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6. Area under the normal distribution The following figure shows the normal distribution with the proportion of the area under the normal curve contained within one, two, and three standard deviations of the mean. The last proportion on each side, 0.13%, depicts the remaining area under the curve. Specifically, 0.13% of the area under the standard normal distribution is located above z-score values greater than the mean (p) plus three standard deviations (+3o). Also, because the normal distribution is symmetrical, 0.13% of the area under the standard normal distribution is located below z-score values less than the mean (H) minus three standard deviations (-30).

6. Area under the normal distribution The following figure shows the normal distribution with the proportion of the area under the normal curve contained within one, two, and three standard deviations of the mean. The last proportion on each side, 0.13%, depicts the remaining area under the curve. Specifically, 0.13% of the area under the standard normal distribution is located above z-score values greater than the mean (p) plus three standard deviations (+3o). Also, because the normal distribution is symmetrical, 0.13% of the area under the standard normal distribution is located below z-score values less than the mean (H) minus three standard deviations (-30).

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
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### 6. Area Under the Normal Distribution The following figure shows the normal distribution with the proportion of the area under the normal curve contained within one, two, and three standard deviations of the mean. The last proportion on each side, 0.13%, depicts the remaining area under the curve. Specifically, 0.13% of the area under the standard normal distribution is located above z-score values greater than the mean (μ) plus three standard deviations (+3σ). Also, because the normal distribution is symmetrical, 0.13% of the area under the standard normal distribution is located below z-score values less than the mean (μ) minus three standard deviations (−3σ). #### Diagram Explanation The diagram is a bell-shaped curve representing a normal distribution. Key details include: - **Mean (μ):** Located at the center of the distribution. - **Standard Deviations:** - From μ to +1σ and -1σ, each containing 34.13% of the data. - From +1σ to +2σ and -1σ to -2σ, each containing 13.59%. - From +2σ to +3σ and -2σ to -3σ, each containing 2.15%. - Beyond ±3σ, each tail contains 0.13%, depicting the extreme ends of the distribution. Use the figure to help you answer the following questions. The National Assessment of Educational Progress (NAEP) is a nationwide assessment of students' proficiency in nine subjects: mathematics, reading, writing, science, the arts, civics, economics, geography, and US history. The main NAEP assessments are conducted annually on samples of students from grades 4, 8, and 12. In 2002, the reading scores for female students had a mean of 269 with a standard deviation of 33. Assume that these scores are normally distributed with the given mean and standard deviation. A score of 170 is **three standard deviations** below the mean, while a score of 368 is **three standard deviations** above the mean. This illustration explains how scores are distributed around the mean in a normal distribution.
Transcribed Image Text:### 6. Area Under the Normal Distribution The following figure shows the normal distribution with the proportion of the area under the normal curve contained within one, two, and three standard deviations of the mean. The last proportion on each side, 0.13%, depicts the remaining area under the curve. Specifically, 0.13% of the area under the standard normal distribution is located above z-score values greater than the mean (μ) plus three standard deviations (+3σ). Also, because the normal distribution is symmetrical, 0.13% of the area under the standard normal distribution is located below z-score values less than the mean (μ) minus three standard deviations (−3σ). #### Diagram Explanation The diagram is a bell-shaped curve representing a normal distribution. Key details include: - **Mean (μ):** Located at the center of the distribution. - **Standard Deviations:** - From μ to +1σ and -1σ, each containing 34.13% of the data. - From +1σ to +2σ and -1σ to -2σ, each containing 13.59%. - From +2σ to +3σ and -2σ to -3σ, each containing 2.15%. - Beyond ±3σ, each tail contains 0.13%, depicting the extreme ends of the distribution. Use the figure to help you answer the following questions. The National Assessment of Educational Progress (NAEP) is a nationwide assessment of students' proficiency in nine subjects: mathematics, reading, writing, science, the arts, civics, economics, geography, and US history. The main NAEP assessments are conducted annually on samples of students from grades 4, 8, and 12. In 2002, the reading scores for female students had a mean of 269 with a standard deviation of 33. Assume that these scores are normally distributed with the given mean and standard deviation. A score of 170 is **three standard deviations** below the mean, while a score of 368 is **three standard deviations** above the mean. This illustration explains how scores are distributed around the mean in a normal distribution.
**Transcription for Educational Website:** --- **Figure Overview:** The figure presents a normal distribution curve with specific percentages highlighted at various standard deviations from the mean (µ): - At ±1σ, the percentage is 13.59% for each side. - At ±2σ, it is 2.15% for each side. - At ±3σ, it is 0.13% for each side. **Explanation and Context:** The National Assessment of Educational Progress (NAEP) is a nationwide assessment evaluating students' proficiency in nine subjects: mathematics, reading, writing, science, the arts, civics, economics, geography, and US history. The main NAEP assessments are conducted annually on samples of students from grades 4, 8, and 12. In 2002, the reading scores for female students had a mean of 269 with a standard deviation of 33. Assume these scores are normally distributed with the given mean and standard deviation. - A score of 170 is **three standard deviations** below the mean, while a score of 368 is **three standard deviations** above the mean. This means that the percentage of female students with scores between 170 and 368 is **99.74%**. - A score of 302 is *________* above the mean. As a result, the percentage of female students with scores below 302 is *________*. From this data, you can infer that 97.72% of female students have scores above *________*. ---
Transcribed Image Text:**Transcription for Educational Website:** --- **Figure Overview:** The figure presents a normal distribution curve with specific percentages highlighted at various standard deviations from the mean (µ): - At ±1σ, the percentage is 13.59% for each side. - At ±2σ, it is 2.15% for each side. - At ±3σ, it is 0.13% for each side. **Explanation and Context:** The National Assessment of Educational Progress (NAEP) is a nationwide assessment evaluating students' proficiency in nine subjects: mathematics, reading, writing, science, the arts, civics, economics, geography, and US history. The main NAEP assessments are conducted annually on samples of students from grades 4, 8, and 12. In 2002, the reading scores for female students had a mean of 269 with a standard deviation of 33. Assume these scores are normally distributed with the given mean and standard deviation. - A score of 170 is **three standard deviations** below the mean, while a score of 368 is **three standard deviations** above the mean. This means that the percentage of female students with scores between 170 and 368 is **99.74%**. - A score of 302 is *________* above the mean. As a result, the percentage of female students with scores below 302 is *________*. From this data, you can infer that 97.72% of female students have scores above *________*. ---
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