Math / Data & Statistics

QuestionAccording to flightstats.com, American Airlines flights from Dallas to Chicago are on time $80 \%$ of the time. Suppose 25 flights are randomly selected, and the number of on-time flights is recorded. (a) Explain why this is a binomial experiment. (b) Determine the values of $n$ and $p$ . (c) Find and interpret the probability that exactly 15 flights are on time. (d) Find and interpret the probability that fewer than 15 flights are on time. (e) Find and interpret the probability that at least 15 flights are on time. (f) Find and interpret the probability that between 13 and 15 flights, inclusive, are on time. (c) Using the binomial distribution, the probability that exactly 15 flights are on time is $\square$ (Round to four decimal places as needed.) Interpret the probability. In 100 trials of this experiment, it is expected that about (Round to the nearest whole number as needed.) $\square$ will result in exactly 15 flights being on time. (d) Using the binomial distribution, the probability that fewer than 15 flights are on time is (Round to four decimal places as needed.) $\square$ Interpret the probability. In 100 trials of this experiment, it is expected that about $\square$ will result in fewer than 15 flights being on time. (Round to the nearest whole number as needed.)

Studdy Solution

STEP 1

1. Each flight is an independent trial.
2. The probability of a flight being on time is constant at $80\%$ .
3. The number of flights selected is 25.

STEP 2

1. Explain why this is a binomial experiment.
2. Determine the values of $n$ and $p$ .
3. Calculate the probability that exactly 15 flights are on time.
4. Calculate the probability that fewer than 15 flights are on time.
5. Calculate the probability that at least 15 flights are on time.
6. Calculate the probability that between 13 and 15 flights, inclusive, are on time.

STEP 3

A binomial experiment is characterized by: - A fixed number of trials. - Each trial has two possible outcomes: success or failure. - The probability of success is the same for each trial. - The trials are independent.
In this scenario: - The fixed number of trials is 25 flights. - Each flight can either be on time (success) or not on time (failure). - The probability of a flight being on time is $0.80$ . - Each flight's status is independent of the others.
Thus, this is a binomial experiment.

STEP 4

Determine the values of $n$ and $p$ :
- $n = 25$ (the number of trials) - $p = 0.80$ (the probability of success)

STEP 5

Calculate the probability that exactly 15 flights are on time using the binomial probability formula:
$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$
Where: - $n = 25$ - $k = 15$ - $p = 0.80$
$P(X = 15) = \binom{25}{15} (0.80)^{15} (0.20)^{10}$
Calculate this probability.

STEP 6

Calculate the probability that fewer than 15 flights are on time:
$P(X < 15) = \sum_{k=0}^{14} \binom{25}{k} (0.80)^k (0.20)^{25-k}$
Calculate this probability.

STEP 7

Calculate the probability that at least 15 flights are on time:
$P(X \geq 15) = 1 - P(X < 15)$
Calculate this probability.

STEP 8

Calculate the probability that between 13 and 15 flights, inclusive, are on time:
$P(13 \leq X \leq 15) = \sum_{k=13}^{15} \binom{25}{k} (0.80)^k (0.20)^{25-k}$
Calculate this probability.
Solution for (c): Using the binomial distribution, the probability that exactly 15 flights are on time is approximately $0.0346$ .
Interpretation: In 100 trials of this experiment, it is expected that about 3 or 4 trials will result in exactly 15 flights being on time.
Solution for (d): Using the binomial distribution, the probability that fewer than 15 flights are on time is approximately $0.0207$ .
Interpretation: In 100 trials of this experiment, it is expected that about 2 trials will result in fewer than 15 flights being on time.

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Studdy Solution

STEP 1

1. Each flight is an independent trial.
2. The probability of a flight being on time is constant at $80\%$ .
3. The number of flights selected is 25.

STEP 2

STEP 3

STEP 4

Determine the values of $n$ and $p$ :
- $n = 25$ (the number of trials) - $p = 0.80$ (the probability of success)

STEP 5

Calculate the probability that exactly 15 flights are on time using the binomial probability formula:
$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$
Where: - $n = 25$ - $k = 15$ - $p = 0.80$
$P(X = 15) = \binom{25}{15} (0.80)^{15} (0.20)^{10}$
Calculate this probability.

STEP 6

Calculate the probability that fewer than 15 flights are on time:
$P(X < 15) = \sum_{k=0}^{14} \binom{25}{k} (0.80)^k (0.20)^{25-k}$
Calculate this probability.

STEP 7

Calculate the probability that at least 15 flights are on time:
$P(X \geq 15) = 1 - P(X < 15)$
Calculate this probability.

STEP 8

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Studdy Solution

STEP 1

1. Each flight is an independent trial.2. The probability of a flight being on time is constant at 80%80\%80%.3. The number of flights selected is 25.

STEP 2

STEP 3

STEP 4

Determine the values of nnn and ppp:- n=25n = 25n=25 (the number of trials) - p=0.80p = 0.80p=0.80 (the probability of success)

STEP 5

STEP 6

Calculate the probability that fewer than 15 flights are on time:P(X<15)=∑k=014(25k)(0.80)k(0.20)25−k P(X < 15) = \sum_{k=0}^{14} \binom{25}{k} (0.80)^k (0.20)^{25-k} P(X<15)=k=0∑14​(k25​)(0.80)k(0.20)25−kCalculate this probability.

STEP 7

Calculate the probability that at least 15 flights are on time:P(X≥15)=1−P(X<15) P(X \geq 15) = 1 - P(X < 15) P(X≥15)=1−P(X<15)Calculate this probability.

STEP 8

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1. Each flight is an independent trial.
2. The probability of a flight being on time is constant at $80\%$ .
3. The number of flights selected is 25.

Determine the values of $n$ and $p$ :
- $n = 25$ (the number of trials) - $p = 0.80$ (the probability of success)

Calculate the probability that fewer than 15 flights are on time:
$P(X < 15) = \sum_{k=0}^{14} \binom{25}{k} (0.80)^k (0.20)^{25-k}$
Calculate this probability.

Calculate the probability that at least 15 flights are on time:
$P(X \geq 15) = 1 - P(X < 15)$
Calculate this probability.