MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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**Normal Distribution Problems** 1. **Problem 10:** - **Given:** - Mean (μ) = 26 - Standard Deviation (σ) = 2.3 - **Task:** - Find the probability of a score above 29.6. 2. **Problem 11:** - **Given:** - Mean (μ) = 100 - Standard Deviation (σ) = 25 - **Task:** - Find the score that separates the top 80% from the bottom 20%. **Explanation for Educational Website:** 1. **Understanding Problem 10:** To find the probability of a score above a certain value in a normal distribution, we use the Z-score formula which is given by: \[ Z = \frac{X - \mu}{\sigma} \] where \( X \) is the value for which we are finding the probability, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. - Calculate the Z-score for X = 29.6: \[ Z = \frac{29.6 - 26}{2.3} \approx 1.57 \] - Use the Z-table or a calculator to find the area to the right of Z = 1.57, which gives you the probability of a score above 29.6. 2. **Understanding Problem 11:** To find the score that separates the top 80% from the bottom 20%, we need to find the corresponding Z-score for the 20th percentile. - The Z-score for the 20th percentile is approximately -0.84 (from Z-tables or standard normal distribution tables). - Use the Z-score formula in reverse to find X: \[ X = \mu + (Z \times \sigma) \] \[ X = 100 + (-0.84 \times 25) \approx 79 \] By solving such problems, students can gain a better understanding of how the properties of the normal distribution apply in real-world scenarios, enhancing their statistical analysis skills.
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Transcribed Image Text:**Normal Distribution Problems** 1. **Problem 10:** - **Given:** - Mean (μ) = 26 - Standard Deviation (σ) = 2.3 - **Task:** - Find the probability of a score above 29.6. 2. **Problem 11:** - **Given:** - Mean (μ) = 100 - Standard Deviation (σ) = 25 - **Task:** - Find the score that separates the top 80% from the bottom 20%. **Explanation for Educational Website:** 1. **Understanding Problem 10:** To find the probability of a score above a certain value in a normal distribution, we use the Z-score formula which is given by: \[ Z = \frac{X - \mu}{\sigma} \] where \( X \) is the value for which we are finding the probability, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. - Calculate the Z-score for X = 29.6: \[ Z = \frac{29.6 - 26}{2.3} \approx 1.57 \] - Use the Z-table or a calculator to find the area to the right of Z = 1.57, which gives you the probability of a score above 29.6. 2. **Understanding Problem 11:** To find the score that separates the top 80% from the bottom 20%, we need to find the corresponding Z-score for the 20th percentile. - The Z-score for the 20th percentile is approximately -0.84 (from Z-tables or standard normal distribution tables). - Use the Z-score formula in reverse to find X: \[ X = \mu + (Z \times \sigma) \] \[ X = 100 + (-0.84 \times 25) \approx 79 \] By solving such problems, students can gain a better understanding of how the properties of the normal distribution apply in real-world scenarios, enhancing their statistical analysis skills.
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