Question

The weight of an organ in adult males has a​ bell-shaped distribution with a mean of 320 grams and a standard deviation of 45 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 275 grams and 365 ​grams?​(c) What percentage of organs weighs less than 275 grams or more than 365 ​grams?​(d) What percentage of organs weighs between 230 grams and 365 ​grams?

Answer 1

The weight of an organ in adult males has a bell-shaped distribution. Using the empirical rule, this answer explains how to determine the weights that about 99.7% of the organs will be between, the percentage of organs that weigh between given weights, the percentage of organs that weigh less than or more than given weights, and the percentage of organs that weigh between given weights.

Step-by-step explanation:

a) About 99.7% of organs will be between what weights?

To determine the weights that about 99.7% of organs will fall between, we can use the empirical rule. According to the empirical rule, for a bell-shaped distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations. Since the mean weight is 320 grams and the standard deviation is 45 grams, we can calculate the desired range as follows:

Lower limit: 320 - (3 x 45) = 320 - 135 = 185 grams

Upper limit: 320 + (3 x 45) = 320 + 135 = 455 grams

Therefore, about 99.7% of organs will weigh between 185 grams and 455 grams.

b) What percentage of organs weighs between 275 grams and 365 grams?

To determine the percentage of organs that weigh between 275 grams and 365 grams, we need to calculate the z-scores for the given weights and then use the standard normal distribution table or calculator to find the corresponding probabilities. The z-score for 275 grams is calculated as:

Z = (275 - 320) / 45 = -1

The z-score for 365 grams is calculated as:

Z = (365 - 320) / 45 = 1

Using the standard normal distribution table or calculator, we can find the probability for a z-score of -1 (approximately 0.1587) and a z-score of 1 (approximately 0.8413). To find the probability between these two z-scores, we subtract the smaller probability from the larger probability:

P(275 < x < 365) = 0.8413 - 0.1587 = 0.6826 (or 68.26%)

c) What percentage of organs weighs less than 275 grams or more than 365 grams?

To determine the percentage of organs that weigh less than 275 grams or more than 365 grams, we can subtract the probability calculated in part b from 1, since the total probability must add up to 1:

P(x < 275 or x > 365) = 1 - P(275 < x < 365) = 1 - 0.6826 = 0.3174 (or 31.74%)

d) What percentage of organs weighs between 230 grams and 365 grams?

To determine the percentage of organs that weigh between 230 grams and 365 grams, we need to calculate the z-scores for the lower and upper limits and then use the standard normal distribution table or calculator to find the corresponding probabilities. The z-score for 230 grams is calculated as:

Z = (230 - 320) / 45 = -2

The z-score for 365 grams is calculated as:

Z = (365 - 320) / 45 = 1

Using the standard normal distribution table or calculator, we can find the probability for a z-score of -2 (approximately 0.0228) and a z-score of 1 (approximately 0.8413). To find the probability between these two z-scores, we subtract the smaller probability from the larger probability:

P(230 < x < 365) = 0.8413 - 0.0228 = 0.8185 (or 81.85%)

Answer 2

a) 99.7%

b) 68%

c) 68% of organs weighs between 275 and 365 grams, so 32% of organs weighs will be less than 275 or more than 365 grams.

d) 81.5% of organs weighs between 290 grams and 365 grams.

Step-by-step explanation:

A) 99.7% of organs will be between 3 standard deviation from the mean.

`320 - 3 * 45 = 185`

`320 + 3 * 45 = 455`

So 99.7% of organs will be between 185 and 455.

B) 275 grams and 365 grams are 1 standard deviation from the mean.

From empirical rule about 68% data falls within 1 standard deviation from the mean.

So 68% of organs weighs btwn 275 grams and 365 grams.

C) Since 68% of organs weighs between 275 and 365 grams, so 32% of organs weighs will be less than 275 or more than 365 grams.

D) 230 is 2 standard deviation below the mean and 365 is one standard deviation above the mean.

According to the empirical rule, 95% of the observation lies within 2 standard deviations of the mean. Therefore 5% lies outside 2 standard deviations of the mean.

So 95%/2 = 47.5% of organs weighs between the mean and 230 grams.

According to the empirical rule about 68% data falls within one standard deviation from the mean.

so, 68%/2 = 34% of organs weighs between the mean and 365 grams.

So total = 47.5% + 34% = 81.5% of organs weighs between 290 grams and 365 grams.