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?

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%.