Question

the weight of an organ in adult males has a bell - shaped distribution with a mean of 350 grams and a standard deviation of 35 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 245 grams and 455 grams? (c) what percentage of organs weighs less than 245 grams or more than 455 grams? (d) what percentage of organs weighs between 245 grams and 385 grams? (a) 315 and 385 grams (use ascending order.) (b) 99.7% (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.)

Explanation:

Step1: Recall empirical rule for normal distribution

The empirical rule states that for a normal - distribution: about 68% of the data lies within 1 standard deviation of the mean ($\mu\pm\sigma$), about 95% lies within 2 standard deviations ($\mu\pm2\sigma$), and about 99.7% lies within 3 standard deviations ($\mu\pm3\sigma$). Given $\mu = 350$ grams and $\sigma=35$ grams.

Step2: Calculate values for 2 - standard - deviation range

For part (c), the lower value is $\mu - 2\sigma=350 - 2\times35=350 - 70 = 280$ grams and the upper value is $\mu + 2\sigma=350+2\times35 = 350 + 70=420$ grams. The range from 245 grams ($\mu - 3\sigma$) to 455 grams ($\mu + 3\sigma$) contains about 99.7% of the data. The range from 280 grams to 420 grams contains about 95% of the data.
The proportion of data between 245 and 280 grams is $\frac{99.7 - 95}{2}=2.35\%$, and the proportion of data between 420 and 455 grams is also 2.35%. So the proportion of data between 245 and 455 grams is 95%+2.35%+2.35% = 99.7%.

Step3: Calculate proportion outside 2 - standard - deviation range

For part (d), the proportion of data outside the range of 245 grams and 455 grams is $100 - 99.7=0.3\%$.

Step4: Calculate proportion between 245 and 385 grams

The value 385 grams is $\mu+\sigma$. The proportion of data between $\mu - 3\sigma$ (245 grams) and $\mu$ is 49.85% (half of 99.7%), and the proportion of data between $\mu$ and $\mu+\sigma$ is 34% (half of 68%). So the proportion of data between 245 grams and 385 grams is 49.85%+34% = 83.85%.

Answer:

(c) 99.7
(d) 0.3