Question

A certain flight arrives on time

8383

percent of the time. Suppose

147147

flights are randomly selected. Use the normal approximation to the binomial to approximate the probability that ​(a) exactly

128128

flights are on time.​(b) at least

128128

flights are on time.​(c) fewer than

127127

flights are on time.​(d) between

127127

and

132132​,

inclusive are on time.

Answer 1

Here we have

n=147 and p=0.83

Since np = 122.01 and n(1-p) = 24.99 both are greater than 5 so we can use normal approximation here.

Using normal approximation, X has approximately normal distribution with mean and SD as follows:

(a)

The z-score for X= 128- 0.5 = 127.5 is

The z-score for X= 128 + 0.5 = 128.5 is

(a)

Using z table, the probability that ​exactly 128 flights are on time is

(b)

The z-score for X= 128- 0.5 = 127.5 is

Using z table, the probability that ​at least 128 flights are on time is

(c)

The z-score for X= 127- 0.5 = 126.5 is

Using z table, the probability that ​fewer than 127 flights are on time is

(d)

The z-score for X= 127- 0.5 = 126.5 is

The z-score for X= 132+ 0.5 = 132.5 is

The required probability is