To solve this problem, we calculate the probability that the total weight of luggage for 80 passengers is between 1200 and 2400 pounds, given that the weight of luggage follows a normal distribution with a mean of 50 pounds and standard deviation of 20.63 pounds.

Step 1: Define the total weight distribution

The total weight of the luggage for 80 passengers is the sum of independent normally distributed random variables. For a normal distribution:

$\text{Mean of total weight}, \mu_{\text{total}} = n \cdot \mu = 80 \cdot 50 = 4000 \, \text{pounds}$

$\text{Standard deviation of total weight}, \sigma_{\text{total}} = \sqrt{n} \cdot \sigma = \sqrt{80} \cdot 20.63 \approx 184.23 \, \text{pounds}.$

Step 2: Standardize the values

We standardize the weights $1200$ and $2400$ using the z-score formula:

$z = \frac{x - \mu_{\text{total}}}{\sigma_{\text{total}}}$

For $x = 1200$: $z_1 = \frac{1200 - 4000}{184.23} = \frac{-2800}{184.23} \approx -15.2$

For $x = 2400$: $z_2 = \frac{2400 - 4000}{184.23} = \frac{-1600}{184.23} \approx -8.7$

Step 3: Calculate probabilities

From standard normal distribution tables or a calculator: $P(Z < -15.2) \approx 0 \quad \text{and} \quad P(Z < -8.7) \approx 0$

Step 4: Interpret the result

The probability that the total weight of the luggage for 80 passengers is between $1200$ and $2400$ pounds is essentially 0 because the calculated z-scores are so extreme (far from the mean).

Final Answer:

$\text{The probability is approximately } 0.0 \%.$

Do you want to explore similar calculations or have more details? Here are 5 related questions:

1. How would the probability change if the standard deviation was smaller?
2. What is the probability that the total weight exceeds 4500 pounds for 80 passengers?
3. How is the central limit theorem applied in this problem?
4. Can we calculate probabilities for individual luggage weights falling within a specific range?
5. How does changing the sample size (number of passengers) affect the distribution of total weight?

Tip: Use z-score tables or statistical software for quick evaluations of probabilities for extreme z-scores.