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

See similar textbooks

Related questions

Concept explainers

Question

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.

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.

Expert Solution

Trending now

This is a popular solution!

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Continuous Probability Distribution$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.