The mean percentage of people 65 and older in populations across all U.S. countries is 0.18 and the standard deviation is 0.04. Fill in the normal curve below with values for u and I, and label each interval and the percentage of data each comprises, based on the normal approximation of those values. What are the values of each one on the normal curve? ### Part 2 – Estimate Values Using the Empirical RuleOne reason to use a normal distribution model is to make estimates about data, especially data that are not readily available. The model gives statisticians confidence that the mean is equal to the median and that the empirical rule holds true. This rule, also called the "68-95-99.7 rule," states that for normal distributions:- Approximately 68 percent of the data are within one standard deviation, σ, of the mean, μ.- Approximately 95 percent of the data are within two standard deviations, σ, of the mean, μ.- Approximately 99.7 percent of the data are within three standard deviations, σ, of the mean, μ. **Understanding the Normal Distribution Curve**The image is a graphical representation of the normal distribution, also known as the bell curve. This curve is a visual depiction of how data points are distributed around the mean in a dataset that follows a normal distribution.### Key Components of the Graph:1. **Horizontal Axis (Standard Deviations):** - Labeled as \( \mu - 3\sigma \), \( \mu - 2\sigma \), \( \mu - \sigma \), \( \mu \), \( \mu + \sigma \), \( \mu + 2\sigma \), and \( \mu + 3\sigma \). - \( \mu \) represents the mean (average) of the dataset. - \( \sigma \) represents the standard deviation, which measures the dispersion of the data points from the mean.2. **Vertical Axis:** - This axis represents the probability density, which indicates the likelihood of each value occurring in the dataset.### Key Areas of the Graph:1. **The Center of the Curve (\( \mu \)):** - The peak of the curve is at the mean (\( \mu \)), where the majority of data points are concentrated.2. **One Standard Deviation (\( \mu \pm \sigma \)):** - Around 68% of the data lies within one standard deviation of the mean (between \( \mu - \sigma \) and \( \mu + \sigma \)). 3. **Two Standard Deviations (\( \mu \pm 2\sigma \)):** - Around 95% of the data falls within two standard deviations of the mean (between \( \mu - 2\sigma \) and \( \mu + 2\sigma \)). 4. **Three Standard Deviations (\( \mu \pm 3\sigma \)):** - About 99.7% of the data is within three standard deviations of the mean (between \( \mu - 3\sigma \) and \( \mu + 3\sigma \)).### Significance of the Normal Distribution:- **Symmetry:** The curve is symmetric around the mean, meaning that the data is evenly distributed on both sides of the mean.- **Bell Shape:** The shape of the curve resembles a bell, with rapid decrease in probability as you move away from the mean.- **Practical Applications:** This

Given Data : Population Mean,μ = 0.18 Population Standard Deviation,σ = 0.04

Answered: O art 2 - Estimate Values Using the…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

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The mean percentage of people 65 and older in populations across all U.S. countries is 0.18 and the standard deviation is 0.04. Fill in the normal curve below with values for u and I, and label each interval and the percentage of data each comprises, based on the normal approximation of those values. What are the values of each one on the normal curve?

### Part 2 – Estimate Values Using the Empirical Rule

One reason to use a normal distribution model is to make estimates about data, especially data that are not readily available. The model gives statisticians confidence that the mean is equal to the median and that the empirical rule holds true. This rule, also called the "68-95-99.7 rule," states that for normal distributions:

- Approximately 68 percent of the data are within one standard deviation, σ, of the mean, μ.
- Approximately 95 percent of the data are within two standard deviations, σ, of the mean, μ.
- Approximately 99.7 percent of the data are within three standard deviations, σ, of the mean, μ.

**Understanding the Normal Distribution Curve**

The image is a graphical representation of the normal distribution, also known as the bell curve. This curve is a visual depiction of how data points are distributed around the mean in a dataset that follows a normal distribution.

### Key Components of the Graph:

1. **Horizontal Axis (Standard Deviations):**
- Labeled as \( \mu - 3\sigma \), \( \mu - 2\sigma \), \( \mu - \sigma \), \( \mu \), \( \mu + \sigma \), \( \mu + 2\sigma \), and \( \mu + 3\sigma \).
- \( \mu \) represents the mean (average) of the dataset.
- \( \sigma \) represents the standard deviation, which measures the dispersion of the data points from the mean.

2. **Vertical Axis:**
- This axis represents the probability density, which indicates the likelihood of each value occurring in the dataset.

### Key Areas of the Graph:

1. **The Center of the Curve (\( \mu \)):**
- The peak of the curve is at the mean (\( \mu \)), where the majority of data points are concentrated.

2. **One Standard Deviation (\( \mu \pm \sigma \)):**
- Around 68% of the data lies within one standard deviation of the mean (between \( \mu - \sigma \) and \( \mu + \sigma \)).

3. **Two Standard Deviations (\( \mu \pm 2\sigma \)):**
- Around 95% of the data falls within two standard deviations of the mean (between \( \mu - 2\sigma \) and \( \mu + 2\sigma \)).

4. **Three Standard Deviations (\( \mu \pm 3\sigma \)):**
- About 99.7% of the data is within three standard deviations of the mean (between \( \mu - 3\sigma \) and \( \mu + 3\sigma \)).

### Significance of the Normal Distribution:

- **Symmetry:** The curve is symmetric around the mean, meaning that the data is evenly distributed on both sides of the mean.
- **Bell Shape:** The shape of the curve resembles a bell, with rapid decrease in probability as you move away from the mean.
- **Practical Applications:** This

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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