Below, 

n

 is the sample size, p

 is the population proportion of successes, and X

 is the number of successes in the sample. Use the normal approximation and the Cumulative Normal Distribution Table to find the probability. Round the answer to at least four decimal places.=

n

=

78

, 

p

0.43

P

<

X

40


To solve this problem, we will use the normal approximation to the binomial distribution. The binomial distribution with parameters \(n = 78\) and \(p = 0.43\) can be approximated by a normal distribution when \(n\) is large. 

The key steps are:

1. **Mean (\( \mu \)) of the binomial distribution:**
   \[
   \mu = np = 78 \times 0.43 = 33.54
   \]

2. **Standard deviation (\( \sigma \)) of the binomial distribution:**
   \[
   \sigma = \sqrt{np(1 - p)} = \sqrt{78 \times 0.43 \times (1 - 0.43)} = \sqrt{78 \times 0.43 \times 0.57} \approx 4.3835
   \]

3. **Apply the continuity correction:**  
   Since we are approximating a discrete distribution with a continuous one, we add 0.5 to account for the continuity correction. The problem asks to find \(P(X < 40)\), so we approximate it by finding \(P(X < 39.5)\).

4. **Convert to z-score:**  
   The z-score is calculated by the formula:
   \[
   z = \frac{X - \mu}{\sigma}
   \]
   For \(X = 39.5\):
   \[
   z = \frac{39.5 - 33.54}{4.3835} \approx 1.3584
   \]

5. **Find the cumulative probability from the z-score:**  
   Using the standard normal distribution table, we find the cumulative probability corresponding to \(z = 1.3584\). From the table, we get:
   \[
   P(Z < 1.3584) \approx 0.9123
   \]

Thus, the probability \(P(X < 40)\) is approximately **0.9123**.

### Would you like further details or have any questions?

### Related Questions:
1. What is the general rule for applying normal approximation to binomial distribution?
2. How is the continuity correction applied in normal approximation?
3. What if the value of \( p \) was much closer to 0 or 1? How would that affect the normal approximation?
4. How do we calculate z-scores for different values of \( X \) in this context?
5. Could this problem be solved without using the normal approximation?

### Tip:
Always ensure the sample size \( n \) is large enough when using normal approximation, a common rule of thumb is that both \( np \) and \( n(1-p) \) should be at least 5.

Below, n is the sample size, p is the population proportion of successes, and X is the number of successes in the sample. Use the normal approximation and the Cumulative Normal Distribution Table to find the probability. Round the answer to at least four decimal places. n = 78, p = 0.43, P(X < 40).

Explore the steps to solve a binomial probability problem using the normal approximation with continuity correction. Given n = 78 and p = 0.43, find P(X < 40) by calculating the mean, standard deviation, and z-score. Suitable for Grades 11-12 or college-level statistics students.

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