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 99.7% 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% within $\mu\pm1\sigma$
• 95% within $\mu\pm2\sigma$
• 99.7% within $\mu\pm3\sigma$

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

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Part (a)

Step1: Calculate 3σ range

Lower bound: $\mu - 3\sigma = 320 - 3\times15$
Upper bound: $\mu + 3\sigma = 320 + 3\times15$

Step2: Compute values

Lower: $320 - 45 = 275$
Upper: $320 + 45 = 365$

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Part (b)

Step1: Map values to σ range

$290 = 320 - 2\times15 = \mu-2\sigma$
$350 = 320 + 2\times15 = \mu+2\sigma$

Step2: Apply empirical rule

95% of data lies in $\mu\pm2\sigma$

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Part (c)

Step1: Use complement rule

Total data = 100%, subtract % from (b)
Percentage = $100\% - 95\%$

Step2: Compute result

$100\% - 95\% = 5\%$

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Part (d)

Step1: Map values to σ range

$305 = 320 - 1\times15 = \mu-1\sigma$
$350 = 320 + 2\times15 = \mu+2\sigma$

Step2: Split into segments

% = (% between $\mu-\sigma$ and $\mu$) + (% between $\mu$ and $\mu+2\sigma$)
$\frac{68\%}{2} + \frac{95\%}{2} = 34\% + 47.5\%$

Step3: Sum the segments

$34\% + 47.5\% = 81.5\%$

Answer:

(a) 275 and 365 grams
(b) 95%
(c) 5%
(d) 81.5%