Questions: If n p ≥ 5 and n q ≥ 5, estimate P( fewer than 4 ) with n=14 and p=0.4 by using the normal distribution as an approximation to the binomial distribution; if n p<5 or n q<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. A. P( fewer than 4)= (Round to four decimal places as needed) B. The normal approximation is not suitable.

If n p ≥ 5 and n q ≥ 5, estimate P( fewer than 4 ) with n=14 and p=0.4 by using the normal distribution as an approximation to the binomial distribution; if n p<5 or n q<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. A. P( fewer than 4)= (Round to four decimal places as needed) B. The normal approximation is not suitable.

Transcript text: If $n p \geq 5$ and $n q \geqslant 5$, estimate $P($ fewer than 4 ) with $n=14$ and $p=0.4$ by using the normal distribution as an approximation to the binomial distribution; if $n p<5$ or $n q<5$, 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. A. $P($ fewer than 4$)=$ $\square$ (Round to four decimal places as needed) B. The normal approximation is not suitable.

Solution

Solution Steps

To solve this problem, we need to determine if the normal approximation to the binomial distribution is suitable. This is done by checking if both $ n \times p $ and $ n \times q $ are greater than or equal to 5, where $ q = 1 - p $. If the conditions are met, we can use the normal distribution to approximate the probability of getting fewer than 4 successes. We calculate the mean and standard deviation of the binomial distribution, then use these to find the z-score for 3.5 (continuity correction for "fewer than 4") and finally use the z-score to find the probability from the standard normal distribution.

Step 1: Check Normal Approximation Suitability

To determine if the normal approximation to the binomial distribution is suitable, we calculate $ n \times p $ and $ n \times q $: \[ n = 14, \quad p = 0.4, \quad q = 1 - p = 0.6 \] Calculating: \[ n \times p = 14 \times 0.4 = 5.6 \quad \text{and} \quad n \times q = 14 \times 0.6 = 8.4 \] Since both $ n \times p \geq 5 $ and $ n \times q \geq 5 $, the normal approximation is suitable.

Step 2: Calculate Mean and Standard Deviation

The mean $ \mu $ and standard deviation $ \sigma $ of the binomial distribution are given by: \[ \mu = n \times p = 5.6, \quad \sigma = \sqrt{n \times p \times q} = \sqrt{14 \times 0.4 \times 0.6} \] Calculating $ \sigma $: \[ \sigma = \sqrt{14 \times 0.4 \times 0.6} = \sqrt{3.36} \approx 1.833 \]

Step 3: Apply Continuity Correction and Calculate Z-Score

To find $ P(\text{fewer than } 4) $, we apply continuity correction: \[ x = 3.5 \] Calculating the z-score: \[ z = \frac{x - \mu}{\sigma} = \frac{3.5 - 5.6}{1.833} \approx -1.148 \]

Step 4: Find Probability Using Z-Score

Using the z-score, we find the probability: \[ P(Z < -1.148) \approx 0.1260 \]