Suppose x has a binomial probability distribution with n = 200 and p = 0.60. Use the normal approximation to the binomial to find P (X > 130).
To find P(X > 130) using the normal approximation to the binomial, we can use the following steps:
Step 1: Calculate the mean (μ) and standard deviation (σ) of the binomial distribution. For a binomial distribution, the mean is given by μ = n * p and the standard deviation is given by σ = sqrt(n * p * (1 - p)).
Given n = 200 and p = 0.60, we can calculate:
μ = n * p = 200 * 0.60 = 120
σ = sqrt(n * p * (1 - p)) = sqrt(200 * 0.60 * 0.40) = 7.4833 (approximated to four decimal places).
Step 2: Convert the binomial distribution to the standard normal distribution using the z-score formula. The z-score is calculated as z = (x - μ) / σ, where x is the number of successes.
For x = 130, the z-score is:
z = (130 - 120) / 7.4833 ≈ 1.3395.
Step 3: Find the probability P(X > 130) using the standard normal distribution table or a calculator. Since we want the probability to the right of 130, we need to find 1 - P(Z ≤ z).
Using the standard normal distribution table or a calculator, you can find that P(Z ≤ 1.3395) ≈ 0.9119. Therefore, P(X > 130) ≈ 1 - 0.9119 = 0.0881 (or 8.81%).
So, the approximate probability of X being greater than 130 is 0.0881 (or 8.81%).