Question 1172799: Delta International delivers approximately one mil-
lion packages a day between East Asia and the United
States. A random sample of the daily number of pack-
age delivery failures over the past six months pro-
vided the following results: 15, 10, 8, 16, 12, 11, 9, 8,
12, 9, 10, 8, 7, 16, 14, 12, 10, 9, 8, 11.
unusual about the operations during these days and,
thus, the results can be considered typical. Using these
data and your understanding of the delivery process
answer the following:
a. What probability model should be used and why?
b. What is the probability of 10 or more failed deliv-
eries on a typical future day?
c. What is the probability of less than 6 failed deliveries?
d. Find the number of failures such that the probabil-
ity of exceeding this number is 10% or less.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let's analyze this problem step-by-step.
**a. Probability Model**
* **Poisson Distribution:**
* The Poisson distribution is the appropriate probability model here.
* **Why?**
* We are dealing with the number of occurrences (delivery failures) within a fixed interval (a day).
* The events (failures) are assumed to be independent.
* The average rate of failures is relatively small compared to the total number of deliveries (one million packages).
* The events occur randomly.
**b. Probability of 10 or More Failed Deliveries**
1. **Calculate the Mean (λ):**
* Sum the data: 15 + 10 + 8 + 16 + 12 + 11 + 9 + 8 + 12 + 9 + 10 + 8 + 7 + 16 + 14 + 12 + 10 + 9 + 8 + 11 = 215
* Divide by the number of data points (20): 215 / 20 = 10.75
* λ = 10.75
2. **Poisson Probability Formula:**
* P(X = k) = (e^(-λ) * λ^k) / k!
* Where:
* X = number of failures
* k = specific number of failures
* λ = mean (10.75)
* e ≈ 2.71828
3. **Calculate P(X ≥ 10):**
* P(X ≥ 10) = 1 - P(X < 10)
* P(X < 10) = P(X = 0) + P(X = 1) + ... + P(X = 9)
* It is easier to use a calculator or statistical software to calculate this.
* Using a calculator or software, we find that P(X<10) = 0.4633.
* P(X ≥ 10) = 1-0.4633 = 0.5367.
* Therefore, the probability of 10 or more failed deliveries is approximately 0.5367.
**c. Probability of Less Than 6 Failed Deliveries**
1. **Calculate P(X < 6):**
* P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
* Again, using a calculator or statistical software:
* P(X < 6) = 0.0579.
* Therefore, the probability of less than 6 failed deliveries is approximately 0.0579.
**d. Number of Failures with 10% or Less Probability of Exceeding**
1. **Find k such that P(X > k) ≤ 0.10:**
* This is equivalent to finding k such that P(X ≤ k) ≥ 0.90.
* We need to use a Poisson cumulative distribution table or a calculator/software to find the value of k.
* Using statistical software, we find that when k=15, P(X<=15) = 0.9079. When k=14, P(X<=14)=0.8494.
* Therefore the number of failures is 15.
**Summary:**
* **a.** Poisson distribution
* **b.** Approximately 0.5367
* **c.** Approximately 0.0579
* **d.** 15 failures
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