A survey found that women's heights are normally distributed with mean 63.5 in and standard deviation 2.5 in. A branch of the military requires women's heights to be between 58 in and 80 in.a. Find the percentage of women meeting the height requirement. Are many women being denied the opportunity to join this branch of the military because they are too short or too tall?b. If this branch of the military changes the height requirements so that all women are eligible except the shortest 1% and the tallest 2%, what are the new height requirements? ### Positive z Scores: Standard Normal (z) Distribution Table#### OverviewThis table displays the cumulative area from the left under the standard normal curve for positive z scores. This is useful for determining probabilities and critical values in statistical analyses.#### Table Structure- **Rows and Columns**: The table is organized with z scores increasing from 0.0 to 3.09 along the row headers, with additional decimals from .00 to .09 across the column headers.- **Values**: Each cell in the table represents the cumulative probability from the left of the standard normal distribution for a given z score (intersection of row and column).For example:- A z score of 0.00 corresponds to a cumulative area of 0.5000.- A z score of 1.23 corresponds to a cumulative area of 0.8907.#### Important Calculations- For z scores above 3.49, use 0.9999 for the cumulative area.- **Interpolation**: - For specific z scores like 1.645, use interpolation to find more accurate cumulative areas. - $ z = 1.645 $ is approximately 0.9500. - $ z = 2.575 $ is approximately 0.9950.#### Common Critical ValuesDifferent levels of confidence correspond to specific critical z scores:- A 90% confidence level corresponds to a critical z value of 1.645.- A 95% confidence level corresponds to a critical z value of 1.96.- A 99% confidence level corresponds to a critical z value of 2.575.#### Visual RepresentationThe blue bell curve diagram at the top represents the standard normal distribution. The shading under the curve indicates the area (probability) left of a specific z score.This table is a fundamental tool in statistical data analysis, particularly for tasks involving hypothesis testing, confidence intervals, and probability calculations. **Negative z-Scores****Standard Normal (z) Distribution: Cumulative Area from the LEFT**This table shows the cumulative area under the standard normal curve to the left of specified z-scores, which are negative in this case. This table is used to find probabilities associated with standard normal distribution.**z-Score Table**- **Columns**: Represent the hundredths place of the z-score.- **Rows**: Represent the tenths and units place of the z-score.- **Values**: Indicate the cumulative probability to the left of the z-score.**Example Calculations:**- For a z-score of -3.4, use the row for -3.4 and the column for .00, resulting in a cumulative area of 0.0003.- For a z-score of -2.57, using the interpolation, the cumulative area is approximately 0.0050.**Graph**The graph at the top right shows a normal distribution curve with a shaded area to the left, representing the area under the curve up to a certain z-score.**Special Notes**- For values of z below -3.49, use 0.0001 for the area.- Use common values for interpolation: - z-score: -1.645 has an area of 0.0500 - z-score: -2.575 has an area of 0.0050This table and graph are essential for understanding probabilities in statistics, particularly those that fall below the mean in a normal distribution.

It is given that the distribution of women’s height follows normal distribution with mean 63.5 in…

Answered: A survey found that women's heights…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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A survey found that women's heights are normally distributed with mean 63.5 in and standard deviation 2.5 in. A branch of the military requires women's heights to be between 58 in and 80 in.

a. Find the percentage of women meeting the height requirement. Are many women being denied the opportunity to join this branch of the military because they are too short or too tall?

b. If this branch of the military changes the height requirements so that all women are eligible except the shortest 1% and the tallest 2%, what are the new height requirements?

### Positive z Scores: Standard Normal (z) Distribution Table

#### Overview
This table displays the cumulative area from the left under the standard normal curve for positive z scores. This is useful for determining probabilities and critical values in statistical analyses.

#### Table Structure
- **Rows and Columns**: The table is organized with z scores increasing from 0.0 to 3.09 along the row headers, with additional decimals from .00 to .09 across the column headers.

- **Values**: Each cell in the table represents the cumulative probability from the left of the standard normal distribution for a given z score (intersection of row and column).

For example:
- A z score of 0.00 corresponds to a cumulative area of 0.5000.
- A z score of 1.23 corresponds to a cumulative area of 0.8907.

#### Important Calculations
- For z scores above 3.49, use 0.9999 for the cumulative area.

- **Interpolation**:
- For specific z scores like 1.645, use interpolation to find more accurate cumulative areas.
- $ z = 1.645 $ is approximately 0.9500.
- $ z = 2.575 $ is approximately 0.9950.

#### Common Critical Values
Different levels of confidence correspond to specific critical z scores:
- A 90% confidence level corresponds to a critical z value of 1.645.
- A 95% confidence level corresponds to a critical z value of 1.96.
- A 99% confidence level corresponds to a critical z value of 2.575.

#### Visual Representation
The blue bell curve diagram at the top represents the standard normal distribution. The shading under the curve indicates the area (probability) left of a specific z score.

This table is a fundamental tool in statistical data analysis, particularly for tasks involving hypothesis testing, confidence intervals, and probability calculations.

**Negative z-Scores**

**Standard Normal (z) Distribution: Cumulative Area from the LEFT**

This table shows the cumulative area under the standard normal curve to the left of specified z-scores, which are negative in this case. This table is used to find probabilities associated with standard normal distribution.

**z-Score Table**

- **Columns**: Represent the hundredths place of the z-score.
- **Rows**: Represent the tenths and units place of the z-score.
- **Values**: Indicate the cumulative probability to the left of the z-score.

**Example Calculations:**

- For a z-score of -3.4, use the row for -3.4 and the column for .00, resulting in a cumulative area of 0.0003.
- For a z-score of -2.57, using the interpolation, the cumulative area is approximately 0.0050.

**Graph**

The graph at the top right shows a normal distribution curve with a shaded area to the left, representing the area under the curve up to a certain z-score.

**Special Notes**

- For values of z below -3.49, use 0.0001 for the area.
- Use common values for interpolation:
- z-score: -1.645 has an area of 0.0500
- z-score: -2.575 has an area of 0.0050

This table and graph are essential for understanding probabilities in statistics, particularly those that fall below the mean in a normal distribution.

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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