Answer:
a) 99.7% of the organs will be between 250 and 420 grams.
b) 95% of the organs weighs between 280 grams and 400 grams.
c) 5% of the organs weighs less than 280 grams or more than 400 grams.
d) 97.5% of the organs weighs between 250 grams and 400 grams.
Step-by-step explanation:
The mean value of the weight of an organ is 340 grams and the standard deviation is 30 grams, so
.
(a) About 99.7% of organs will be between what weights?
The Empirical Rule states that 99.7% of the values of X of a set with mean
and standard deviation
belong to the following interval
![\mu - 3\sigma \leq X \leq \mu + 3\sigma](https://tex.z-dn.net/?f=%5Cmu%20-%203%5Csigma%20%5Cleq%20X%20%5Cleq%20%5Cmu%20%2B%203%5Csigma)
We have that
, so:
![\mu - 3\sigma \leq X \leq \mu + 3\sigma](https://tex.z-dn.net/?f=%5Cmu%20-%203%5Csigma%20%5Cleq%20X%20%5Cleq%20%5Cmu%20%2B%203%5Csigma)
![340 - 3(30) \leq X \leq 340 + 3(30)](https://tex.z-dn.net/?f=340%20-%203%2830%29%20%5Cleq%20X%20%5Cleq%20340%20%2B%203%2830%29)
![250 \leq X \leq 420](https://tex.z-dn.net/?f=250%20%5Cleq%20X%20%5Cleq%20420)
99.7% of the organs will be between 250 and 420 grams.
(b) What percentage of organs weighs between 280 grams and 400 grams?
280 grams is 2 standard deviations below the mean.
400 grams is 2 standard deviations above the mean.
The Empirical rule states that 95% of the weights of the organs are between 2 standard deviations of the mean, so
95% of the organs weighs between 280 grams and 400 grams.
(c) What percentage of organs weighs less than 280 grams or more than 400 grams?
95% of the organs weighs between 280 grams and 400 grams. This means that 5% of the organs weighs less than 280 grams or more than 400 grams.
(d) What percentage of organs weighs between 250 grams and 400 grams?
250 grams is 3 standard deviations below the mean.
The Empirical rule states that 50% of the weights are above the mean, and 50% are below. Of those that are below, 100% weights at least 3 standard deviations below the mean.
400 grams is 2 standard deviations above the mean.
Of the values that are above the mean, 95% of them are at most 2 standard deviations above the mean.
So:
![P = 0.5 + 0.95(0.5) = 0.5 + 0.475 = 0.975](https://tex.z-dn.net/?f=P%20%3D%200.5%20%2B%200.95%280.5%29%20%3D%200.5%20%2B%200.475%20%3D%200.975)
97.5% of the organs weighs between 250 grams and 400 grams.