The mean of the population 50 225 Its normally distributed A

The mean of the population =50, ²=25 ,It\'s normally distributed. Accuqire a sample with n=100.

1)What\'s the possibility of the the mean of the sample is lower tha 47?

2)What\'s the possibility of the the mean of the sample fall into the range of 49~51?

Solution

1. Let Y be the random variable denoting the mean of sample.
   Therefore, Pr ( Y < 47 ) = Pr [ ( Y - 50 ) / 5 < ( 47 - 50 ) / (5 / 10) ]
   = Pr ( z < - 6 )
= 1 - Pr ( z < 6 )
   = 1 - ( 6 ) = 1 - 1 ~ 0  

2. Let Y be the random variable denoting the mean of sample.
Therefore, Pr ( 49 < Y < 51 ) = Pr ( Y < 51 ) - Pr ( Y < 49 )
  
Computation of Pr ( Y < 51 ) :
Pr [ ( Y - 50) / ( 5 / 10 ) < ( 51 - 50 ) / ( 5 / 10 ) ]
= ( 2 ) = 0.97725
Computation of Pr ( Y < 49 ) :
Pr [ ( Y - 50) / ( 5 / 10 ) < ( 49 - 50 ) / ( 5 / 10 ) ]
=   ( - 2 ) = 1 -   ( 2 )
Therefore, required probability is :
( 2 ) - 1 + ( 2 ) = (0.97725  * 2) - 1 = 0.9545