Question

The weight of an organ in adult males has a​ bell-shaped distribution with a mean of 350 grams and a standard deviation of 35 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 280 grams and 420 ​grams? ​(c) What percentage of organs weighs less than 280 grams or more than 420 ​grams? ​(d) What percentage of organs weighs between 245 grams and 385 ​grams?

Answer 1

a) About 99.7% of the organs will be within 245 and 455 grams.

b) 95% weighs between 280 grams and 420 ​grams.

c) 5% of the organs weighs less than 280 grams or more than 420 ​grams

d) 68% weighs between 245 grams and 385 ​grams.

Explanation:

The empirical rule tells us that is expected to have 68% of the data within 1 standard deviation from the mean, 95% in the interval of 2 standard deviations and 99.7% in the interval of 3 standard deviations.

In this case, the mean is 350 grams and the standard deviation is 35 grams.

a) This corresponds to ±3 standard deviations.

`LB=\mu-3\sigma=350-3*35=350-105=245\\\\ UB=\mu+3\sigma=350+3*35=350+105=455`

b) We have to calculate how many standard deviations correspond to this interval.

`z=(420-350)/(35)=(70)/(35)=2`

For 2 standard deviations, 95% of the data willl fall within the interval.

c) As 95% lies within 240 and 420 grams, then 1-0.95=0.05=5% lies outside this bounds.

d) We have to calculate how many standard deviations correspond to this interval.

`z=(385-350)/(35)=(35)/(35)=1`

For 1 standard deviations, 68% of the data willl fall within the interval.