Question

The weight of an organ in adult males has a bell-shaped distribution with a mean of 300grams and a standard deviation of 20 grams. Use the empirical rule to determine the following.

(a) About 95% of organs will be between what weights?
(b) What percentage of organs weighs between 280 grams and 320grams?
(c) What percentage of organs weighs less than 280 grams or more than 320 grams?
(d) What percentage of organs weighs between 240 grams and 340 grams?

Answer 1

a) Within 260 grams and 340 grams.

b) 68%

c) 32%

d) 97.35%

Explanation:

The empirical rule 68-95-99.7 for bell-shaped distributions tells us that:

• Approximately 68% of the data is within 1 standard deviation from the mean.

• Approximately 95% of the data is within 2 standard deviation from the mean.

• Approximately 99.7% of the data is within 3 standard deviation from the mean.

a) The data that covers 95% of the organs is within 2 standard deviations (z=±2).

Then we can calculate the bounds as:

`X_1=\mu+z_1\cdot\sigma=300+-2\cdot 20=300+-40=260 \\\\X_2=\mu+z_2\cdot\sigma=300+2\cdot 20=300+40=340`

b) We have to calculate the number of deviations from the mean (z-score) we have for the values X=280 and X=320.

`z_1=(X_1-\mu)/(\sigma)=(280-300)/(20)=(-20)/(20)=-1\\\\\\z_2=(X_2-\mu)/(\sigma)=(320-300)/(20)=(20)/(20)=1\\\\\\`

As there are the bounds for one standard devaition, it is expected tht 68% of the data will be within 280 grams and 320 grams.

c) This interval is complementary from the interval in point b, so it is expected that (100-68)%=32% of the organs weighs less than 280 grams or more than 320 grams.

d) We apply the same as point b but with X=240 and X=340 as bounds.

`z_1=(X_1-\mu)/(\sigma)=(240-300)/(20)=(-60)/(20)=-3\\\\\\z_2=(X_2-\mu)/(\sigma)=(340-300)/(20)=(40)/(20)=2\\\\\\`

The lower bound is 3 deviations under the mean, so it is expected that (99.7/2)=49.85% of the data will be within this value and the mean.

The upper bound is 2 deviations above the mean, so it is expected that (95/2)=47.5% of the data will be within the mean and this value.

Then, within 240 grams and 340 grams will be (49.85+47.5)=97.35% of the organs.