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Scores on a statistics final in a large class were normally distributed with a mean of 73 and a standard deviation of 6. Use the empirical rule to answer the following questions. a. Estimate the proportion of the scores less than 67. b. Estimate the proportion of the scores more than 85. c. Estimate the proportion of the scores between 61 and 79.

Scores on a statistics final in a large class were normally distributed with a mean of 73 and a standard deviation of 6. Use the empirical rule to answer the following questions. a. Estimate the proportion of the scores less than 67. b. Estimate the proportion of the scores more than 85. c. Estimate the proportion of the scores between 61 and 79.

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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**Understanding Normal Distribution and the Empirical Rule** **Context:** Scores on a statistics final in a large class were normally distributed with a mean of 73 and a standard deviation of 6. Use the empirical rule to answer the following questions: **Questions:** a. Estimate the proportion of the scores less than 67. b. Estimate the proportion of the scores more than 85. c. Estimate the proportion of the scores between 61 and 79. **Explanation of Concepts:** The normal distribution is a probability distribution that is symmetric about the mean. The empirical rule, also known as the 68-95-99.7 rule, states that: - About 68% of the data falls within one standard deviation of the mean. - About 95% of the data falls within two standard deviations of the mean. - About 99.7% of the data falls within three standard deviations of the mean. In this context: - The mean (µ) is 73. - The standard deviation (σ) is 6. Each of the questions can be answered by applying the empirical rule to determine the proportion of scores within specified ranges of the normal distribution. **Graph/Diagram Explanation:** A typical graph for this problem would be a bell-shaped curve centered at 73 (the mean), with markers at one, two, and three standard deviations away from the mean in both directions (left and right). These markers would represent 67, 61, and 55 to the left of the mean, and 79, 85, and 91 to the right of the mean. Such a graphical representation helps in visualizing and estimating the proportions asked in the questions. **Detailed Breakdown of Questions:** **Question a:** - \(67 = 73 - 1 \cdot 6\) 67 is one standard deviation below the mean. According to the empirical rule, approximately 68% of the data lies between one standard deviation below and one standard deviation above the mean. This implies that about 32% of the data lies outside this range. Since the normal distribution is symmetric, half of this 32% (which is 16%) lies below one standard deviation below the mean. **Answer:** The proportion of scores less than 67 is approximately 16%. **Question b:** - \(85 = 73 + 2 \cdot 6\) 85 is two standard deviations above the mean. According
Transcribed Image Text:**Understanding Normal Distribution and the Empirical Rule** **Context:** Scores on a statistics final in a large class were normally distributed with a mean of 73 and a standard deviation of 6. Use the empirical rule to answer the following questions: **Questions:** a. Estimate the proportion of the scores less than 67. b. Estimate the proportion of the scores more than 85. c. Estimate the proportion of the scores between 61 and 79. **Explanation of Concepts:** The normal distribution is a probability distribution that is symmetric about the mean. The empirical rule, also known as the 68-95-99.7 rule, states that: - About 68% of the data falls within one standard deviation of the mean. - About 95% of the data falls within two standard deviations of the mean. - About 99.7% of the data falls within three standard deviations of the mean. In this context: - The mean (µ) is 73. - The standard deviation (σ) is 6. Each of the questions can be answered by applying the empirical rule to determine the proportion of scores within specified ranges of the normal distribution. **Graph/Diagram Explanation:** A typical graph for this problem would be a bell-shaped curve centered at 73 (the mean), with markers at one, two, and three standard deviations away from the mean in both directions (left and right). These markers would represent 67, 61, and 55 to the left of the mean, and 79, 85, and 91 to the right of the mean. Such a graphical representation helps in visualizing and estimating the proportions asked in the questions. **Detailed Breakdown of Questions:** **Question a:** - \(67 = 73 - 1 \cdot 6\) 67 is one standard deviation below the mean. According to the empirical rule, approximately 68% of the data lies between one standard deviation below and one standard deviation above the mean. This implies that about 32% of the data lies outside this range. Since the normal distribution is symmetric, half of this 32% (which is 16%) lies below one standard deviation below the mean. **Answer:** The proportion of scores less than 67 is approximately 16%. **Question b:** - \(85 = 73 + 2 \cdot 6\) 85 is two standard deviations above the mean. According
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