The weight of an organ in adult males has a​ bell-shaped distribution with a mean of 340 grams and a standard deviation of 15 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 325 grams and 355 ​grams?
​(c) What percentage of organs weighs less than 325 grams or more than 355 ​grams?
​(d) What percentage of organs weighs between 310 grams and 385 ​grams?

To solve this problem, we will apply the **Empirical Rule**, which is used to approximate percentages in a **bell-shaped (normal) distribution**. The rule states:

1. About 68% of the data falls within 1 standard deviation of the mean.
2. About 95% of the data falls within 2 standard deviations of the mean.
3. About 99.7% of the data falls within 3 standard deviations of the mean.

### Given:
- **Mean (μ) = 340 grams**
- **Standard Deviation (σ) = 15 grams**

---

### (a) About 95% of organs will be between what weights?

According to the Empirical Rule, 95% of the data lies within **2 standard deviations** from the mean:

\[
\text{Lower bound} = \mu - 2\sigma = 340 - 2(15) = 340 - 30 = 310 \text{ grams}
\]
\[
\text{Upper bound} = \mu + 2\sigma = 340 + 2(15) = 340 + 30 = 370 \text{ grams}
\]

So, about **95%** of organs weigh between **310 grams and 370 grams**.

---

### (b) What percentage of organs weighs between 325 grams and 355 grams?

We need to find how many standard deviations 325 and 355 grams are from the mean:

\[
z_{\text{325}} = \frac{325 - 340}{15} = \frac{-15}{15} = -1 \quad (\text{1 standard deviation below the mean})
\]
\[
z_{\text{355}} = \frac{355 - 340}{15} = \frac{15}{15} = 1 \quad (\text{1 standard deviation above the mean})
\]

According to the Empirical Rule, about **68%** of the data lies within **1 standard deviation** from the mean. Therefore, the percentage of organs that weigh between **325 grams and 355 grams** is **68%**.

---

### (c) What percentage of organs weighs less than 325 grams or more than 355 grams?

Since we already know that **68%** of the data lies between **325 grams and 355 grams** (1 standard deviation from the mean), the remaining percentage of data lies outside this range:

\[
100\% - 68\% = 32\%
\]

Therefore, the percentage of organs that weigh less than **325 grams** or more than **355 grams** is **32%**.

---

### (d) What percentage of organs weighs between 310 grams and 385 grams?

We need to calculate how many standard deviations 310 grams and 385 grams are from the mean:

\[
z_{\text{310}} = \frac{310 - 340}{15} = \frac{-30}{15} = -2 \quad (\text{2 standard deviations below the mean})
\]
\[
z_{\text{385}} = \frac{385 - 340}{15} = \frac{45}{15} = 3 \quad (\text{3 standard deviations above the mean})
\]

From the Empirical Rule:
- **95%** of the data lies within 2 standard deviations (310 to 370 grams).
- **99.7%** of the data lies within 3 standard deviations (310 to 385 grams).

So, the percentage of organs that weigh between **310 grams and 385 grams** is **99.7%**.

---

### Summary:
- (a) About **95%** of organs weigh between **310 grams and 370 grams**.
- (b) **68%** of organs weigh between **325 grams and 355 grams**.
- (c) **32%** of organs weigh less than **325 grams** or more than **355 grams**.
- (d) **99.7%** of organs weigh between **310 grams and 385 grams**.

---

Do you have any questions or want further details on any of these points?

---

Here are five related questions for deeper understanding:
1. What percentage of organs weighs between 295 grams and 385 grams?
2. What percentage of organs weighs more than 370 grams?
3. What is the probability that an organ weighs between 310 grams and 340 grams?
4. How does the Empirical Rule apply to skewed distributions?
5. If the standard deviation was 20 grams, how would the range in part (a) change?

**Tip:** The Empirical Rule only applies to normal distributions. For other distributions, more advanced techniques like z-scores or percentiles are required!

The weight of an organ in adult males has a​ bell-shaped distribution with a mean of 340 grams and a standard deviation of 15 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 325 grams and 355 ​grams? ​(c) What percentage of organs weighs less than 325 grams or more than 355 ​grams? ​(d) What percentage of organs weighs between 310 grams and 385 ​grams?

Use the empirical rule to solve a problem involving the weight of organs in adult males with a bell-shaped distribution. Learn how to calculate ranges for 95% and 99.7% of the data, find percentages between given weights, and interpret z-scores. Suitable for students in Grades 9-12.

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