Question

Suppose that X is a binomial random variable with n = 200 and p = 0.4.

A) Approximate the probability that x is less than or equal to 70
B) Approximate the probability that x is greater than 70 and less than 90.
C) Approximate the probability x = 80

Answer 1

To approximate the probabilities, we can use the normal distribution approximation, provided that np > 5 and n(1-p) > 5. In this case, we have np = 80 and n(1-p) = 120, satisfying the conditions for approximation.

Step-by-step explanation:

In this case, X is a binomial random variable with parameters n = 200 and p = 0.4. To approximate the probabilities, we can use the normal distribution approximation, provided that np > 5 and n(1-p) > 5. In this case, np = 200(0.4) = 80 and n(1-p) = 200(0.6) = 120, satisfying the conditions for approximation.

A) To approximate the probability that x is less than or equal to 70, we can use the normal approximation with mean μ = np = 80 and standard deviation σ = √(np(1-p)) = √(200(0.4)(0.6)) ≈ 6.93. We can then find the standardized score z = (70 - μ) / σ and use the standard normal distribution table or calculator to find the corresponding probability.

B) To approximate the probability that x is greater than 70 and less than 90, we can find the cumulative probability for x = 70 and x = 89 using the normal distribution approximation, and then subtract the two probabilities.