According to an airline, a particular flight is on time 75% of the time. Suppose 140 flights are randomly selected and the number of on time flights is recorded.

Determine if you should use the Binomial model or Normal model to solve this problem. show work and write a sentence justifying your decision.

Find the mean or expected number of on time flights.

Find the standard deviation, round to 3 decimal places.

Find Q1, round to 3 decimal places.

Find the probability that at least 110 flights are on time.

To determine whether to use the Binomial model or Normal model, we need to check if the conditions for each are met.

1. Binomial model conditions:
a. The number of trials, n, is fixed: In this case, n is fixed at 140 flights.
b. The trials are independent: It is reasonable to assume that one flight being on time does not affect the others.
c. The probability of success, p, is the same for all trials: According to the airline, the probability of a flight being on time is 75% for all flights.
d. The number of successes, x, is counted: We are interested in the number of on-time flights.

2. Normal model conditions:
a. np ≥ 10 and n(1-p) ≥ 10: We need to check if these conditions are satisfied.

Now let's calculate the mean, standard deviation, Q1, and find the probability.

Mean:
The mean can be found using the formula: mean = n * p. Here, n = 140 flights and p = 0.75 (75%).
Mean = 140 * 0.75 = 105 on-time flights.

Standard Deviation:
The standard deviation can be found using the formula: standard deviation = sqrt(n * p * (1 - p)).
Standard Deviation = sqrt(140 * 0.75 * (1 -0.75)) = sqrt(26.25) ≈ 5.123 (rounded to 3 decimal places).

Q1 (First Quartile):
The first quartile is the value below which 25% of the data falls. We can find this by using the cumulative distribution function (CDF) of the binomial distribution with parameters n, p, and x = Q1.

Now let's find the probability that at least 110 flights are on time:
Here we need to calculate the probability of getting 110 or more on-time flights out of 140. This can be done using the binomial probability formula or by using cumulative distribution function (CDF) of the binomial distribution with parameters n, p, and x = 110.

In this case, since np = 140 * 0.75 = 105 and n(1-p) = 140 * 0.25 = 35, both conditions for the Normal model are satisfied.

Therefore, for this problem, we will use the Normal model to make calculations because the conditions are met.

To summarize our decision, we will use the Normal model because we have a large sample size (n) and both conditions for the Normal model are satisfied.

Remember that the Binomial model is appropriate when the number of trials is fixed and meets the conditions mentioned earlier. The Normal model is used when the Binomial model conditions are not met and when the sample size is large enough (np ≥ 10 and n(1-p) ≥ 10).