Question

the weight of an organ in adult males has a bell - shaped distribution with a mean of 320 grams and a standard deviation of 15 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 350 grams?
(c) what percentage of organs weighs less than 290 grams or more than 350 grams?
(d) what percentage of organs weighs between 305 grams and 350 grams?
(a) (square) and (square) grams (use ascending order.)

Explanation:

Step1: Recall Empirical Rule basics

For bell-shaped distributions:

• 68% of data lies within $\mu \pm 1\sigma$
• 95% of data lies within $\mu \pm 2\sigma$
• 99.7% of data lies within $\mu \pm 3\sigma$

Given $\mu = 320$, $\sigma = 15$

Step2: Solve part (a): 68% weight range

Calculate $\mu - 1\sigma$ and $\mu + 1\sigma$
$\mu - 1\sigma = 320 - 15 = 305$
$\mu + 1\sigma = 320 + 15 = 335$

Step3: Solve part (b): % between 290 & 350

Check $\mu \pm 2\sigma$:
$\mu - 2\sigma = 320 - 2\times15 = 290$
$\mu + 2\sigma = 320 + 2\times15 = 350$
This matches the range, so use 95% rule.

Step4: Solve part (c): % <290 or >350

Total data is 100%. Subtract % between 290-350:
$100\% - 95\% = 5\%$

Step5: Solve part (d): % between 305 & 350

Break into two segments: 305-320 and 320-350.
305 is $\mu - 1\sigma$, so 305-320 is $\frac{68\%}{2}=34\%$
350 is $\mu + 2\sigma$, so 320-350 is $\frac{95\%}{2}=47.5\%$
Sum the two percentages: $34\% + 47.5\% = 81.5\%$

Answer:

(a) 305 and 335 grams
(b) 95%
(c) 5%
(d) 81.5%