Questions: The weight of an organ in adult males has a bell-shaped distribution with a mean of 300 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 230 grams and 370 grams? (c) What percentage of organs weighs less than 230 grams or more than 370 grams? (d) What percentage of organs weighs between 265 grams and 405 grams?

Solution Steps

The empirical rule states that for a bell-shaped distribution:

• Approximately \( 99.7\% \) of the data lies within \( 3 \) standard deviations of the mean.

Given:

• Mean (\( \mu \)) = \( 300 \) grams
• Standard deviation (\( \sigma \)) = \( 35 \) grams

The range for \( 99.7\% \) of the data is: \[ \mu - 3\sigma \quad \text{to} \quad \mu + 3\sigma \] Substitute the values: \[ 300 - 3(35) = 195 \quad \text{and} \quad 300 + 3(35) = 405 \] Thus, \( 99.7\% \) of organs will be between \( 195 \) grams and \( 405 \) grams.

First, convert the weights to z-scores: \[ z = \frac{X - \mu}{\sigma} \] For \( 230 \) grams: \[ z = \frac{230 - 300}{35} = -2 \] For \( 370 \) grams: \[ z = \frac{370 - 300}{35} = 2 \] Using the empirical rule:

• Approximately \( 95\% \) of the data lies within \( 2 \) standard deviations of the mean.

Thus, \( 95\% \) of organs weigh between \( 230 \) grams and \( 370 \) grams.

From Step 2, we know that \( 95\% \) of organs lie between \( 230 \) grams and \( 370 \) grams. Therefore, the percentage of organs outside this range is: \[ 100\% - 95\% = 5\% \] Thus, \( 5\% \) of organs weigh less than \( 230 \) grams or more than \( 370 \) grams.

Final Answer