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A binomial experiment is given. Decide whether you can use the normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why. A survey of adults found that 46% have used a multivitamin in the past 12 months. You randomly select 40 adults and ask them if they have used a multivitamin in the past 12 months. Select the correct answer below and, if necessary, fill in the answer boxes within your choice. OA. No, because np < 5. OB. No, because nq < 5. OC. Yes, the mean is and the standard deviation is (Round to two decimal places as needed.) C

A binomial experiment is given. Decide whether you can use the normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why. A survey of adults found that 46% have used a multivitamin in the past 12 months. You randomly select 40 adults and ask them if they have used a multivitamin in the past 12 months. Select the correct answer below and, if necessary, fill in the answer boxes within your choice. OA. No, because np < 5. OB. No, because nq < 5. OC. Yes, the mean is and the standard deviation is (Round to two decimal places as needed.) C

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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### Approximating a Binomial Distribution Using Normal Distribution **Scenario:** A binomial experiment is described where a survey of adults reveals that 46% have used a multivitamin in the past 12 months. A random selection of 40 adults is then made, and they are asked if they have used a multivitamin in the past 12 months. **Task:** Decide whether the normal distribution can be used to approximate the binomial distribution. If appropriate, find the mean and standard deviation. If not, explain why. 1. **Compute \( np \):** - \( n = 40 \) (number of trials) - \( p = 0.46 \) (probability of success) - \( np = 40 \times 0.46 = 18.4 \) 2. **Compute \( nq \):** - \( q = 1 - p = 1 - 0.46 = 0.54 \) - \( nq = 40 \times 0.54 = 21.6 \) Both values \( np \) and \( nq \) should be at least 5 to use the normal approximation to the binomial distribution. **Possible Answers:** A. **No, because \( np < 5 \).** > This is incorrect because \( np = 18.4 \) which is greater than 5. B. **No, because \( nq < 5 \).** > This is incorrect because \( nq = 21.6 \) which is greater than 5. C. **Yes, the mean is \( \mu \) and the standard deviation is \( \sigma \).** - **Mean (\(\mu\)):** - Find \( \mu \): - \( \mu = np = 18.4 \) - **Standard Deviation (\(\sigma\)):** - Find \( \sigma \): - \( \sigma = \sqrt{npq} = \sqrt{40 \times 0.46 \times 0.54} = \sqrt{9.936} \approx 3.15 \) - **Fill in the values:** \[ \text{Mean} = 18.4 \quad \text{and} \quad \text{Standard Deviation} = 3.15 \quad (\text{rounded to
Transcribed Image Text:### Approximating a Binomial Distribution Using Normal Distribution **Scenario:** A binomial experiment is described where a survey of adults reveals that 46% have used a multivitamin in the past 12 months. A random selection of 40 adults is then made, and they are asked if they have used a multivitamin in the past 12 months. **Task:** Decide whether the normal distribution can be used to approximate the binomial distribution. If appropriate, find the mean and standard deviation. If not, explain why. 1. **Compute \( np \):** - \( n = 40 \) (number of trials) - \( p = 0.46 \) (probability of success) - \( np = 40 \times 0.46 = 18.4 \) 2. **Compute \( nq \):** - \( q = 1 - p = 1 - 0.46 = 0.54 \) - \( nq = 40 \times 0.54 = 21.6 \) Both values \( np \) and \( nq \) should be at least 5 to use the normal approximation to the binomial distribution. **Possible Answers:** A. **No, because \( np < 5 \).** > This is incorrect because \( np = 18.4 \) which is greater than 5. B. **No, because \( nq < 5 \).** > This is incorrect because \( nq = 21.6 \) which is greater than 5. C. **Yes, the mean is \( \mu \) and the standard deviation is \( \sigma \).** - **Mean (\(\mu\)):** - Find \( \mu \): - \( \mu = np = 18.4 \) - **Standard Deviation (\(\sigma\)):** - Find \( \sigma \): - \( \sigma = \sqrt{npq} = \sqrt{40 \times 0.46 \times 0.54} = \sqrt{9.936} \approx 3.15 \) - **Fill in the values:** \[ \text{Mean} = 18.4 \quad \text{and} \quad \text{Standard Deviation} = 3.15 \quad (\text{rounded to
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