2 The weights of envelopes sent from an insurance office are

2. The weights of envelopes sent from an insurance office are normally distributed with mean = 12 ounces and standard deviation = 3.7 ounces. The mail room clerk would like to know the average weight of 20 envelopes. What is the probability that the mean weight x is: a. b. lighter than 10 ounces? P( ) = P( ) = _______________ c. heavier than 13 ounces? P( ) = P( ) = _______________ d. between 10 and 13 ounces? P( ) = P( ) = _______________

Solution

Mean ( u ) =12
Standard Deviation ( sd )=3.7
Number ( n ) = 20
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  
b)
P(X < 10) = (10-12)/3.7/ Sqrt ( 20 )
= -2/0.8273= -2.4174
= P ( Z <-2.4174) From Standard NOrmal Table
= 0.0078                  

c)
P(X > 13) = (13-12)/3.7/ Sqrt ( 20 )
= 1/0.827= 1.2087
= P ( Z >1.2087) From Standard Normal Table
= 0.1134                  

d)
To find P(a <= Z <=b) = F(b) - F(a)
P(X < 10) = (10-12)/3.7/ Sqrt ( 20 )
= -2/0.8273
= -2.4174
= P ( Z <-2.4174) From Standard Normal Table
= 0.00782
P(X < 13) = (13-12)/3.7/ Sqrt ( 20 )
= 1/0.8273 = 1.2087
= P ( Z <1.2087) From Standard Normal Table
= 0.88661
P(10 < X < 13) = 0.88661-0.00782 = 0.8788