Question

The weight of an organ in adult males has a​ bell-shaped distribution with a mean of 325 grams and a standard deviation of 20 grams. Use the empirical rule to determine the following. ​(a) About 99.7 ​% of organs will be between what​ weights? ​(b) What percentage of organs weighs between 305 grams and 345 ​grams? ​(c) What percentage of organs weighs less than 305 grams or more than 345 ​grams? ​(d) What percentage of organs weighs between 305 grams and 385 ​grams?

Answer 1

(a)

= 325

= 20

By empirical rule, 99.7% organs ill lie between

i.e.

325 - (3 X 20) = 325 - 60 = 265

325 + (3 X 20) = 325 + 60 = 385

So, answer is:

between 265 and 385

(b)

To find P(305 < X < 345)

Case 1: For X from 305 to mid value:
Z = (305 - 325)/20 = - 1

Case 2: For X from mid value to 345:
Z = (345 - 325)/20 = 1

By empirical rule:

Percentage is 68%

(c)

Percentage between less than 305 and more than 345 = 100% - 68% = 32%

(d)
P(305 < X <385):

Case 1: For X from 305 to mid value:
Z = (305 - 325)/20 = - 1

By Empirical Rule, area from -1 to 0 is area = 0.68/2 = 0.34

Case 2:For X from mid value to 385:

Z = (385 - 325)/20

    = 3

By Empirical Rule, area from 0 to 3 is area = 0.997/2 = 0.4985

So,

Percentage = 0.34 + 0.4985 = 0.8385 = 83.85%

So,

Answer is:

83.85 %