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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Question
Need help with 8 d and e please
**Normal Distribution Problem Set** Consider \( x \) as a normally distributed continuous random variable with a mean (\( \mu \)) of 32 and a standard deviation (\( \sigma \)) of 6. Evaluate the following probabilities and questions: a. **What is the probability that \( x \) is between 23 and 30.5?** - Calculation: \( z = \frac{23-32}{6} \) and \( z = \frac{30.5-32}{6} \) - **Result:** 0.0318 - **Graph Explanation:** This is represented by a shaded area on a normal distribution curve between 23 and 30.5. b. **What is the probability that \( x \) is between 17 and 32?** - Calculation: \( z = \frac{17-32}{6} \) and \( z = \frac{32-32}{6} \) - **Result:** 0.49379 - **Graph Explanation:** The shaded area on the normal distribution curve represents this probability range, between 17 and 32. c. **What percentage of the population will be greater than 35?** - **Result:** 30.9% - **Graph Explanation:** The shaded area above \( x = 35 \) shows this portion of the population in the normal distribution curve. d. **What is the value of \( x \) where approximately 45% of the population lies less than \( x \)?** - **Calculation:** \( x = \mu + z\sigma \) where \( z \) for 45th percentile is found - **Graph Explanation:** The shaded area under the curve represents the 45th percentile mark. e. **What is the value of \( x \) where approximately 35% of the population lies less than \( x \)?** - **Graph Explanation:** This shows the point on the curve where the area to the left is 35%. These exercises help to understand the properties of a normal distribution curve and the calculation of probabilities for given ranges.
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Transcribed Image Text:**Normal Distribution Problem Set** Consider \( x \) as a normally distributed continuous random variable with a mean (\( \mu \)) of 32 and a standard deviation (\( \sigma \)) of 6. Evaluate the following probabilities and questions: a. **What is the probability that \( x \) is between 23 and 30.5?** - Calculation: \( z = \frac{23-32}{6} \) and \( z = \frac{30.5-32}{6} \) - **Result:** 0.0318 - **Graph Explanation:** This is represented by a shaded area on a normal distribution curve between 23 and 30.5. b. **What is the probability that \( x \) is between 17 and 32?** - Calculation: \( z = \frac{17-32}{6} \) and \( z = \frac{32-32}{6} \) - **Result:** 0.49379 - **Graph Explanation:** The shaded area on the normal distribution curve represents this probability range, between 17 and 32. c. **What percentage of the population will be greater than 35?** - **Result:** 30.9% - **Graph Explanation:** The shaded area above \( x = 35 \) shows this portion of the population in the normal distribution curve. d. **What is the value of \( x \) where approximately 45% of the population lies less than \( x \)?** - **Calculation:** \( x = \mu + z\sigma \) where \( z \) for 45th percentile is found - **Graph Explanation:** The shaded area under the curve represents the 45th percentile mark. e. **What is the value of \( x \) where approximately 35% of the population lies less than \( x \)?** - **Graph Explanation:** This shows the point on the curve where the area to the left is 35%. These exercises help to understand the properties of a normal distribution curve and the calculation of probabilities for given ranges.
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