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?

Question

Mathematics

Kian Nicholson

18 February, 01:42

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?

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Answers (1)

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Kayden Salinas · Answer 1

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.