The distribution of weights for the population of males in t

The distribution of weights for the population of males in the United States is normal with mean = 172.2 pounds and standard deviation = 29.8 pounds.

a) What is the probability that among five males selected at random from the population, at least one will have a weight between 130 and 210 pounds?

b) If repeated samples of size 100 are selected from this population, list 3 properties for the distribution of the sample mean (Xbar).

Solution

a)
Mean ( u ) =172.2
Standard Deviation ( sd )=29.8
Number ( n ) = 5
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  
a)
To find P(a <= Z <=b) = F(b) - F(a)
P(X < 130) = (130-172.2)/29.8/ Sqrt ( 5 )
= -42.2/13.327
= -3.1665
= P ( Z <-3.1665) From Standard Normal Table
= 0.00077
P(X < 210) = (210-172.2)/29.8/ Sqrt ( 5 )
= 37.8/13.327 = 2.8364
= P ( Z <2.8364) From Standard Normal Table
= 0.99772
P(130 < X < 210) = 0.99772-0.00077 = 0.9969