Problem 18 - Algebra Form and Function

Question

Table 17.18 shows the number of flights carrying a given number of passengers over a ten-week time period. $$\begin{array}{c|c|c|c|c} \hline \text { Passengers } & 0-50 & 51-100 & 101-150 & 151-200 \\ \hline \text { Flights } & 3 & 10 & 16 & 20 \\ \hline \text { Passengers } & 201-250 & 251-300 & 301-350 & \\ \hline \text { Flights } & 15 & 4 & 2 & \\ \hline \end{array}$$ (a) What is the probability, given as a percentage, that a flight picked at random from this group of flights carried more than 200 passengers? (b) What is the probability that a passenger picked at random from among this group of passengers was on a plane carrying more than 200 passengers?

Step-by-Step Solution

Answer

Answer: (a) The probability that a random flight carried more than 200 passengers is 30%. (b) The probability that a random passenger was on a plane carrying more than 200 passengers is approximately 44.76%.

Step 1: Determine the total number of flights

To find the total number of flights, sum the number of flights for each passenger range: \(3+10+16+20+15+4+2=70\).

Step 2: Calculate probability of a random flight carrying more than 200 passengers

To find this probability, divide the number of flights carrying more than 200 passengers by the total number of flights: Probability = \(\frac{\text{flights with more than 200 passengers}}{\text{total flights}}\) Probability = \(\frac{15+4+2}{70} = \frac{21}{70}=0.3\) In percentage form, the probability is: \(0.3 \times 100\% = 30\%\).

Step 3: Determine the total number of passengers for each range

To find the total number of passengers in each range, multiply the number of flights by the midpoint of the range: 0-50: \(3 \times 25 = 75\) 51-100: \(10 \times 75 = 750\) 101-150: \(16 \times 125 = 2000\) 151-200: \(20 \times 175 = 3500\) 201-250: \(15 \times 225 = 3375\) 251-300: \(4 \times 275 = 1100\) 301-350: \(2 \times 325 = 650\)

Step 4: Calculate the probability that a random passenger was on a plane carrying more than 200 passengers

Sum the total number of passengers from step 3: \(75+750+2000+3500+3375+1100+650=11450\). Now, divide the number of passengers on planes with more than 200 passengers by the total number of passengers: Probability = \(\frac{\text{passengers on planes with more than 200 passengers}}{\text{total passengers}}\) Probability = \(\frac{3375+1100+650}{11450} = \frac{5125}{11450}\approx 0.4476\) In percentage form, the probability is: \(0.4476 \times 100\% \approx 44.76\%\). The final answers are: (a) The probability that a random flight carried more than 200 passengers is \(30\%\). (b) The probability that a random passenger was on a plane carrying more than 200 passengers is approximately \(44.76\%\).

Key Concepts

Probability Calculation

Probability in statistics is a measure of the likelihood of an event occurring. It's used to make predictions based on data. In this exercise, we calculate probabilities to understand passenger distributions on flights.

To find the probability that a randomly chosen flight carried more than 200 passengers, we first determine how many flights meet this criterion. This involves calculating the ratio of these specific flights to the total number of flights. Let's break it down:

• Total flights: 70, determined by adding all flights across passenger ranges (3 + 10 + 16 + 20 + 15 + 4 + 2).
• Flights with more than 200 passengers: 21, calculated as the sum of flights where passenger numbers exceed 200 (15 + 4 + 2).
• Probability is calculated by dividing flights with more than 200 passengers by total flights: \( \frac{21}{70} \).

This results in a probability of \( 0.3 \), or 30%, when expressed as a percentage. Understanding this calculation is essential for making data-driven decisions in airline operations.

Passenger Data Analysis

Passenger data analysis involves scrutinizing the number of passengers across various flights to gain insights. This analysis helps airlines adjust strategies to increase efficiency and passenger satisfaction.

In the exercise, we analyzed the number of passengers across different flight ranges. Each range corresponds to a specific number of flights. To further analyze this data, we calculate the total number of passengers per range by multiplying each range's midpoint by its respective flight count.

• Example calculation for 0-50 passengers range is: 3 flights \( \times \) midpoint 25 = 75 passengers.
• Similar calculations are done for other ranges like 51-100, 101-150, and so on, resulting in respective passenger totals.

These calculations enable us to determine the total passengers and facilitate deeper analysis. We can then assess how many passengers traveled on flights with more than 200 passengers, offering insights into traveler patterns and preferences.

Statistical Methods

Statistical methods allow us to make sense of data by providing tools to analyze, interpret, and predict based on real-world datasets. In this context, we have used these methods to deduce probabilities.

One important statistical method is determining the total population (in this case, passengers) within specific segments and using this data for probability calculation. This involves adding calculated passengers across all ranges, thereby obtaining a comprehensive view of total passengers involved, which was 11,450.

Next, to find the probability that a randomly picked passenger was on flights carrying more than 200 passengers, we perform another calculation:

• Calculate total passengers on flights exceeding 200 by adding those from the 201-250, 251-300, and 301-350 ranges: 3,375 + 1,100 + 650 = 5,125.
• Determine the probability by dividing this figure by the total 11,450 passengers: \( \frac{5,125}{11,450} \approx 0.4476 \).

Expressed as a percentage, the probability is approximately 44.76%. Statistical methods in exercises like this help students understand how data interpretation leads to actionable insights.