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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,

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.4: Distributions Of Data
Problem 22PFA
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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%.
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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