The weight of an organ in adult males has a bell-shaped distribution with a mean of 310 grams and a standard deviation of 50 grams. Use the empirical rule (a) About \( 68 \% \) of organs will be between what weights? (b) What percentage of organs weighs between 210 grams and 410 grams? (c) What percentage of organs weighs less than 210 grams or more than 410 grams? (d) What percentage of organs weighs between 160 grams and 410 grams? (a) \( \square \) and \( \square \) grams (Use ascending order.)

To solve this problem, we will use the empirical rule, which states that for a normal distribution: 1. About \( 68\% \) of the data falls within one standard deviation of the mean. 2. About \( 95\% \) of the data falls within two standard deviations of the mean. 3. About \( 99.7\% \) of the data falls within three standard deviations of the mean. Given: - Mean (\( \mu \)) = 310 grams - Standard Deviation (\( \sigma \)) = 50 grams ### (a) About \( 68\% \) of organs will be between what weights? Using the empirical rule: - One standard deviation from the mean is calculated as: - Lower limit: \( \mu - \sigma = 310 - 50 = 260 \) grams - Upper limit: \( \mu + \sigma = 310 + 50 = 360 \) grams Thus, about \( 68\% \) of organs will weigh between **260 grams** and **360 grams**. ### (b) What percentage of organs weighs between 210 grams and 410 grams? To find this, we need to determine how many standard deviations away from the mean these weights are: - For 210 grams: \[ z = \frac{210 - 310}{50} = \frac{-100}{50} = -2 \] - For 410 grams: \[ z = \frac{410 - 310}{50} = \frac{100}{50} = 2 \] According to the empirical rule, about \( 95\% \) of the data falls within two standard deviations of the mean. Therefore, the percentage of organs that weigh between 210 grams and 410 grams is approximately **95%**. ### (c) What percentage of organs weighs less than 210 grams or more than 410 grams? Since \( 95\% \) of the organs weigh between 210 grams and 410 grams, the remaining percentage will be in the tails: \[ 100\% - 95\% = 5\% \] This \( 5\% \) is split between the two tails (less than 210 grams and more than 410 grams), so: \[ \text{Percentage in each tail} = \frac{5\%}{2} = 2.5\% \] Thus, the percentage of organs that weigh less than 210 grams or more than 410 grams is **5%**. ### (d) What percentage of organs weighs between 160 grams and 410 grams? First, we calculate how many standard deviations 160 grams is from the mean: - For 160 grams: \[ z = \frac{160 - 310}{50} = \frac{-150}{50} = -3 \] According to the empirical rule, about \( 99.7\% \) of the data falls within three standard deviations of the mean. Therefore, the percentage of organs that weigh between 160 grams and 410 grams is approximately **99.7%**. ### Summary of Answers: (a) \( 260 \) and \( 360 \) grams (b) \( 95\% \) (c) \( 5\% \) (d) \( 99.7\% \)