Question

Suppose x is a binomial random variable with n = 200 and p = 0.4. Approximate the probability that x is greater than 70 and less than 90.

A) 0.215
B) 0.462
C) 0.576
D) 0.324

Answer 1

To approximate the probability that x is greater than 70 and less than 90, we use the normal approximation to the binomial distribution. The probability is approximately 0.215.

Step-by-step explanation:

To approximate the probability that x is greater than 70 and less than 90, we can use the normal approximation to the binomial distribution. First, we calculate the mean (μ): μ = np = 200 * 0.4 = 80. The standard deviation (σ) is given by σ = √(npq) = √(200 * 0.4 * 0.6) ≈ 6.93. Now, we can standardize the values of 70 and 90 using the z-score formula: z_70 = (70 - 80) / 6.93 ≈ -1.44 and z_90 = (90 - 80) / 6.93 ≈ 1.44. Next, we can use a standard normal distribution table or a calculator to find the probability associated with these z-scores. The probability that x is greater than 70 and less than 90 is approximately the difference of these two probabilities: P(70 < x < 90) ≈ P(z < 1.44) - P(z < -1.44) ≈ 0.576 - 0.362 ≈ 0.215. Therefore, the correct answer is A) 0.215.