Problem 48 - Algebra and Trigonometry with Analytic Geometry

Question

Starting work times In a survey about what time people go to work, it was found that \(8.2\) million people go to work between midnight and 6 A.M., \(60.4\) million between 6 A.M. and 9 A.M., and \(18.3\) million between 9 A.M. and midnight. (a) Find the probability that a person goes to work between 6 A.M. and midnight. (b) Determine the probability that a person goes to work between midnight and \(6 \mathrm{~A} . \mathrm{M}\).

Step-by-Step Solution

Answer

(a) 90.5%; (b) 9.44%

Step 1: Total Number of People

Sum the total number of people surveyed: \[ 8.2 + 60.4 + 18.3 = 86.9 \, \text{million} \]

Step 2: Probability Calculation for 6 A.M. to Midnight

Calculate the probability that a person goes to work between 6 A.M. and midnight. Sum the number of people going to work in this interval: \[ 60.4 + 18.3 = 78.7 \, \text{million} \]Then divide by the total number of people: \[ \frac{78.7}{86.9} \approx 0.905 \text{ or } 90.5\% \]

Step 3: Probability Calculation for Midnight to 6 A.M.

Calculate the probability that a person goes to work between midnight and 6 A.M., using the number of people for this interval: \[ 8.2 \text{ million} \]Then divide by the total number of people: \[ \frac{8.2}{86.9} \approx 0.0944 \text{ or } 9.44\% \]

Key Concepts

Understanding Survey Analysis

Survey analysis is a crucial part of understanding large sets of data. It helps in decoding the behavior or preferences of a significant group of people. For example, when analyzing the survey about work starting times, it gives insight into when most individuals start their working day. This can be particularly useful for various industries, like transportation planning or energy management, to create more efficient schedules or allocate resources appropriately. Here’s how survey analysis works in this context:

• Gather a large sample of data from the population you're interested in studying. This data needs to be accurately representative to yield meaningful results.
• Categorize and segregate the data into meaningful intervals or groups, such as time intervals when people start work, as seen in the exercise.
• Analyze the grouped data to uncover patterns or preferences—which time frames are most popular, for instance—as highlighted by which interval has more people.

This methodical approach makes survey analysis a powerful tool to inform decision-making.

Basics of Probability Calculation

Probability calculation is a fundamental aspect of statistics that helps in assessing the likelihood of an event occurring. In this exercise, we're dealing with the probability of a person starting work during certain time intervals. To find these probabilities effectively, follow these basic steps:

• Identify the total number of events, or in this case, people—a total of 86.9 million.
• For a specific event (such as starting work between 6 A.M. and midnight), count how many people fall within that category.
• Finally, divide the number of people in the desired category by the total number of people. This division results in a probability between 0 and 1, which can also be expressed as a percentage by multiplying by 100.

In our case, calculating the probabilities for each interval provides specific insights:
The probability for the 6 A.M. to midnight interval was 90.5%, indicating most people work within this time frame. This is a significant finding from the data.

Importance of Time Intervals in Data

Time intervals in data analysis allow us to categorize events that occur over time, making it easier to understand patterns within the data.
In the given survey, the division into time intervals such as midnight to 6 A.M., 6 A.M. to 9 A.M., and 9 A.M. to midnight helps in efficiently organizing the data concerning work start times. Here’s why this approach is helpful:

• **Clarity:** By breaking down data into intervals, it's simpler to analyze and compare specific parts. This interval system allows you to see different peaks in work start times.
• **Granularity:** Detailed intervals provide a more refined view of when events occur, which can lead to better decision-making. Smaller intervals give more precise insights, whereas broader intervals may generalize excessive data.
• **Relevance:** Specific intervals can be linked to corresponding social patterns or business needs, making the data more actionable for practical uses, like scheduling shifts or services.

Using time intervals in survey data like this uncovers fundamental behavioral trends, proving their importance in designing effective solutions.