A certain flight arrives on time 82 percent of the time. Suppose 198flights are randomly selected. Use the normal approximation to the binomial to approximate the probability that​(a) exactly 172 flights are on time.​(b) at least 172 flights are on time.​(c) fewer than148 flights are on time.​(d) between 148 and 168​, inclusive are on time.

Question
Asked Oct 26, 2019
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A certain flight arrives on time 82 percent of the time. Suppose 198
flights are randomly selected. Use the normal approximation to the binomial to approximate the probability that
​(a) exactly 172 flights are on time.
​(b) at least 172 flights are on time.
​(c) fewer than148 flights are on time.
​(d) between 148 and 168​, inclusive are on time.
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Expert Answer

Step 1

It is given that 82% of the time the flight arrives on time. Therefore, probability of a selected flight being on time is 0.82.

The mean of binomial random variable is given and calculated as follows:

np
=(198) (0.82)
=162.36
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np =(198) (0.82) =162.36

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Step 2

The standard deviation of binomial random variable is given and calculated as follows:

np (1-p)
= (198) (0.82) (1-0.82)
(198) (0.82) (0.18)
=5.406
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np (1-p) = (198) (0.82) (1-0.82) (198) (0.82) (0.18) =5.406

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Step 3

By the normal approximation, the random variable X, number of flights being on time follows normal distribution with mean 162.36 and standard deviation 5.406.

(a).

The probability that exactly 172 flights b...

=NORM.DIST (172.5,162.36,5.406, TRUE)-NORM.DIST(171.5, 162.36,5.406, TRUE)
Е
G
Н
I
0.01509704
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=NORM.DIST (172.5,162.36,5.406, TRUE)-NORM.DIST(171.5, 162.36,5.406, TRUE) Е G Н I 0.01509704

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