Assuming that the heights of college women are normally distributed with mean 62 inches and standard deviation 3.5 inches, answer the following questions. (Hint: Use the figure below with mean ? and standard deviation ?.) ### Understanding the Distribution of College Women's HeightsAssuming that the heights of college women are normally distributed with a mean (μ) of 62 inches and a standard deviation (σ) of 3.5 inches, we can analyze and understand these heights using the normal distribution curve.#### Diagram: Area Under a Normal CurveThis diagram illustrates the distribution of data under a normal curve. The curve is symmetric around the mean (μ), and areas under the curve correspond to the percentage of the population within specific ranges.Here are the key areas under the normal curve:- **μ ± σ (1 standard deviation from the mean)**: 68% of the population- **μ ± 2σ (2 standard deviations from the mean)**: 95% of the population- **μ ± 3σ (3 standard deviations from the mean)**: 99.7% of the populationThe diagram shows these intervals visually with color-coded sections and their corresponding percentages:- From μ - 3σ to μ - 2σ: 2.35%- From μ - 2σ to μ - σ: 13.5%- From μ - σ to μ: 34%- From μ to μ + σ: 34%- From μ + σ to μ + 2σ: 13.5%- From μ + 2σ to μ + 3σ: 2.35%### QuestionsUsing the information from the diagram, we can now answer some specific questions regarding the distribution of heights among college women.1. **What percentage of women are taller than 62 inches?** - Explanation: Since 62 inches is the mean (μ), 50% of the population will be taller than the mean in a normal distribution. - **Answer: 50%**2. **What percentage of women are shorter than 62 inches?** - Explanation: Similarly, 50% of the population will be shorter than the mean. - **Answer: 50%**3. **What percentage of women are between 58.5 inches and 65.5 inches?** - Explanation: 58.5 inches is one standard deviation below the mean (62 - 3.5 = 58.5) and 65.5 inches is one standard deviation above the mean (62 + 3.5 = 65.5). According to the diagram, 68

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Answered: Assuming that the heights of college…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

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Assuming that the heights of college women are normally distributed with mean 62 inches and standard deviation 3.5 inches, answer the following questions. (Hint: Use the figure below with mean ? and standard deviation ?.)

### Understanding the Distribution of College Women's Heights

Assuming that the heights of college women are normally distributed with a mean (μ) of 62 inches and a standard deviation (σ) of 3.5 inches, we can analyze and understand these heights using the normal distribution curve.

#### Diagram: Area Under a Normal Curve
This diagram illustrates the distribution of data under a normal curve. The curve is symmetric around the mean (μ), and areas under the curve correspond to the percentage of the population within specific ranges.

Here are the key areas under the normal curve:
- **μ ± σ (1 standard deviation from the mean)**: 68% of the population
- **μ ± 2σ (2 standard deviations from the mean)**: 95% of the population
- **μ ± 3σ (3 standard deviations from the mean)**: 99.7% of the population

The diagram shows these intervals visually with color-coded sections and their corresponding percentages:
- From μ - 3σ to μ - 2σ: 2.35%
- From μ - 2σ to μ - σ: 13.5%
- From μ - σ to μ: 34%
- From μ to μ + σ: 34%
- From μ + σ to μ + 2σ: 13.5%
- From μ + 2σ to μ + 3σ: 2.35%

### Questions
Using the information from the diagram, we can now answer some specific questions regarding the distribution of heights among college women.

1. **What percentage of women are taller than 62 inches?**
- Explanation: Since 62 inches is the mean (μ), 50% of the population will be taller than the mean in a normal distribution.
- **Answer: 50%**

2. **What percentage of women are shorter than 62 inches?**
- Explanation: Similarly, 50% of the population will be shorter than the mean.
- **Answer: 50%**

3. **What percentage of women are between 58.5 inches and 65.5 inches?**
- Explanation: 58.5 inches is one standard deviation below the mean (62 - 3.5 = 58.5) and 65.5 inches is one standard deviation above the mean (62 + 3.5 = 65.5). According to the diagram, 68

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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