Questions: The weight of an organ in adult males has a bell-shaped distribution with a mean of 330 grams and a standard deviation of 20 grams. Use the empirical rule to determine the following. (a) About 68% of organs will be between what weights? (b) What percentage of organs weighs between 290 grams and 370 grams? (c) What percentage of organs weighs less than 290 grams or more than 370 grams? (d) What percentage of organs weighs between 310 grams and 390 grams?

Solution Steps

The empirical rule, also known as the 68-95-99.7 rule, states that for a normal distribution:

• Approximately 68% of the data falls within one standard deviation of the mean.
• Approximately 95% of the data falls within two standard deviations of the mean.
• Approximately 99.7% of the data falls within three standard deviations of the mean.

Given:

• Mean (\(\mu\)) = 330 grams
• Standard deviation (\(\sigma\)) = 20 grams

For 68% of the data, we calculate the range as: \[ \mu - \sigma = 330 - 20 = 310 \text{ grams} \] \[ \mu + \sigma = 330 + 20 = 350 \text{ grams} \]

The range 290 to 370 grams is within two standard deviations from the mean: \[ \mu - 2\sigma = 330 - 2 \times 20 = 290 \text{ grams} \] \[ \mu + 2\sigma = 330 + 2 \times 20 = 370 \text{ grams} \]

According to the empirical rule, approximately 95% of the data falls within two standard deviations of the mean.

Since 95% of the data falls between 290 and 370 grams, the percentage of data that falls outside this range is: \[ 100\% - 95\% = 5\% \]

Final Answer