The weight of an organ in adult males has a bell-shaped distribution with a mean of 325 grams and a standard deviation of 15 grams. Use the empirical rule to determin (a) About $68 \%$ of organs will be between what weights?
(b) What percentage-of organs weighs between 280 grams and 370 grams?
(c) What percentage of organs weighs less than 280 grams or more than 370 grams?
(d) What percentage of organs weighs between 310 grams and 370 grams?
(a) and grams (Use ascending order.)
(b) $\square \%$ (Type an integer or a decimal)
(c) $\square \%$ (Type an integer or a decimal.)
(d) $\square \%$ (Type an integer or decimal rounded to two decimal places as needed)
Answer
For the fourth question, we need to find the percentage of organs that weigh between 310 grams and 370 grams. This is equivalent to the mean minus one standard deviation and the mean plus three standard deviations. According to the empirical rule, this should be about \(\boxed{81.5\%}\).
Step 1 ：The empirical rule, also known as the 68-95-99.7 rule, states that for a normal distribution, almost all data falls within three standard deviations of the mean. Specifically, 68% of data falls within one standard deviation, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.
Step 2 ：For the first question, we need to find the weights that 68% of organs will fall between. This is simply the mean plus and minus one standard deviation. So, the weights that 68% of organs will fall between are \(\boxed{310}\) grams and \(\boxed{340}\) grams.
Step 3 ：For the second question, we need to find the percentage of organs that weigh between 280 grams and 370 grams. This is equivalent to the mean minus three standard deviations and the mean plus three standard deviations. According to the empirical rule, this should be about \(\boxed{99.7\%}\).
Step 4 ：For the third question, we need to find the percentage of organs that weigh less than 280 grams or more than 370 grams. This is the complement of the percentage found in part (b), so we subtract that percentage from 100%. So, about \(\boxed{0.3\%}\) of organs weigh less than 280 grams or more than 370 grams.
Step 5 ：For the fourth question, we need to find the percentage of organs that weigh between 310 grams and 370 grams. This is equivalent to the mean minus one standard deviation and the mean plus three standard deviations. According to the empirical rule, this should be about \(\boxed{81.5\%}\).
