.03804          
Z-statistic = (p̂1 - p̂2)/SE = (   0.034   /   0.0380   ) =   0.90
                  
z-critical value , Z* =        1.9600   [excel formula =NORMSINV(α/2)]      
p-value =        0.3704   [excel formula =2*NORMSDIST(z)]      

Do not reject H0. There is not sufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers

c)

Supermarket 1  appears to have the greater customer satisfaction

level of significance, α =   0.05              
Z critical value =   Z α/2 =    1.960   [excel function: =normsinv(α/2)      
                  
Std error , SE =    SQRT(p̂1 * (1 - p̂1)/n1 + p̂2 * (1-p̂2)/n2) =     0.03809          
margin of error , E = Z*SE =    1.960   *   0.0381   =   0.07465
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    0.034   -   0.0746   =   -0.041
upper limit = (p̂1 - p̂2) + E =    0.034   +   0.0746   =   0.109
                  
so, confidence interval is (   -0.04   < p1 - p2 <   0.11   )