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A set of test scores is normally distributed with a mean of 95 and a standard deviation of 20. Use the 68-95-99.7 rule to find the percentage of scores in each of the following categories. a. Greater than 95 b. Greater than 115 c. Less than 75 d. Less than 135

A set of test scores is normally distributed with a mean of 95 and a standard deviation of 20. Use the 68-95-99.7 rule to find the percentage of scores in each of the following categories. a. Greater than 95 b. Greater than 115 c. Less than 75 d. Less than 135

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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14. find the percentage of scores in each of the following categories. 

### Understanding Normal Distribution in Test Scores A set of test scores is normally distributed with a mean (μ) of 95 and a standard deviation (σ) of 20. Using the 68-95-99.7 rule, also known as the Empirical Rule, we can determine the percentage of scores in various categories: #### The 68-95-99.7 Rule: 1. 68% of the data falls within ±1 standard deviation from the mean 2. 95% of the data falls within ±2 standard deviations from the mean 3. 99.7% of the data falls within ±3 standard deviations from the mean ### Categories and Calculation: a. **Greater than 95** - This is the mean, so 50% of the scores will be greater than 95. b. **Greater than 115** - 115 is one standard deviation above the mean (95 + 20). According to the rule, 68% of the scores fall within ±1σ, so 34% will be above the mean (95 to 115). - The remaining 50% from above the mean split 34% within the first standard deviation and 16% more (50% - 34%) beyond it. - Therefore, 16% of the scores are greater than 115. c. **Less than 75** - 75 is one standard deviation below the mean (95 - 20). From the rule, 68% of scores are within ±1σ, so 34% are between 75 and 95. - Subtracting from 50% below the mean gives 16% below 75 (50% - 34%). d. **Less than 135** - 135 is two standard deviations above the mean (95 + 2*20). From the rule, 95% of scores are within ±2σ, so only 2.5% are beyond each of the tails. - Therefore, 97.5% (100% - 2.5%) of scores are less than 135. e. **Less than 55** - 55 is two standard deviations below the mean (95 - 2*20). Similar to category d, 2.5% are beyond the lower tail. - Therefore, 2.5% of scores are less than 55. f. **
Transcribed Image Text:### Understanding Normal Distribution in Test Scores A set of test scores is normally distributed with a mean (μ) of 95 and a standard deviation (σ) of 20. Using the 68-95-99.7 rule, also known as the Empirical Rule, we can determine the percentage of scores in various categories: #### The 68-95-99.7 Rule: 1. 68% of the data falls within ±1 standard deviation from the mean 2. 95% of the data falls within ±2 standard deviations from the mean 3. 99.7% of the data falls within ±3 standard deviations from the mean ### Categories and Calculation: a. **Greater than 95** - This is the mean, so 50% of the scores will be greater than 95. b. **Greater than 115** - 115 is one standard deviation above the mean (95 + 20). According to the rule, 68% of the scores fall within ±1σ, so 34% will be above the mean (95 to 115). - The remaining 50% from above the mean split 34% within the first standard deviation and 16% more (50% - 34%) beyond it. - Therefore, 16% of the scores are greater than 115. c. **Less than 75** - 75 is one standard deviation below the mean (95 - 20). From the rule, 68% of scores are within ±1σ, so 34% are between 75 and 95. - Subtracting from 50% below the mean gives 16% below 75 (50% - 34%). d. **Less than 135** - 135 is two standard deviations above the mean (95 + 2*20). From the rule, 95% of scores are within ±2σ, so only 2.5% are beyond each of the tails. - Therefore, 97.5% (100% - 2.5%) of scores are less than 135. e. **Less than 55** - 55 is two standard deviations below the mean (95 - 2*20). Similar to category d, 2.5% are beyond the lower tail. - Therefore, 2.5% of scores are less than 55. f. **
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