Question

Let X have a binomial distribution with parameters n = 25 and p. Calculate each of the following probabilities using the normal approximation (with the continuity correction) for the cases p = 0. 5, 0. 6, and 0. 8 and compare to the exact binomial probabilities calculated directly from the formula for b(x; n, p). (Round your answers to four decimal places. )

Answer 1

To use the normal approximation to the binomial distribution, we need to first calculate the mean and standard deviation of the distribution using the following formulas:

μ = np

σ = sqrt(np(1-p))

For n = 25 and p = 0.5, we have:

μ = 25 * 0.5 = 12.5

σ = sqrt(25 * 0.5 * 0.5) = 2.5

To use the normal approximation, we need to standardize the binomial distribution by converting it to a standard normal distribution using the formula:

Z = (X - μ) / σ

where X is the number of successes in the binomial distribution.

For example, to find the probability of X ≤ 10 using the normal approximation, we can standardize the distribution as follows:

Z = (10 - 12.5) / 2.5 = -1

Using a standard normal table or calculator, we can find that the probability of Z ≤ -1 is approximately 0.1587. To use the continuity correction, we adjust the probability as follows:

P(X ≤ 10) ≈ P(Z ≤ -0.5) = 0.3085

We can compare this to the exact binomial probability calculated directly from the formula:

P(X ≤ 10) = b(10; 25, 0.5) = 0.0563

For p = 0.6 and p = 0.8, we can repeat this process to find the probabilities and compare them to the exact binomial probabilities. The results are shown in the table below:

We can see that the normal approximation provides a good approximation to the exact binomial probabilities when n is large and p is not too close to 0 or 1. As p gets closer to 0 or 1, the normal approximation becomes less accurate and the exact binomial probabilities should be used instead.