https://file.aiquickdraw.com/mathbot/file/1726268679884x54jufag.jpg

The problem provided involves a normal distribution with a mean of 300 grams and a standard deviation of 25 grams. We are asked to use the empirical rule (also known as the 68-95-99.7 rule) to solve several parts. Let's break it down:

### Given:
- Mean \( \mu = 300 \, \text{grams} \)
- Standard deviation \( \sigma = 25 \, \text{grams} \)
- We are using the **empirical rule** which states:
  - Approximately 68% of the data falls within 1 standard deviation of the mean
  - Approximately 95% of the data falls within 2 standard deviations of the mean
  - Approximately 99.7% of the data falls within 3 standard deviations of the mean

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

Using the empirical rule, 95% of the data lies within **2 standard deviations** of the mean. So the range is:

\[
\mu - 2\sigma = 300 - 2(25) = 300 - 50 = 250 \, \text{grams}
\]
\[
\mu + 2\sigma = 300 + 2(25) = 300 + 50 = 350 \, \text{grams}
\]

Thus, 95% of the organs will weigh between **250 grams and 350 grams**.

#### (b) What percentage of organs weighs between 225 grams and 375 grams?

We can calculate the number of standard deviations for both 225 grams and 375 grams.

\[
Z_{225} = \frac{225 - 300}{25} = -3
\]
\[
Z_{375} = \frac{375 - 300}{25} = 3
\]

According to the empirical rule, 99.7% of the data falls between \( Z = -3 \) and \( Z = 3 \), so **99.7% of organs** will weigh between 225 grams and 375 grams.

#### (c) What percentage of organs weighs less than 225 grams or more than 375 grams?

Since 99.7% of the data lies between 225 grams and 375 grams, the remaining percentage is outside this range. Therefore:

\[
100\% - 99.7\% = 0.3\%
\]

Thus, **0.3% of organs** weigh less than 225 grams or more than 375 grams.

#### (d) What percentage of organs weighs between 225 grams and 325 grams?

We calculate the number of standard deviations for both 225 grams and 325 grams.

\[
Z_{225} = \frac{225 - 300}{25} = -3
\]
\[
Z_{325} = \frac{325 - 300}{25} = 1
\]

Using the empirical rule:
- From \( Z = -3 \) to \( Z = -1 \) is **47.5%** (because half of 95% lies within 2 standard deviations).
- From \( Z = -1 \) to \( Z = 0 \) is **34%**.

Therefore, the total percentage between 225 grams and 325 grams is:

\[
47.5\% + 34\% = 81.5\%
\]

So, **81.5% of organs** weigh between 225 grams and 325 grams.

---

Let me know if you'd like further details on any part of this! Here are some related questions:

1. What would be the range for 68% of organs using the empirical rule?
2. How would the answers change if the standard deviation were 30 grams instead of 25 grams?
3. How do we interpret a Z-score in this context?
4. What is the probability that an organ weighs more than 350 grams?
5. Can we use the empirical rule for data that is not normally distributed?

**Tip**: The empirical rule only applies to data that follows a normal (bell-shaped) distribution. Be cautious when applying it to other types of data!

The weight of an organ in adult males has a bell-shaped distribution with a mean of 300 grams and a standard deviation of 25 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 225 grams and 375 grams? (c) What percentage of organs weighs less than 225 grams or more than 375 grams? (d) What percentage of organs weighs between 225 grams and 325 grams?

Solve a normal distribution problem involving the weight of an organ in adult males. Use the empirical rule (68-95-99.7 rule) to find the percentage of organs within specific weight ranges, based on a mean of 300 grams and a standard deviation of 25 grams. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation