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Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities. Identify any unusual events. Explain. In a survey of U.S. college students, 62% said that if their college offers new programs tailored to the new economy, it will make them more likely to re-enroll. You randomly select 41 college students. Find the probability that the number who are more likely to enroll if their college offers new programs tailored to the new economy is (a) exactly 25, (b) at least 30, and (c) at most 24.

Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities. Identify any unusual events. Explain. In a survey of U.S. college students, 62% said that if their college offers new programs tailored to the new economy, it will make them more likely to re-enroll. You randomly select 41 college students. Find the probability that the number who are more likely to enroll if their college offers new programs tailored to the new economy is (a) exactly 25, (b) at least 30, and (c) at most 24.

Big Ideas Math A Bridge To Success Algebra 1: Student Edition 2015
Big Ideas Math A Bridge To Success Algebra 1: Student Edition 2015
1st Edition
ISBN:9781680331141
Author:HOUGHTON MIFFLIN HARCOURT
Publisher:HOUGHTON MIFFLIN HARCOURT
Chapter11: Data Analysis And Displays
Section11.3: Shapes Of Distributions
Problem 8Q
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**Title: Understanding the Normal Approximation to the Binomial Distribution** **Description:** This lesson will explore whether the normal distribution can approximate the binomial distribution in specific scenarios. We will analyze given data, perform necessary calculations, and discuss how to sketch applicable graphs. The process involves determining the circumstances under which normal approximation is suitable, and when it is not, we employ the binomial distribution to calculate required probabilities. ### Problem Statement **Scenario:** In a survey of U.S. college students, 62% said that if their college offers new programs tailored to the new economy, it will make them more likely to re-enroll. You randomly select 41 college students. **Tasks:** 1. Determine the probability that the number who are more likely to enroll if their college offers new programs tailored to the new economy is: - **Exactly 25** - **At least 30** - **At most 24** ### Steps to Determine the Utilization of Normal Distribution: 1. **Check the Conditions:** - The normal distribution can be used to approximate the binomial distribution when both \( np \) and \( n(1-p) \) are greater than or equal to 5. Here: - \( n = 41 \) - \( p = 0.62 \) Calculate: - \( np = 41 \times 0.62 = 25.42 \) - \( n(1-p) = 41 \times 0.38 = 15.58 \) Both values are greater than 5, so we can use the normal approximation. 2. **Normal Approximation:** - Mean (\( \mu \)) of the binomial distribution: \( \mu = np = 25.42 \) - Standard deviation (\( \sigma \)) of the binomial distribution: \( \sigma = \sqrt{np(1-p)} = \sqrt{25.42 \times 0.38} = \sqrt{9.6596} \approx 3.11 \) 3. **Convert to Z-Scores and Find Probabilities:** - For **exactly 25**: - Use continuity correction: \( P(24.5 < X < 25.5) \) - Convert to Z-scores: \( Z = \frac{24.5 - 25.
Transcribed Image Text:**Title: Understanding the Normal Approximation to the Binomial Distribution** **Description:** This lesson will explore whether the normal distribution can approximate the binomial distribution in specific scenarios. We will analyze given data, perform necessary calculations, and discuss how to sketch applicable graphs. The process involves determining the circumstances under which normal approximation is suitable, and when it is not, we employ the binomial distribution to calculate required probabilities. ### Problem Statement **Scenario:** In a survey of U.S. college students, 62% said that if their college offers new programs tailored to the new economy, it will make them more likely to re-enroll. You randomly select 41 college students. **Tasks:** 1. Determine the probability that the number who are more likely to enroll if their college offers new programs tailored to the new economy is: - **Exactly 25** - **At least 30** - **At most 24** ### Steps to Determine the Utilization of Normal Distribution: 1. **Check the Conditions:** - The normal distribution can be used to approximate the binomial distribution when both \( np \) and \( n(1-p) \) are greater than or equal to 5. Here: - \( n = 41 \) - \( p = 0.62 \) Calculate: - \( np = 41 \times 0.62 = 25.42 \) - \( n(1-p) = 41 \times 0.38 = 15.58 \) Both values are greater than 5, so we can use the normal approximation. 2. **Normal Approximation:** - Mean (\( \mu \)) of the binomial distribution: \( \mu = np = 25.42 \) - Standard deviation (\( \sigma \)) of the binomial distribution: \( \sigma = \sqrt{np(1-p)} = \sqrt{25.42 \times 0.38} = \sqrt{9.6596} \approx 3.11 \) 3. **Convert to Z-Scores and Find Probabilities:** - For **exactly 25**: - Use continuity correction: \( P(24.5 < X < 25.5) \) - Convert to Z-scores: \( Z = \frac{24.5 - 25.
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