Final answer:
To approximate P(X ≤ 15) for a binomial random variable X ~ B(200, 0.08) using normal distribution, calculate the mean (μ = np) and standard deviation (σ = √npq), apply the continuity correction by using 15 + 0.5, then find the corresponding z-score and look up the probability in the standard normal distribution table or use a calculator.
Step-by-step explanation:
Approximating Binomial Distribution with Normal Distribution
To find the approximate probability of X being less than or equal to 15, we will use the normal approximation to the binomial distribution. Given X ~ B(200, 0.08), the mean (μ) and standard deviation (σ) of the binomial distribution are calculated using μ = np and σ = √(npq) respectively, where q = 1 - p. In this case:
μ = 200 * 0.08 = 16
σ = √(200 * 0.08 * (1 - 0.08)) = √(200 * 0.08 * 0.92) ≈ 3.697
Before using the normal approximation, we apply the continuity correction factor by adjusting the value of X we're interested in. Since we're looking for P(X ≤ 15), we use 15 + 0.5 = 15.5. We then calculate the z-score using the formula:
Z = (X - μ) / σ = (15.5 - 16) / 3.697 ≈ -0.135
Now, look up the probability associated with this z-score from the standard normal distribution table, or use a calculator.
It is important to ensure that np and nq are greater than 5 to apply this approximation effectively. Lastly, we use a calculator or statistical table to determine P(X ≤ 15).