Final answer:
The probability for the specific given values is approximately 0.871.
Step-by-step explanation:
To use the normal approximation to the binomial, we need to check if np and nq are both greater than 5.
In this case, n = 50 and p = 0.8, so np = 50 * 0.8 = 40 and nq = 50 * 0.2 = 10.
Since both np and nq are greater than 5, we can proceed with the approximation.
To approximate the probability for X = 44, we use the formula for a normal distribution with mean:
μ = np = 50 * 0.8 = 40 and,
Standard deviation σ = √(npq) = √(50 * 0.8 * 0.2)
= 4.
Actually, since we are looking for the probability that is less than or equal to 44, we should use the value 44.5 instead of 44.
Using a standard normal distribution table or a calculator, we can find the z-score for 44.5.
The z-score is given by z = (x - μ) / σ
= (44.5 - 40) / 4
= 1.125.
Using the z-score table, we find that the probability of a z-score less than or equal to 1.125 is approximately 0.871.