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.