### Binomial Probability Estimation Using Normal Approximation**Problem Statement:**If $ np \geq 5 $ and $ nq \geq 5 $, estimate $ P(\text{fewer than 3}) $ with $ n = 14 $ 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:- **A.** $ P(\text{fewer than 3}) = $ [Answer Box] - (Round to four decimal places as needed.)- **B.** The normal approximation is not suitable.---**Explanation:**- **Step 1:** Calculate $ np $ and $ nq $ - $ np = 14 \cdot 0.4 = 5.6 $ - $ nq = 14 \cdot (1 - 0.4) = 14 \cdot 0.6 = 8.4 $ - **Step 2:** Check the conditions - Both $ np $ and $ nq $ are greater than or equal to 5, so the normal approximation is suitable. - **Step 3:** Use the normal approximation - The mean $ \mu = np = 5.6 $ - The standard deviation $ \sigma = \sqrt{npq} = \sqrt{14 \cdot 0.4 \cdot 0.6} \approx 1.833 $- **Step 4:** Convert the binomial variable to a standard normal variable - Using continuity correction, $ P(X < 3) $ converts to $ P(X < 2.5) $ - Standardize: $ Z = \frac{2.5 - 5.6}{1.833} \approx -1.691 $- **Step 5:** Find the corresponding probability on the standard normal distribution table - $ P(Z < -1.691) \approx 0.0455 $So, theoretically, the answer to choice **A** would be approximately $ 0.0455 \ ### Using Normal Distribution to Approximate Binomial Probabilities#### Problem Statement:Given: - \( n = 13 $- $ p = 0.5 $If $ np \geq 5 $ and $ nq \geq 5 $, estimate $ P(\text{at least 8}) $ using the normal distribution as an approximation to the binomial distribution. If $ np < 5 $ or $ nq < 5 $, state that the normal approximation is not suitable. Here, $ q = 1 - p = 0.5 $.#### Choices:1. **Choice A**: \[ P(\text{at least 8}) = \ \_\_\_\_ \] (Round to three decimal places as needed.)2. **Choice B**: \[ \text{The normal distribution cannot be used.} \]#### Notes:- To apply the normal approximation to the binomial distribution, calculate $ np $ and $ nq $. Check if both are ≥ 5.- If both conditions are met, use the normal distribution to approximate the binomial probability.- If either $ np $ or $ nq $ is less than 5, use choice B stating that the normal distribution is not suitable.

Identify the correct option to obtain the probability that fewer than 3. The correct option is…

If np 5 and nq s5, estimate P(fewer than 3) with n=D14 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. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. P(fewer than 3)= (Round to four decimal places as needed.) O B. The normal approximation is not suitable.

If np 5 and nq s5, estimate P(fewer than 3) with n=D14 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. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. P(fewer than 3)= (Round to four decimal places as needed.) O B. The normal approximation is not suitable.

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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Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.

Question

### Binomial Probability Estimation Using Normal Approximation

**Problem Statement:**

If $ np \geq 5 $ and $ nq \geq 5 $, estimate $ P(\text{fewer than 3}) $ with $ n = 14 $ 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:

- **A.** $ P(\text{fewer than 3}) = $ [Answer Box]
- (Round to four decimal places as needed.)

- **B.** The normal approximation is not suitable.

---

**Explanation:**

- **Step 1:** Calculate $ np $ and $ nq $
- $ np = 14 \cdot 0.4 = 5.6 $
- $ nq = 14 \cdot (1 - 0.4) = 14 \cdot 0.6 = 8.4 $

- **Step 2:** Check the conditions
- Both $ np $ and $ nq $ are greater than or equal to 5, so the normal approximation is suitable.

- **Step 3:** Use the normal approximation
- The mean $ \mu = np = 5.6 $
- The standard deviation $ \sigma = \sqrt{npq} = \sqrt{14 \cdot 0.4 \cdot 0.6} \approx 1.833 $

- **Step 4:** Convert the binomial variable to a standard normal variable
- Using continuity correction, $ P(X < 3) $ converts to $ P(X < 2.5) $
- Standardize: $ Z = \frac{2.5 - 5.6}{1.833} \approx -1.691 $

- **Step 5:** Find the corresponding probability on the standard normal distribution table
- $ P(Z < -1.691) \approx 0.0455 $

So, theoretically, the answer to choice **A** would be approximately \( 0.0455 \

### Using Normal Distribution to Approximate Binomial Probabilities

#### Problem Statement:

Given:
- $ n = 13 $
- $ p = 0.5 $

If $ np \geq 5 $ and $ nq \geq 5 $, estimate $ P(\text{at least 8}) $ using the normal distribution as an approximation to the binomial distribution. If $ np < 5 $ or $ nq < 5 $, state that the normal approximation is not suitable.

Here, $ q = 1 - p = 0.5 $.

#### Choices:

1. **Choice A**:
\[
P(\text{at least 8}) = \ \_\_\_\_
\]
(Round to three decimal places as needed.)

2. **Choice B**:
\[
\text{The normal distribution cannot be used.}
\]

#### Notes:
- To apply the normal approximation to the binomial distribution, calculate $ np $ and $ nq $. Check if both are ≥ 5.
- If both conditions are met, use the normal distribution to approximate the binomial probability.
- If either $ np $ or $ nq $ is less than 5, use choice B stating that the normal distribution is not suitable.

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