Questions: Question 19 of 20 This quiz: 20 point(s) possible This question: 1 Submit quiz point(s) possible The weight of an organ in adult males has a bell-shaped distribution with a mean of 350 grams and a standard deviation of 50 grams. Use the empirical rule to determine the following. (a) About 95% of organs will be between what weights? (b) What percentage of organs weighs between 300 grams and 400 grams? (c) What percentage of organs weighs less than 300 grams or more than 400 grams? (d) What percentage of organs weighs between 250 grams and 500 grams? (a) and grams (Use ascending order.) (b) % (Type an integer or a decimal.) (c) % (Type an integer or a decimal.) (d) % (Type an integer or decimal rounded to two decimal places as needed.)

Solution Steps

According to the empirical rule, approximately 95% of the data in a normal distribution falls within two standard deviations from the mean. Given the mean \(\mu = 350\) grams and the standard deviation \(\sigma = 50\) grams, we calculate the range as follows:

\[ \text{Lower bound} = \mu - 2\sigma = 350 - 2 \times 50 = 250 \, \text{grams} \]

\[ \text{Upper bound} = \mu + 2\sigma = 350 + 2 \times 50 = 450 \, \text{grams} \]

Thus, about 95% of organs will weigh between 250 grams and 450 grams.

To find the percentage of organs weighing between 300 grams and 400 grams, we calculate the probability that a value falls within this range using the standard normal distribution. The Z-scores for 300 grams and 400 grams are calculated as follows:

\[ Z_{\text{start}} = \frac{300 - 350}{50} = -1.0 \]

\[ Z_{\text{end}} = \frac{400 - 350}{50} = 1.0 \]

The probability \(P\) that an organ weighs between 300 grams and 400 grams is given by:

\[ P = \Phi(Z_{\text{end}}) - \Phi(Z_{\text{start}}) = \Phi(1.0) - \Phi(-1.0) = 0.68 \]

Thus, 68% of organs weigh between 300 grams and 400 grams.

The percentage of organs weighing less than 300 grams or more than 400 grams is the complement of the probability calculated in Step 2:

\[ \text{Percentage} = 100\% - 68\% = 32\% \]

Thus, 32% of organs weigh less than 300 grams or more than 400 grams.

Final Answer