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One population has a mean of u = 50 and a standard deviation of a = 15, and a different population has a mean of u = 50 and a standard deviation of 0 = 5. Refer to the images below to answer the questions that follow.

One population has a mean of u = 50 and a standard deviation of a = 15, and a different population has a mean of u = 50 and a standard deviation of 0 = 5. Refer to the images below to answer the questions that follow.

MATLAB: An Introduction with Applications
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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--- One population has a mean of μ = 50 and a standard deviation of σ = 15, and a different population has a mean of μ = 50 and a standard deviation of σ = 5. Refer to the images below to answer the questions that follow. ![Chart](chart_image) The image contains a bell curve (normal distribution curve). It is symmetrical and centered around the mean (μ) value of 50. The curve represents a population with a standard deviation, although the image does not specify which of the populations (standard deviation of 15 or 5) it belongs to. --- **Explanation of the Graph:** - The normal distribution curve, depicted in red, is centered around the mean, μ, which is 50 in this case. - The peak of the bell curve at the mean indicates the highest frequency of data points around this value. - The width and spread of the curve would differ depending on the standard deviation. However, the image only shows one curve without specifying if it represents the standard deviation of ±15 or ±5. - A larger standard deviation (σ = 15) would make the curve wider and flatter. - A smaller standard deviation (σ = 5) would make the curve narrower and taller. For accurate distinction, additional information or separate images with labels indicating the specific standard deviations would be required. --- Please refer to the next section for the follow-up questions based on the provided graph and statistical information.
Transcribed Image Text:--- One population has a mean of μ = 50 and a standard deviation of σ = 15, and a different population has a mean of μ = 50 and a standard deviation of σ = 5. Refer to the images below to answer the questions that follow. ![Chart](chart_image) The image contains a bell curve (normal distribution curve). It is symmetrical and centered around the mean (μ) value of 50. The curve represents a population with a standard deviation, although the image does not specify which of the populations (standard deviation of 15 or 5) it belongs to. --- **Explanation of the Graph:** - The normal distribution curve, depicted in red, is centered around the mean, μ, which is 50 in this case. - The peak of the bell curve at the mean indicates the highest frequency of data points around this value. - The width and spread of the curve would differ depending on the standard deviation. However, the image only shows one curve without specifying if it represents the standard deviation of ±15 or ±5. - A larger standard deviation (σ = 15) would make the curve wider and flatter. - A smaller standard deviation (σ = 5) would make the curve narrower and taller. For accurate distinction, additional information or separate images with labels indicating the specific standard deviations would be required. --- Please refer to the next section for the follow-up questions based on the provided graph and statistical information.
### Understanding Probability Distributions In this section, we will explore two different probability distributions graphically depicted in the image above. These distributions are crucial in statistics for understanding data behavior and making predictions. #### Graph Descriptions: 1. **Top Graph - Leptokurtic Distribution:** - This graph shows a probability distribution that is more peaked than a normal distribution. - The graph has a sharp peak around the mean value, indicating a higher probability of data points near the mean. - The x-axis ranges from 20 to 80, with the peak centered around 50. 2. **Bottom Graph - Platykurtic Distribution:** - This graph shows a probability distribution that is flatter than a normal distribution. - It indicates that data points are more dispersed away from the mean. - Like the top graph, the x-axis ranges from 20 to 80, with a central peak around the value of 50. However, this peak is much broader. #### Key Differences: - **Kurtosis**: The top graph (leptokurtic) has higher kurtosis, meaning it has heavy tails and a sharper peak. The bottom graph (platykurtic) has lower kurtosis, meaning it has lighter tails and a flatter peak. - **Width of the Peak**: The sharpness of the peak in the leptokurtic distribution indicates low variability. In contrast, the broader peak in the platykurtic distribution implies higher variability and more spread out data points. Understanding these distributions helps in analyzing datasets and inferring patterns. Different kurtosis levels affect predictions and the reliability of statistical models built using such data. ##### Application: - In finance, a leptokurtic distribution may indicate a higher risk of extreme returns, while a platykurtic distribution might suggest a more stable investment with lower chances of extreme results. - In quality control, leptokurtic distributions can indicate consistency in manufacturing processes, whereas platykurtic distributions might signal variability that needs addressing. By recognizing these patterns, professionals in various fields can make informed decisions based on the statistical properties of their data.
Transcribed Image Text:### Understanding Probability Distributions In this section, we will explore two different probability distributions graphically depicted in the image above. These distributions are crucial in statistics for understanding data behavior and making predictions. #### Graph Descriptions: 1. **Top Graph - Leptokurtic Distribution:** - This graph shows a probability distribution that is more peaked than a normal distribution. - The graph has a sharp peak around the mean value, indicating a higher probability of data points near the mean. - The x-axis ranges from 20 to 80, with the peak centered around 50. 2. **Bottom Graph - Platykurtic Distribution:** - This graph shows a probability distribution that is flatter than a normal distribution. - It indicates that data points are more dispersed away from the mean. - Like the top graph, the x-axis ranges from 20 to 80, with a central peak around the value of 50. However, this peak is much broader. #### Key Differences: - **Kurtosis**: The top graph (leptokurtic) has higher kurtosis, meaning it has heavy tails and a sharper peak. The bottom graph (platykurtic) has lower kurtosis, meaning it has lighter tails and a flatter peak. - **Width of the Peak**: The sharpness of the peak in the leptokurtic distribution indicates low variability. In contrast, the broader peak in the platykurtic distribution implies higher variability and more spread out data points. Understanding these distributions helps in analyzing datasets and inferring patterns. Different kurtosis levels affect predictions and the reliability of statistical models built using such data. ##### Application: - In finance, a leptokurtic distribution may indicate a higher risk of extreme returns, while a platykurtic distribution might suggest a more stable investment with lower chances of extreme results. - In quality control, leptokurtic distributions can indicate consistency in manufacturing processes, whereas platykurtic distributions might signal variability that needs addressing. By recognizing these patterns, professionals in various fields can make informed decisions based on the statistical properties of their data.
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