Question

The population mean and standard deviation are given below. Find the indicated probability and determine whether a sample mean in the given range below would be considered unusual. If​ convenient, use technology to find the probability.

For a sample of

nequals=3131​,

find the probability of a sample mean being less than

12 comma 74912,749

or greater than

12 comma 75212,752

when

muμequals=12 comma 74912,749

and

sigmaσequals=1.51.5.

For the given​ sample, the probability of a sample mean being less than

12 comma 74912,749

or greater than

12 comma 75212,752

is

nothing.

Answer 1

µ =    12749                                      
σ =    1.5                                      
n=   31                                      
we need to calculate probability for ,                                          
12749   ≤ X ≤    12752                                  
X1 =    12749   ,    X2 =   12752                          
                                          
Z1 =   (X1 - µ )/(σ/√n) = (   12749   -   12749   ) / (   1.5   / √   31   ) =   0.00  
Z2 =   (X2 - µ )/(σ/√n) = (   12752   -   12749   ) / (   1.5   / √   31   ) =   11.14  
                                          
P (   12749   < X <    12752   ) =    P (    0.00   < Z <    11.14   )       
                                          
= P ( Z <    11.14   ) - P ( Z <   0.00   ) =    1.0000   -    0.5000   = 0.50000

required probability = 1-0.5000 = 0.50 (answer)

The sample mean would not be considered unusual because there is probability less than 0.05 of the sample mean being within this range.