Apply Procedure 6.3 on page 289 to approximate the required...

Apply Procedure 6.3 on page 289 to approximate the required binomial probabilities. On-Time Airlines. The Office of Aviation Enforcement and Procecdings (OAEP) publishes important consumer information about airlines in Air Travel Consumer Report. For the 12 months ending June $30,2008,73.7 \%$ of all flights arriving at U.S. airports arrived "on time." meaning no more than 15 minutes late at the arrival gate. The Boston airport reported 10,288 arrivals during June 2008 . If the overall percentage of on-time flights applies to Boston, what is the probability that, during June $2008,$ the number of on-time flights to Boston a. exceeded $7600 ?$ b. was between 7450 and $7550,$ inclusive?

Apply Procedure 6.3 on page 289 to approximate the required binomial probabilities.
On-Time Airlines. The Office of Aviation Enforcement and Procecdings (OAEP) publishes important consumer information about airlines in Air Travel Consumer Report. For the
12 months ending June $30,2008,73.7 \%$ of all flights arriving at U.S. airports arrived "on time." meaning no more than 15 minutes late at the arrival gate. The Boston airport reported 10,288 arrivals during June 2008 . If the overall percentage of on-time flights applies to Boston, what is the probability that, during June $2008,$ the number of on-time flights to Boston
a. exceeded $7600 ?$
b. was between 7450 and $7550,$ inclusive?

Key Concepts

Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success. It is defined by two parameters: the number of trials (n) and the probability of success (p) in each trial, and is used for events with only two possible outcomes, such as success/failure or yes/no outcomes.

Normal Approximation to the Binomial

When the number of trials in a binomial experiment is large, the distribution can be approximated by a normal distribution. This is especially useful in calculating probabilities, as the normal distribution’s properties allow for simpler computations. The normal approximation is valid under conditions typically checked by ensuring that both np and n(1-p) are sufficiently large.

Continuity Correction

The continuity correction is a technique used when a discrete distribution is approximated by a continuous distribution. It involves adjusting the discrete variable by 0.5 in either direction to better estimate the probability for a given interval. This correction improves the accuracy of the normal approximation when calculating probabilities for the binomial distribution.

Central Limit Theorem

The Central Limit Theorem is a fundamental statistical concept that indicates that the sum (or average) of a large number of independent, identically distributed random variables will tend to follow a normal distribution, regardless of the original distribution. This theorem underpins the normal approximation to the binomial distribution when the sample size is large.

Transcript

00:01 So we know that they tell us that the, we have a probability of 0 .737 of a flight being on time.

00:09 And in june of 2008, they said that there were 10 ,288 flights into boston during june of 2008.

00:23 That's a lot of flights.

00:25 And we want to estimate a couple of questions.

00:27 But before we do, we know that this distribution of end.

00:31 Times p and n times 1 minus p are both going to be greater than a equal to 5.

00:36 So this distribution is approximately normal for the number of planes arriving on time.

00:42 We have a very large sample size, ours with a very large n.

00:45 And we know that the mean is going to be 0 .737 is going to be n times p, which is 0 .737 times that 10 ,288.

00:56 10 ,288.

00:59 Again, that's a comma.

01:02 And that comes out to be 7 ,582 .256.

01:08 And i'm actually going to store that value in my calculator is like alpha a.

01:12 That way i don't have to keep typing it in when i convert to z values.

01:15 And now we want to find the standard deviation, which is a square root of n times p.

01:20 So this value times 1 minus p, and then we'll have to make sure that that's square rooted.

01:24 And so we take that value times, this value, times 1 minus .737.

01:34 I get that answer and take it to the .5 or square root it.

01:38 And i find the standard deviation is 44 .6.

01:42 We'll say 5 ,6.

01:44 And i could actually take that value and store it in my calculator.

01:47 I happen to have this one stored as a and i have this one stored as x.

01:53 But you could store it as whatever you want, whichever variability you want, whatever is convenient for you...

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