Suppose x has a distribution with 10 and 7 a If a random s

Suppose x has a distribution with = 10 and = 7.

(a) If a random sample of size n = 34 is drawn, find _x, _x and P(10 x 12). (Round _x to two decimal places and the probability to four decimal places.)

(b) If a random sample of size n = 61 is drawn, find _x, _x and P(10 x 12). (Round _x to two decimal places and the probability to four decimal places.)

Solution

a)
Mean ( u ) =10
Standard Deviation ( sd )=7
Number ( n ) = 34
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  
To find P(a <= Z <=b) = F(b) - F(a)
P(X < 10) = (10-10)/7/ Sqrt ( 34 )
= 0/1.2005
= 0
= P ( Z <0) From Standard Normal Table
= 0.5
P(X < 12) = (12-10)/7/ Sqrt ( 34 )
= 2/1.2005 = 1.666
= P ( Z <1.666) From Standard Normal Table
= 0.95214
P(10 < X < 12) = 0.95214-0.5 = 0.4521                  

b)
To find P(a <= Z <=b) = F(b) - F(a)
P(X < 10) = (10-10)/7/ Sqrt ( 61 )
= 0/0.8963
= 0
= P ( Z <0) From Standard Normal Table
= 0.5
P(X < 12) = (12-10)/7/ Sqrt ( 61 )
= 2/0.8963 = 2.2315
= P ( Z <2.2315) From Standard Normal Table
= 0.98718
P(10 < X < 12) = 0.98718-0.5 = 0.4872