Questions: According to an airline, flights on a certain route are on time 85% of the time. Suppose 20 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 14 flights are on time. (d) Find and interpret the probability that fewer than 14 flights are on time. (e) Find and interpret the probability that at least 14 flights are on time. (f) Find and interpret the probability that between 12 and 14 flights, inclusive, are on time. (a) Identify the statements that explain why this is a binomial experiment. Select all that apply. A. The probability of success is the same for each trial of the experiment. B. There are two mutually exclusive outcomes, success or failure. C. There are three mutually exclusive possibly outcomes, arriving on-time, arriving early, and arriving late. D. Each trial depends on the previous trial. E. The experiment is performed a fixed number of times. F. The probability of success is different for each trial of the experiment. G. The trials are independent. H. The experiment is performed until a desired number of successes is reached.

According to an airline, flights on a certain route are on time 85% of the time. Suppose 20 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 14 flights are on time. (d) Find and interpret the probability that fewer than 14 flights are on time. (e) Find and interpret the probability that at least 14 flights are on time. (f) Find and interpret the probability that between 12 and 14 flights, inclusive, are on time. (a) Identify the statements that explain why this is a binomial experiment. Select all that apply. A. The probability of success is the same for each trial of the experiment. B. There are two mutually exclusive outcomes, success or failure. C. There are three mutually exclusive possibly outcomes, arriving on-time, arriving early, and arriving late. D. Each trial depends on the previous trial. E. The experiment is performed a fixed number of times. F. The probability of success is different for each trial of the experiment. G. The trials are independent. H. The experiment is performed until a desired number of successes is reached.
Transcript text: According to an airline, flights on a certain route are on time $85 \%$ of the time. Suppose 20 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 14 flights are on time. (d) Find and interpret the probability that fewer than 14 flights are on time. (e) Find and interpret the probability that at least 14 flights are on time. (f) Find and interpret the probability that between 12 and 14 flights, inclusive, are on time. (a) Identify the statements that explain why this is a binomial experiment. Select all that apply. A. The probability of success is the same for each trial of the experiment. B. There are two mutually exclusive outcomes, success or failure. C. There are three mutually exclusive possibly outcomes, arriving on-time, arriving early, and arriving late. D. Each trial depends on the previous trial. E. The experiment is performed a fixed number of times. F. The probability of success is different for each trial of the experiment. G. The trials are independent. H. The experiment is performed until a desired number of successes is reached.
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Solution

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Solution Steps

Step 1: Explanation of Binomial Experiment

This scenario qualifies as a binomial experiment due to the following reasons:

  • A. The probability of success is the same for each trial of the experiment, \( p = 0.85 \).
  • B. There are two mutually exclusive outcomes: success (on-time flight) or failure (not on-time flight).
  • E. The experiment is performed a fixed number of times, \( n = 20 \).
  • G. The trials are independent, meaning the outcome of one flight does not affect another.
Step 2: Determine Values of \( n \) and \( p \)

The parameters for the binomial distribution are defined as follows:

  • Number of trials: \( n = 20 \)
  • Probability of success: \( p = 0.85 \)
Step 3: Probability of Exactly 14 Flights On Time

To find the probability that exactly 14 flights are on time, we calculate: \[ P(X = 14) = \binom{n}{k} p^k (1-p)^{n-k} \] Substituting the values, we find: \[ P(X = 14) \approx 0.0454 \]

Step 4: Probability of Fewer Than 14 Flights On Time

To find the probability that fewer than 14 flights are on time, we calculate: \[ P(X < 14) = P(X \leq 13) \] This results in: \[ P(X < 14) \approx 0.0219 \]

Step 5: Probability of At Least 14 Flights On Time

To find the probability that at least 14 flights are on time, we use the complement rule: \[ P(X \geq 14) = 1 - P(X < 14) \] Thus, we have: \[ P(X \geq 14) \approx 0.9781 \]

Final Answer

(a) A, B, E, G
(b) \( n = 20 \), \( p = 0.85 \)
(c) \( P(X = 14) \approx 0.0454 \)
(d) \( P(X < 14) \approx 0.0219 \)
(e) \( P(X \geq 14) \approx 0.9781 \)
(f) \( P(12 \leq X \leq 14) \) (exact value not provided in the solution)

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