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2. The average charitable contribution itemized per income tax return in Pennsylvania is $792. Suppose that the distribution of contributions is normal with a standard deviation of $103. Find the limits for the middle 50% of contributions.

2. The average charitable contribution itemized per income tax return in Pennsylvania is $792. Suppose that the distribution of contributions is normal with a standard deviation of $103. Find the limits for the middle 50% of contributions.

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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### Problem 2: Charitable Contributions in Pennsylvania **Problem Description:** The average charitable contribution itemized per income tax return in Pennsylvania is $792. Suppose that the distribution of contributions is normal with a standard deviation of $103. Find the limits for the middle 50% of contributions. **Solution Outline:** 1. **Identify the Mean and Standard Deviation:** - Mean ($\mu$): $792 - Standard Deviation ($\sigma$): $103 2. **Understanding the Middle 50%:** - The middle 50% of a normal distribution is centered around the mean, covering from the 25th percentile to the 75th percentile. 3. **Find the Z-scores for the 25th and 75th Percentiles:** - The Z-score for the 25th percentile is approximately -0.675. - The Z-score for the 75th percentile is approximately +0.675. 4. **Convert Z-scores to Contribution Values:** - Use the Z-score formula: \(X = \mu + Z\sigma\) - For 25th percentile: \(X_{25} = 792 + (-0.675)(103) \approx 723.45\) - For 75th percentile: \(X_{75} = 792 + (0.675)(103) \approx 860.55\) **Conclusion:** The limits for the middle 50% of charitable contributions itemized per income tax return in Pennsylvania are approximately $723.45 and $860.55.
Transcribed Image Text:### Problem 2: Charitable Contributions in Pennsylvania **Problem Description:** The average charitable contribution itemized per income tax return in Pennsylvania is $792. Suppose that the distribution of contributions is normal with a standard deviation of $103. Find the limits for the middle 50% of contributions. **Solution Outline:** 1. **Identify the Mean and Standard Deviation:** - Mean ($\mu$): $792 - Standard Deviation ($\sigma$): $103 2. **Understanding the Middle 50%:** - The middle 50% of a normal distribution is centered around the mean, covering from the 25th percentile to the 75th percentile. 3. **Find the Z-scores for the 25th and 75th Percentiles:** - The Z-score for the 25th percentile is approximately -0.675. - The Z-score for the 75th percentile is approximately +0.675. 4. **Convert Z-scores to Contribution Values:** - Use the Z-score formula: \(X = \mu + Z\sigma\) - For 25th percentile: \(X_{25} = 792 + (-0.675)(103) \approx 723.45\) - For 75th percentile: \(X_{75} = 792 + (0.675)(103) \approx 860.55\) **Conclusion:** The limits for the middle 50% of charitable contributions itemized per income tax return in Pennsylvania are approximately $723.45 and $860.55.
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