Suppose that 53% of the population favors a particular candidate IP a Random sample of 55 voters is chosen, approximate the probability that at least 29 Pavor a particular Candidate. Use the normal approximation to the binomial with a correction. For Continuity,
Suppose that 53% of the population favors a particular candidate IP a Random sample of 55 voters is chosen, approximate the probability that at least 29 Pavor a particular Candidate. Use the normal approximation to the binomial with a correction. For Continuity,
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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![### Probability Approximation Using Normal Distribution
**Problem Statement:**
Suppose that 53% of the population favors a particular candidate. If a random sample of 55 voters is chosen, approximate the probability that at least 29 favor a particular candidate. Use the normal approximation to the binomial with a correction for continuity.
**Solution Explanation:**
To solve this problem, follow these steps:
1. **Identify the Parameters:**
- Population proportion (p) = 0.53
- Sample size (n) = 55
2. **Calculate the Mean (μ) and Standard Deviation (σ) of the Binomial Distribution:**
- Mean (μ) = np = 55 * 0.53 = 29.15
- Standard Deviation (σ) = √(np(1-p)) = √(55 * 0.53 * 0.47) ≈ 3.72
3. **Apply the Normal Approximation:**
- Use the correction for continuity by adjusting the count to at least 28.5 for approximating 29.
4. **Convert to a Standard Normal Distribution (Z-score):**
- \( Z = \frac{X - \mu}{\sigma} \)
- For \(X = 28.5\):
\( Z = \frac{28.5 - 29.15}{3.72} \approx -0.17 \)
5. **Find the Probability Using Z-table:**
- Determine the probability that Z is less than -0.17.
- Using Z-tables or a statistical calculator:
\( P(Z < -0.17) ≈ 0.4325 \)
Since we want to find the probability of at least 29 voters favoring the candidate, we need:
\[ P(X ≥ 29) = 1 - P(X < 29) \]
\[ P(X ≥ 29) = 1 - P(Z < -0.17) \]
\[ P(X ≥ 29) ≈ 1 - 0.4325 ≈ 0.5675 \]
**Conclusion:**
The approximate probability that at least 29 out of 55 voters favor the particular candidate is about 0.5675 or 56.75%.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2561c632-65dd-4d60-9cc7-c4695e3617a7%2F33533a44-55c9-40f5-a7b5-5b91d65686a8%2Fkuif9cu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Probability Approximation Using Normal Distribution
**Problem Statement:**
Suppose that 53% of the population favors a particular candidate. If a random sample of 55 voters is chosen, approximate the probability that at least 29 favor a particular candidate. Use the normal approximation to the binomial with a correction for continuity.
**Solution Explanation:**
To solve this problem, follow these steps:
1. **Identify the Parameters:**
- Population proportion (p) = 0.53
- Sample size (n) = 55
2. **Calculate the Mean (μ) and Standard Deviation (σ) of the Binomial Distribution:**
- Mean (μ) = np = 55 * 0.53 = 29.15
- Standard Deviation (σ) = √(np(1-p)) = √(55 * 0.53 * 0.47) ≈ 3.72
3. **Apply the Normal Approximation:**
- Use the correction for continuity by adjusting the count to at least 28.5 for approximating 29.
4. **Convert to a Standard Normal Distribution (Z-score):**
- \( Z = \frac{X - \mu}{\sigma} \)
- For \(X = 28.5\):
\( Z = \frac{28.5 - 29.15}{3.72} \approx -0.17 \)
5. **Find the Probability Using Z-table:**
- Determine the probability that Z is less than -0.17.
- Using Z-tables or a statistical calculator:
\( P(Z < -0.17) ≈ 0.4325 \)
Since we want to find the probability of at least 29 voters favoring the candidate, we need:
\[ P(X ≥ 29) = 1 - P(X < 29) \]
\[ P(X ≥ 29) = 1 - P(Z < -0.17) \]
\[ P(X ≥ 29) ≈ 1 - 0.4325 ≈ 0.5675 \]
**Conclusion:**
The approximate probability that at least 29 out of 55 voters favor the particular candidate is about 0.5675 or 56.75%.
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