A population of porcupines has a normally distributed lifespan with a mean of 5.5 years and a standard deviation of 1.0 years. We collected a sample of 12 porcupines and calculated their mean lifespan to be x¯ = 6.2 years. Which of the standard normal distributions below represent our observed sample mean (purple dashed line) compared to the sampling distribution of sample means (red line) for a sample size of 12?a. Ab. Bc. Cd. D ### Understanding Z-Scores and Probability DistributionsIn this section, we will explore a set of graphs that illustrate the concept of Z-scores and their corresponding positions within a standard normal distribution curve.#### Graph DescriptionsEach graph (A, B, C, and D) displays a standard normal distribution curve (in red), which represents a bell-shaped probability distribution that is symmetric around the mean (μ = 0) with a standard deviation (σ = 1). These distributions are used to calculate probabilities and identify outliers in data sets.Each graph includes a vertical dashed line (in purple) set at different Z-score values, highlighting how the Z-score relates to the distribution curve.- **Graph A:** - The vertical purple dashed line is positioned at approximately Z = 0.5. - This indicates a Z-score of 0.5, which means the point is 0.5 standard deviations above the mean of the distribution. - **Graph B:** - The vertical purple dashed line is positioned at approximately Z = 1. - This indicates a Z-score of 1, meaning the point is 1 standard deviation above the mean of the distribution. - **Graph C:** - The vertical purple dashed line is positioned at approximately Z = 1.5. - This indicates a Z-score of 1.5, meaning the point is 1.5 standard deviations above the mean of the distribution. - **Graph D:** - The vertical purple dashed line is positioned at approximately Z = 2. - This indicates a Z-score of 2, meaning the point is 2 standard deviations above the mean of the distribution.#### Interpreting Z-ScoresThe Z-score is a measure of how many standard deviations an element is from the mean. It is used in statistics to determine the relative position of a data point within a data set. - A positive Z-score indicates that the data point is above the mean.- A negative Z-score indicates that the data point is below the mean.- A Z-score of 0 indicates that the data point is exactly at the mean.#### ApplicationUnderstanding and visualizing Z-scores in the context of a normal distribution is crucial in fields such as psychology, finance, and natural sciences, where it is essential to determine how unusual or typical a particular value is within a given set of data. By examining the area under the

Answered: A population of porcupines has a…

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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Question

A population of porcupines has a normally distributed lifespan with a mean of 5.5 years and a standard deviation of 1.0 years. We collected a sample of 12 porcupines and calculated their mean lifespan to be x¯ = 6.2 years. Which of the standard normal distributions below represent our observed sample mean (purple dashed line) compared to the sampling distribution of sample means (red line) for a sample size of 12?

a. A

b. B

c. C

d. D

### Understanding Z-Scores and Probability Distributions

In this section, we will explore a set of graphs that illustrate the concept of Z-scores and their corresponding positions within a standard normal distribution curve.

#### Graph Descriptions

Each graph (A, B, C, and D) displays a standard normal distribution curve (in red), which represents a bell-shaped probability distribution that is symmetric around the mean (μ = 0) with a standard deviation (σ = 1). These distributions are used to calculate probabilities and identify outliers in data sets.

Each graph includes a vertical dashed line (in purple) set at different Z-score values, highlighting how the Z-score relates to the distribution curve.

- **Graph A:**
- The vertical purple dashed line is positioned at approximately Z = 0.5.
- This indicates a Z-score of 0.5, which means the point is 0.5 standard deviations above the mean of the distribution.

- **Graph B:**
- The vertical purple dashed line is positioned at approximately Z = 1.
- This indicates a Z-score of 1, meaning the point is 1 standard deviation above the mean of the distribution.

- **Graph C:**
- The vertical purple dashed line is positioned at approximately Z = 1.5.
- This indicates a Z-score of 1.5, meaning the point is 1.5 standard deviations above the mean of the distribution.

- **Graph D:**
- The vertical purple dashed line is positioned at approximately Z = 2.
- This indicates a Z-score of 2, meaning the point is 2 standard deviations above the mean of the distribution.

#### Interpreting Z-Scores

The Z-score is a measure of how many standard deviations an element is from the mean. It is used in statistics to determine the relative position of a data point within a data set.

- A positive Z-score indicates that the data point is above the mean.
- A negative Z-score indicates that the data point is below the mean.
- A Z-score of 0 indicates that the data point is exactly at the mean.

#### Application

Understanding and visualizing Z-scores in the context of a normal distribution is crucial in fields such as psychology, finance, and natural sciences, where it is essential to determine how unusual or typical a particular value is within a given set of data. By examining the area under the

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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