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A quick approximation is sometimes useful when an exact answer is not required. (a) Given any normally distributed random variable, x, with mean, u, and standard deviation, a, use the Empirical Rule to estimate the following probabilities. (Hint: Don't forget that probabilities should be reported as numbers between 0 and 1.) P(x 2 u + 30) = P(x 2 H- 20) - P(u - o

A quick approximation is sometimes useful when an exact answer is not required. (a) Given any normally distributed random variable, x, with mean, u, and standard deviation, a, use the Empirical Rule to estimate the following probabilities. (Hint: Don't forget that probabilities should be reported as numbers between 0 and 1.) P(x 2 u + 30) = P(x 2 H- 20) - P(u - o

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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**Title: Using the Empirical Rule for Quick Probability Approximations** A quick approximation is sometimes useful when an exact answer is not required. ### Task: 1. **Probability Estimation with the Empirical Rule:** - Given any normally distributed random variable, \( x \), with mean, \( \mu \), and standard deviation, \( \sigma \), use the Empirical Rule to estimate the following probabilities. - **Note:** Probabilities should be reported as numbers between 0 and 1. - \( P(x < \mu + \sigma) = \) - \( P(x < \mu + 2\sigma) = \) - \( P(\mu - 2\sigma < x < \mu + 2\sigma) = \) - \( P(\mu - 3\sigma < x < \mu + 3\sigma) = \) 2. **Specific Calculation Using Given Values:** - Given that \( x \) has a normal probability density function with \( \mu = 5.5 \) and \( \sigma = 8.374 \), use the Empirical Rule to estimate the probability: \( P(-11.248 < x < 13.874) = \) 3. **Use the normal probability density function to calculate the probability in part (b).** - **Instruction:** Round your answer to four decimal places. - [Input Box for Answer] - Submit Answer **Navigation:** - View Previous Question - View Next Question - Home - My Assignments - Request Extension **Footer:** - Copyright © 1998 - 2021 Cengage Learning, Inc. All Rights Reserved - TERMS OF USE - PRIVACY *Note: Ensure you understand the Empirical Rule, which suggests that for a normal distribution: - Approximately 68% of the data falls within one standard deviation (\( \sigma \)) - Approximately 95% within two standard deviations - Approximately 99.7% within three standard deviations*
Transcribed Image Text:**Title: Using the Empirical Rule for Quick Probability Approximations** A quick approximation is sometimes useful when an exact answer is not required. ### Task: 1. **Probability Estimation with the Empirical Rule:** - Given any normally distributed random variable, \( x \), with mean, \( \mu \), and standard deviation, \( \sigma \), use the Empirical Rule to estimate the following probabilities. - **Note:** Probabilities should be reported as numbers between 0 and 1. - \( P(x < \mu + \sigma) = \) - \( P(x < \mu + 2\sigma) = \) - \( P(\mu - 2\sigma < x < \mu + 2\sigma) = \) - \( P(\mu - 3\sigma < x < \mu + 3\sigma) = \) 2. **Specific Calculation Using Given Values:** - Given that \( x \) has a normal probability density function with \( \mu = 5.5 \) and \( \sigma = 8.374 \), use the Empirical Rule to estimate the probability: \( P(-11.248 < x < 13.874) = \) 3. **Use the normal probability density function to calculate the probability in part (b).** - **Instruction:** Round your answer to four decimal places. - [Input Box for Answer] - Submit Answer **Navigation:** - View Previous Question - View Next Question - Home - My Assignments - Request Extension **Footer:** - Copyright © 1998 - 2021 Cengage Learning, Inc. All Rights Reserved - TERMS OF USE - PRIVACY *Note: Ensure you understand the Empirical Rule, which suggests that for a normal distribution: - Approximately 68% of the data falls within one standard deviation (\( \sigma \)) - Approximately 95% within two standard deviations - Approximately 99.7% within three standard deviations*
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