Question

A mail-order company claims that at least 60% of all orders are mailed within 48 hours. From time to time the quality control department at the company checks whether this promise is fulfilled. Recently the quality control department at this company took a sample of 400 orders and found that 208 of them were mailed within 48 hours of the placement of the orders. Testing at a 1% significance level, can you conclude that the company’s claim is true? Use both the critical value approach and the p-value approach.

Answer 1

Ho :   p ≥ 0.6 (claim)
H1 :   p <   0.6       (Left tail test)          
                          
Level of Significance,   α =    0.01                  
Number of Items of Interest,   x =   208                  
Sample Size,   n =    400                  
                          
Sample Proportion ,    p̂ = x/n =    0.5200                  
                          
Standard Error ,    SE = √( p(1-p)/n ) =    0.0245                  
Z Test Statistic = ( p̂-p)/SE = (   0.5200   -   0.6   ) /   0.0245   =   -3.2660
                          
critical z value =        -2.326   [excel function =NORMSINV(α)]              
                          
p-Value   =   0.0005   [excel function =NORMSDIST(z)]              
Decision:   p-value<α , reject null hypothesis                       
There is enough evidence to reject the claim