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Let's go through the question step by step:

### Part (a):
The problem describes a Poisson distribution where the probability of having \( x \) people entering a security check-in line in a 15-minute interval is modeled by the equation:
\[
P(X = x) = \frac{e^{-16.4}(16.4)^x}{x!}, \quad x = 0, 1, 2, \dots
\]
Where \( 16.4 \) represents the average number of people entering the security check-in line in a 15-minute interval.

**Answer for (a):** The correct explanation for \( 16.4 \) in the model is that it represents **the average number of people who enter a security check-in lineup every 15 minutes**. 

Thus, the correct answer is **Option A.**

### Part (b):
We are asked to compute the probability that 18 people enter the security check-in lineup in a 15-minute interval. The equation for this is already provided. 

The Poisson probability formula is used:
\[
P(X = x) = \frac{e^{-16.4}(16.4)^x}{x!}
\]
For \( x = 18 \), we can compute the probability using the formula.

### Part (c):
We need to compute the probability that at least 5 people enter the lineup in a 5-minute interval. Since the original rate is for a 15-minute interval, we need to adjust the rate to reflect a 5-minute interval.

The adjusted rate for 5 minutes would be:
\[
\lambda_{\text{5min}} = \frac{16.4}{3} \approx 5.467
\]
Now, using the Poisson distribution with this new rate, we can calculate the cumulative probability for \( X \geq 5 \) for a 5-minute interval.

### Part (d):
The problem asks for the probability that the number of people entering the line is between 15 and 18, inclusive. To compute this, we would sum the individual Poisson probabilities for \( X = 15, 16, 17, 18 \) with the given rate for 15-minute intervals.

If you need any assistance with the specific calculations for parts (b), (c), or (d), feel free to let me know! Would you like me to walk through the steps for any of these parts?

The number of people entering a security check-in lineup in a 15-minute interval at a medium-sized airport can be modeled by the Poisson distribution: P(X = x) = (e^(-16.4) * 16.4^x) / x!, where x = 0, 1, 2, .... Compute the following:
(a) What does 16.4 represent in the model?
(b) Compute the probability that 18 people enter the lineup in 15 minutes.
(c) Compute the probability that at least 5 people enter the lineup in a 5-minute interval.
(d) Given that between 15 and 18 people entered, find the probability that 17 people entered.

Learn how to analyze a Poisson distribution for modeling the number of people entering a security check-in in a 15-minute interval. This detailed solution covers probability computations, conditional probabilities, and the significance of the mean value 16.4. Suitable for advanced high school and college students.

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