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Chapter 5: Problem 23

The mean weight of luggage checked by a randomly selected tourist-class passenger flying between two cities on a certain airline is \(40 \mathrm{lb}\), and the standard deviation is \(10 \mathrm{lb}\). The mean and standard deviation for a businessclass passenger are \(30 \mathrm{lb}\) and \(6 \mathrm{lb}\), respectively. a. If there are 12 business-class passengers and 50 tourist-class passengers on a particular flight, what are the expected value of total luggage weight and the standard deviation of total luggage weight? b. If individual luggage weights are independent, normally distributed rv's, what is the probability that total luggage weight is at most \(2500 \mathrm{lb}\) ?

Short Answer

Expert verified
The expected luggage weight is 2360 lb, with a standard deviation of approximately 73.7 lb. The probability that the total weight is at most 2500 lb is 97.17%.

Step by step solution

01

Calculate the Expected Weight for Each Class

The expected total weight of luggage from the business-class passengers is given by multiplying the mean luggage weight per business-class passenger by the number of business-class passengers: \(E_{bus} = 30 \times 12 = 360 \text{ lb}\). Similarly, the expected total weight from the tourist-class passengers is \(E_{tour} = 40 \times 50 = 2000 \text{ lb}\).
02

Calculate the Total Expected Weight

Add the expected weights from both classes to find the total expected weight of luggage on the flight: \(E_{total} = E_{bus} + E_{tour} = 360 + 2000 = 2360 \text{ lb}\).
03

Calculate the Variance for Each Class

Variance for one business-class luggage is \(\sigma_{bus}^2 = 6^2 = 36\). Since variances add, the total variance for the business-class group is \(Var_{bus} = 12 \times 36 = 432\). Similarly, variance for the tourist-class luggage is \(\sigma_{tour}^2 = 10^2 = 100\), and the total variance for the tourist-class group is \(Var_{tour} = 50 \times 100 = 5000\).
04

Calculate Total Variance and Standard Deviation

Add the variances of each class to find the variance of the total luggage weight: \(Var_{total} = Var_{bus} + Var_{tour} = 432 + 5000 = 5432\). The standard deviation is the square root of the total variance: \(\sigma_{total} = \sqrt{5432} \approx 73.694\text{ lb}\).
05

Find the Probability of Total Weight Being at Most 2500 lb

We need to determine the probability that the total weight of the luggage is at most 2500 lb. Since the weights are normally distributed, this is \( P(X \leq 2500) \). We first convert 2500 lb to a standard normal variable using \( Z = \frac{X - E_{total}}{\sigma_{total}} = \frac{2500 - 2360}{73.694} \approx 1.898\). Using the standard normal table or calculator, find \(P(Z \leq 1.898)\). This value is approximately \(0.9717\), meaning there is a 97.17% probability that the total luggage weight is at most 2500 lb.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expected Value
The concept of expected value helps us determine the average outcome we can anticipate for a given situation. Here, we calculate the expected weight of luggage from business-class and tourist-class passengers.
  • For business-class: Mulitply the mean weight per passenger by the number of passengers, resulting in the expected total weight for that group.
  • The same method is applied to the tourist-class passengers.
By adding these values, we obtain the total expected luggage weight for the entire flight. This value serves as a weighted average that accounts for the number of passengers and their respective average weights.
Standard Deviation
Standard deviation is crucial in understanding how much variability exists from the expected value in a dataset. It measures the typical distance between each data point and the mean. In this context, it relates to the luggage weights. To find the standard deviation of the total luggage weight:
  • First, calculate the variance for each class and then for the combined group by adding up the individual variances.
  • The total variance is found by adding together the variances from both classes.
  • The square root of the total variance gives us the standard deviation, providing insight into the spread of total luggage weight around the expected value.
Normal Distribution
The normal distribution, often known as a bell curve, describes how the data is spread out. When data follows a normal distribution, most values cluster around the mean, with symmetry on either side.
  • In our context, individual luggage weights were assumed to follow a normal distribution.
  • Knowing this enables us to calculate probabilities about the total luggage weight by converting weights into standard normal scores, known as Z-scores.
This distribution reflects real-world scenarios where many small, random factors accumulate to create variability, such as in this luggage weight scenario.
Variance
Variance is a measure of the spread between numbers in a data set. In our problem, it quantifies how much the luggage weights differ from the average.
  • For a single piece of luggage, the variance is the standard deviation squared.
  • The total variance for each passenger class is computed by multiplying the variance of a single piece of luggage by the number of passengers.
  • Finally, the overall variance for all passenger luggage on the flight combines these two totals.
Understanding variance helps in predicting fluctuations and risks, which are crucial for managing expectations around total luggage weight.

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Most popular questions from this chapter

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