Final answer:
To find the probability of observing between 220 and 230 successes in a situation with 500 trials and a 16% chance of success in each trial, we can use the normal approximation of the binomial distribution.
Step-by-step explanation:
To find the probability of observing between 220 and 230 successes in a situation with 500 trials and a 16% chance of success in each trial, we can use the normal approximation of the binomial distribution. First, we need to check if the conditions for approximation are met. The conditions are that np > 5 and n(1-p) > 5, where n is the number of trials and p is the probability of success. In this case, np = 500 * 0.16 = 80 and n(1-p) = 500 * 0.84 = 420, both of which are greater than 5.
The mean (μ) of the distribution is μ = np = 80 and the standard deviation (σ) is σ = √(np(1-p)) = √(500 * 0.16 * 0.84) = 7.056.
Next, we can use the normal distribution to calculate the probabilities. To find the probability of observing between 220 and 230 successes, we need to find the z-scores for 220 and 230, and then use the standard normal distribution table or a calculator to find the corresponding probabilities. The z-scores are z1 = (220 - 80) / 7.056 = 17.16 and z2 = (230 - 80) / 7.056 = 15.49. Using a standard normal distribution table or a calculator, we can find the probabilities associated with these z-scores and subtract them to find the final probability.