The weight of an organ in adult males has a bellshaped distr

The weight of an organ in adult males has a bell-shaped distribution with a mean of 325 grams and a standard deviation of 45 grams. Use the empirical rule to determine the following. About 99.7% of organs will be between what weights What percentage of organs weighs between 235 grams and 415 grams What percentage of organs weighs less than 235 grams or more than 415 grams What percentage of organs weighs between 235 grams and 460 grams

Solution

Mean ( u ) =325
Standard Deviation ( sd )=45
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
Practically all of the area under the normal curve is within three standard deviations of the mean. i.e.
(u ± 3*s.d)
(325-45,325+45) = ( 280, 370 )

b)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 235) = (235-325)/45
= -90/45 = -2
= P ( Z <-2) From Standard Normal Table
= 0.02275
P(X < 415) = (415-325)/45
= 90/45 = 2
= P ( Z <2) From Standard Normal Table
= 0.97725
P(235 < X < 415) = 0.97725-0.02275 = 0.9545   ~ 96%
              
c)
To find P( X > a or X < b ) = P ( X > a ) + P( X < b)
P(X < 235) = (235-325)/45
= -90/45= -2
= P ( Z <-2) From Standard Normal Table
= 0.0228
P(X > 415) = (415-325)/45
= 90/45 = 2
= P ( Z >2) From Standard Normal Table
= 0.0228
P( X < 235 OR X > 415) = 0.0228+0.0228 = 0.0455 ~ 5% OR 4.55%
                  
d)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 235) = (235-325)/45
= -90/45 = -2
= P ( Z <-2) From Standard Normal Table
= 0.02275
P(X < 460) = (460-325)/45
= 135/45 = 3
= P ( Z <3) From Standard Normal Table
= 0.99865
P(235 < X < 460) = 0.99865-0.02275 = 0.9759   ~ 97.59%