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If np 25 and nq 25, estimate P(fewer than 4) with n= 13 and p = 0.4 by using the normal distribution as an approximation to the binomial distribution; if np< 5 or ng<5, then state that the normal approximation is not suitable. %3D %3D Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. P(fewer than 4)= (Round to four decimal places as needed.) O B. The normal approximation is not suitable.

If np 25 and nq 25, estimate P(fewer than 4) with n= 13 and p = 0.4 by using the normal distribution as an approximation to the binomial distribution; if np< 5 or ng<5, then state that the normal approximation is not suitable. %3D %3D Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. P(fewer than 4)= (Round to four decimal places as needed.) O B. The normal approximation is not suitable.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 27T
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### Problem Statement: If \( np \geq 5 \) and \( nq \geq 5 \), estimate \( P(\text{fewer than 4}) \) with \( n = 13 \) and \( p = 0.4 \) by using the normal distribution as an approximation to the binomial distribution; if \( np < 5 \) or \( nq < 5 \), then state that the normal approximation is not suitable. ### Instructions: Select the correct choice below and, if necessary, fill in the answer box to complete your choice. #### Options: - **A.** \( P(\text{fewer than 4}) = \_\_\_\_ \) - Note: Round to four decimal places as needed. - **B.** The normal approximation is not suitable. --- ### Explanation of Concepts: - **Binomial Distribution:** A discrete probability distribution of the number of successes in a sequence of n independent experiments, where each experiment has a binary outcome (success/failure) with a constant probability of success, \( p \). - **Normal Approximation:** For a large number of trials \( n \), the binomial distribution \( B(n, p) \) can be approximated by a normal distribution \( N(np, \sqrt{npq}) \) if certain conditions, specifically \( np \geq 5 \) and \( nq \geq 5 \), are met. Here, \( q = 1 - p \). --- - **Given:** - \( n = 13 \) - \( p = 0.4 \) - **Calculate \( np \) and \( nq \):** - \( np = 13 \times 0.4 = 5.2 \) - \( nq = 13 \times 0.6 = 7.8 \) Since both conditions \( np \geq 5 \) and \( nq \geq 5 \) are satisfied, the normal approximation can be used. To find \( P(\text{fewer than 4}) \), convert the binomial variable to a normal variable and use the Z-score: - Mean (\( \mu \)) = \( np \) - Standard Deviation (\( \sigma \)) = \( \sqrt{npq} \) Calculate the Z-score and then find the corresponding probability from the standard
Transcribed Image Text:### Problem Statement: If \( np \geq 5 \) and \( nq \geq 5 \), estimate \( P(\text{fewer than 4}) \) with \( n = 13 \) and \( p = 0.4 \) by using the normal distribution as an approximation to the binomial distribution; if \( np < 5 \) or \( nq < 5 \), then state that the normal approximation is not suitable. ### Instructions: Select the correct choice below and, if necessary, fill in the answer box to complete your choice. #### Options: - **A.** \( P(\text{fewer than 4}) = \_\_\_\_ \) - Note: Round to four decimal places as needed. - **B.** The normal approximation is not suitable. --- ### Explanation of Concepts: - **Binomial Distribution:** A discrete probability distribution of the number of successes in a sequence of n independent experiments, where each experiment has a binary outcome (success/failure) with a constant probability of success, \( p \). - **Normal Approximation:** For a large number of trials \( n \), the binomial distribution \( B(n, p) \) can be approximated by a normal distribution \( N(np, \sqrt{npq}) \) if certain conditions, specifically \( np \geq 5 \) and \( nq \geq 5 \), are met. Here, \( q = 1 - p \). --- - **Given:** - \( n = 13 \) - \( p = 0.4 \) - **Calculate \( np \) and \( nq \):** - \( np = 13 \times 0.4 = 5.2 \) - \( nq = 13 \times 0.6 = 7.8 \) Since both conditions \( np \geq 5 \) and \( nq \geq 5 \) are satisfied, the normal approximation can be used. To find \( P(\text{fewer than 4}) \), convert the binomial variable to a normal variable and use the Z-score: - Mean (\( \mu \)) = \( np \) - Standard Deviation (\( \sigma \)) = \( \sqrt{npq} \) Calculate the Z-score and then find the corresponding probability from the standard
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