Question

A certain flight arrives on time 89 percent of the time. Suppose 133 flights are randomly selected. Use the normal approximation to the binomial to approximate the probability that:

(a) exactly 125 flights are on time.
(b) at least 125 flights are on time.
(c) fewer than 117 flights are on time.
(d) between 117 and 123, inclusive are on time.

Answer 1

To approximate the probability using the normal approximation to the binomial, calculate the mean and standard deviation. Then use the normal distribution to approximate the probability. (a) Use the formula P(X = 125), (b) use the formula P(X >= 125), (c) use the formula P(X < 117), and (d) use the formula P(117 <= X <= 123).

Step-by-step explanation:

To approximate the probability using the normal approximation to the binomial, we need to calculate the mean and standard deviation of the binomial distribution. For (a) exactly 125 flights, the mean is np = 133 * 0.89 = 118.37 and the standard deviation is sqrt(npq) = sqrt(133 * 0.89 * 0.11) = 3.289. Using these values, we can use the normal distribution to approximate the probability.

(a) To find the probability of exactly 125 flights on time, we use the formula for the normal distribution with continuity correction:

P(X = 125) = P(124.5 < X < 125.5) = P((124.5 - 118.37)/3.289 < Z < (125.5 - 118.37)/3.289)

(b) To find the probability that at least 125 flights are on time, we use the formula:

P(X >= 125) = 1 - P(X < 125)

(c) To find the probability that fewer than 117 flights are on time, we use the formula:

P(X < 117) = P(X <= 116.5)

(d) To find the probability that between 117 and 123 inclusive are on time, we use the formula:

P(117 <= X <= 123) = P(116.5 < X < 123.5)