Question

The weight of an organ in adult males has a​ bell-shaped distribution with a mean of 320 grams and a standard deviation of 50 grams. Use the empirical rule to determine the following.
​(a) About 68​% of organs will be between what​ weights? ​
(b) What percentage of organs weighs between 220 grams and 420 ​grams? ​
(c) What percentage of organs weighs less than 220 grams or more than 420 ​grams?
​(d) What percentage of organs weighs between 170 grams and 420 ​grams?

Answer 1

Answer:

a) 68% of organs will be between 270 and 370 grams

b) 95%

c) 5%

d) 97.35%

Step-by-step explanation:

Empirical rule states that

1 standard deviation means a 68% of total area under the normal distribution. On one side is half = 34%.

2 standard deviations means a 95% of total area under the normal distribution. On one side is half = 47.5%.

3 standard deviation means a 99.7% of total area under the normal distribution. On one side is half = 49.85%.

a) Since 1 standard deviation means a 68% of total area under the normal distribution, we have

320 + 50 = 370

320 – 50 = 270

68% of organs will be between 270 and 370 grams

b) Calculate the difference between mean and the percentage of weight of an organ in the problem:

420 – 320 = 100

320 – 220 = 100

Since standard deviation is 50 grams, there is a difference of 2 standard deviations for both sides. According to the rule 95% of the organs are between 200 and 420 grams.  

c) Since 95% of the organs are between 200 and 420 grams, 5% weighs less than 220 grams or more than 420 grams.

d) Calculate the difference between mean and the percentage of weight of an organ in the problem:

420 – 320 = 100

320 – 170 = 150

Since standard deviation is 50 grams, there is a difference of 2 standard deviations for the right side and 2 standard deviations for the left side. According to the rule for 2 standard deviations on one side there is 47.5% and for 3 standard deviations on one side there is 49.85%. so percentage of organs weighs between 170 grams and 420 grams is 47.5% + 49.85% = 97.35%.