Question

The weight of an organ in adult males has a​ bell-shaped distribution with a mean of 320 320 grams and a standard deviation of 35 35 grams. Use the empirical rule to determine the following. ​(a) About 95 95​% of organs will be between what​ weights? ​(b) What percentage of organs weighs between 215 215 grams and 425 425 ​grams? ​(c) What percentage of organs weighs less than 215 215 grams or more than 425 425 ​grams? ​(d) What percentage of organs weighs between 250 250 grams and 355 355 ​grams?

Answer 1

Answer:

a) Between 250 and 390 grams.

b) 99.7% of organs weigh between 215 gras and 425 grams.

c) 0.3% of organs weigh less than 215 grams or more than 425 grams.

d) 81.5% of organs weight between 250 grams and 355 grams.

Step-by-step explanation:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 320

Standard deviation = 35

a) About 95% of organs will be between what​ weights?

By the Empirical Rule, within 2 standard deviations of the mean.

320 - 2*35 = 250

320 + 2*35 = 390

Between 250 and 390 grams.

(b) What percentage of organs weighs between 215 grams and 425 ​grams?

215 = 320 - 3*35

So 215 is three standard deviations below the mean.

425 = 320 + 3*35

So 425 is three standard deviations above the ean.

By the Empirical Rule, 99.7% of organs weigh between 215 gras and 425 grams.

​(c) What percentage of organs weighs less than 215 grams or more than 425 ​grams?

From b), 99.7% of organs weigh between 215 gras and 425 grams.

100 - 99.7 = 0.3

So 0.3% of organs weigh less than 215 grams or more than 425 grams.

(d) What percentage of organs weighs between 250 grams and 355 ​grams?

The normal distribution is symmetric, which means that 50% are below the mean and 50% are above.

250 = 320 - 2*35

So 250 is two standard deviations below the mean. 95% of the measures below the mean are between 250 and the mean.

355 = 320 + 35

So 355 is one standard deviation above the mean. 68% of the measures above the mean are within the mean and 355.

So

0.68*0.5 + 0.95*0.5 = 0.815

81.5% of organs weight between 250 grams and 355 grams.