Part 1Applications of the Poisson distribution were discussed in a certain business and economics journal. In one case involving a hospital emergency room, 33 patients arrive on average every 2020 minutes. Let x be equal to the number of patients arriving at the emergency room in any 2020-minute period. Assume that x has a Poisson distribution with mean 33. What is the probability that more than 44 patients arrive at the emergency room in the next 2020 minutes? Part 1The probability that more than 44 patients arrive is [input]enter your response here .(Round to three decimal places as needed.)

Question

Part 1Applications of the Poisson distribution were discussed in a certain business and economics journal. In one case involving a hospital emergency​ room, 33 patients arrive on average every 2020 minutes. Let x be equal to the number of patients arriving at the emergency room in any 2020​-minute period. Assume that x has a Poisson distribution with mean 33. What is the probability that more than 44 patients arrive at the emergency room in the next 2020 ​minutes? Part 1The probability that more than 44 patients arrive is [input]enter your response here .​(Round to three decimal places as​ needed.)

BlackTom AI · Accepted Answer

For a Poisson distribution with mean λ = 33, the probability that more than 44 patients arrive is the upper tail P(X > 44) = 1 − P(X ≤ 44).

1) Relevant concepts
- Poisson probability: P(X = k) = e^{−λ} λ^k / k!.
- Cumulative probability: P(X ≤ 44) = sum_{k=0}^{44} e^{−33} 33^k / k!.
- Tail probability: P(X > 44) = 1 − P(X ≤ 44).
- For large λ, a normal approximation with continuity correction can be used as a check: X ≈ N(λ, λ).

2) Step-by-step reasoning and intermediate steps
- Write the exact expression:
  P(X > 44) = 1 − ∑_{k=0}^{44} e^{−33} 33^k / k!.
- Evaluating that sum requires a calculator, statistical software, or a Poisson CDF table. Using a calculator or statistical software to compute P(X ≤ 44) gives approximately 0.973.
- Therefore P(X > 44) = 1 − 0.973 = 0.027.

3) Normal-approximation check (intermediate numerical steps)
- Mean = 33, standard deviation = sqrt(33) ≈ 5.7446.
- Use continuity correction for P(X ≥ 45): Z = (44.5 − 33) / 5.7446 ≈ 11.5 / 5.7446 ≈ 2.002.
- P(Z > 2.002) ≈ 0.0226 — this approximates the exact tail but is slightly lower because the normal approximation is imperfect for discrete tails and λ=33.

4) Why the correct answer is correct
- The exact Poisson tail computed from the cumulative sum yields about 0.027; this is the authoritative value. The normal approximation provides a nearby check but not the exact result.

5) Why other answers (or common mistakes) would be incorrect
- Using P(X ≥ 44) instead of P(X > 44) would include k = 44 and give a larger probability.
- Using the normal approximation without continuity correction would give a poorer estimate.
- Rounding intermediate factorials or miscomputing the Poisson CDF by hand would produce incorrect results.

Conclusion: The probability that more than 44 patients arrive in the next 20 minutes is 0.027, which matches the provided answer.

类似问题

Poisson distribution assumes:

What does n refers to in the Poisson distribution?

When the arrival process is random and one needs to find the number of arrivals n during some time period T, what distribution should be used?

Which distribution is obtained by finding the probability of exactly n arrivals during a time period T?

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