Question

If np≥5 and nq≥​5,estimate P(at least 6) with n=13 and p=0.5 by using the normal distribution as an approximation to the binomial​ distribution; if np<5 or nq<​5, then state that the normal approximation is not suitable.

P(at least 6)=

Answer 1

After verifying the conditions for the normal approximation to the binomial distribution are met (np ≥5 and nq ≥5), we use it to estimate P(at least 6) with n=13 and p=0.5. The mean is calculated as 6.5 and the standard deviation as approximately 2.55, and the normal approximation can be applied with these values.

Step-by-step explanation:

The student is asked to estimate P(at least 6) using the normal distribution as an approximation to the binomial distribution, given that n = 13 and p = 0.5. The conditions for this approximation are that both np ≥5 and nq ≥5. We first need to check if the conditions are satisfied. Here n = 13, p = 0.5, and q = 1 - p = 0.5. We find that np = nq = 13 * 0.5 = 6.5, which means the normal approximation can be used.

To apply the normal approximation, we use the correction for continuity and calculate P(X ≥ 5.5) because we are interested in 'at least 6'. We find the mean μ = np = 6.5 and the standard deviation σ = √npq = √(13*0.5*0.5) = √(6.5) ≈ 2.55. Using these parameters, we then look up the cumulative probability in a standard normal distribution table or use a calculator to find the normal approximation.