Question

What is the probability of observing between 220 and 230 successes when approximating a binomial distribution with a large number of trials? (Round your answer to four decimal places.)

Answer 1

To find the probability of observing between 220 and 230 successes when approximating a binomial distribution with a large number of trials, use the normal distribution as an approximation.

Step-by-step explanation:

To approximate a binomial distribution with a large number of trials, we can use the normal distribution as an approximation. In this case, we want to find the probability of observing between 220 and 230 successes. Let's denote the number of trials as n and the probability of success as p. Given that the distribution is large, the mean of the distribution is given by mu = np and the standard deviation is given by sigma = sqrt(np(1-p)).

1. Calculate the mean: mu = n * p
2. Calculate the standard deviation: sigma = sqrt(n * p * (1 - p))
3. Convert the lower and upper bounds of the range to Z-scores using the formula Z = (x - mu) / sigma
4. Look up the Z-scores in the standard normal distribution table to find the corresponding probabilities
5. Subtract the probability of the lower bound from the probability of the upper bound to find the probability of observing between 220 and 230 successes

Remember to round your answer to four decimal places.