Questions: According 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 16 flights are on time. (d) Find and interpret the probability that fewer than 16 flights are on time. (e) Find and interpret the probability that at least 16 flights are on time. (f) Find and interpret the probability that between 14 and 16 flights, inclusive, are on time. (a) Identify the statements that explain why this is a binomial experiment. Select all that apply. A. The experiment is performed a fixed number of times. B. The probability of success is different for each trial of the experiment. C. The experiment is performed until a desired number of successes are reached. D. The trials are independent. E. Each trial depends on the previous trial. F. There are three mutually exclusive possible outcomes, arriving on-time, arriving early, and arriving late. G. The probability of success is the same for each trial of the experiment. H. There are two mutually exclusive outcomes, success (plane arrives on time) or failure (plane does not arrive on time). (b) Using the binomial distribution, determine the values of n and p. n= (Type an integer or a decimal. Do not round.) p= (Type an integer or a decimal. Do not round.) (c) Using the binomial distribution, the probability that exactly 16 flights are on time is . (Round to four decimal places as needed.)

According 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 16 flights are on time. (d) Find and interpret the probability that fewer than 16 flights are on time. (e) Find and interpret the probability that at least 16 flights are on time. (f) Find and interpret the probability that between 14 and 16 flights, inclusive, are on time. (a) Identify the statements that explain why this is a binomial experiment. Select all that apply. A. The experiment is performed a fixed number of times. B. The probability of success is different for each trial of the experiment. C. The experiment is performed until a desired number of successes are reached. D. The trials are independent. E. Each trial depends on the previous trial. F. There are three mutually exclusive possible outcomes, arriving on-time, arriving early, and arriving late. G. The probability of success is the same for each trial of the experiment. H. There are two mutually exclusive outcomes, success (plane arrives on time) or failure (plane does not arrive on time). (b) Using the binomial distribution, determine the values of n and p. n= (Type an integer or a decimal. Do not round.) p= (Type an integer or a decimal. Do not round.) (c) Using the binomial distribution, the probability that exactly 16 flights are on time is . (Round to four decimal places as needed.)
Transcript text: According 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 16 flights are on time. (d) Find and interpret the probability that fewer than 16 flights are on time. (e) Find and interpret the probability that at least 16 flights are on time. (f) Find and interpret the probability that between 14 and 16 flights, inclusive, are on time. (a) Identify the statements that explain why this is a binomial experiment. Select all that apply. A. The experiment is performed a fixed number of times. B. The probability of success is different for each trial of the experiment. C. The experiment is performed until a desired number of successes are reached. D. The trials are independent. E. Each trial depends on the previous trial. F. There are three mutually exclusive possible outcomes, arriving on-time, arriving early, and arriving late. G. The probability of success is the same for each trial of the experiment. H. There are two mutually exclusive outcomes, success (plane arrives on time) or failure (plane does not arrive on time). (b) Using the binomial distribution, determine the values of n and p . $n=\square$ (Type an integer or a decimal. Do not round.) $p=\square$ $\square$ (Type an integer or a decimal. Do not round.) (c) Using the binomial distribution, the probability that exactly 16 flights are on time is $\square$ $\square$. (Round to four decimal places as needed.)
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Solution

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Solution Steps

Step 1: Explanation of the Binomial Experiment

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

  • The experiment is performed a fixed number of times (\( n = 25 \)).
  • The trials are independent, meaning the outcome of one flight does not affect another.
  • The probability of success (on-time flight) remains constant at \( p = 0.8 \) for each trial.
  • There are two mutually exclusive outcomes: success (the flight arrives on time) or failure (the flight does not arrive on time).
Step 2: Values of \( n \) and \( p \)

From the problem statement, we have:

  • \( n = 25 \) (the number of flights)
  • \( p = 0.8 \) (the probability of a flight being on time)
Step 3: Probability of Exactly 16 Flights Being On Time

To find the probability that exactly 16 flights are on time, we use the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

where:

  • \( n = 25 \)
  • \( x = 16 \)
  • \( p = 0.8 \)
  • \( q = 1 - p = 0.2 \)

Calculating this gives:

\[ P(X = 16) = \binom{25}{16} \cdot (0.8)^{16} \cdot (0.2)^{9} \approx 0.0294 \]

Thus, the probability that exactly 16 flights are on time is \( 0.0294 \).

Final Answer

The values are:

  • \( n = 25 \)
  • \( p = 0.8 \)
  • The probability that exactly 16 flights are on time is \( 0.0294 \).

\[ \boxed{n = 25, \, p = 0.8, \, P(X = 16) = 0.0294} \]

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