Problem 17 - Algebra 2 and Trigonometry

Question

To commute to the high school in which Mr. Fedora teaches, he can take either the Line A or the Line B train. Both train stations are the same distance from his house and both stations report that, on average, they run 10 minutes late from the scheduled arrival time. However, the standard deviation for Line A is 1 minute and the standard deviation for Line B is 5 minutes. To arrive at approximately the same time on a regular basis, which train line should Mr. Fedora use? Explain.

Step-by-Step Solution

Answer

Mr. Fedora should use Line A, as it is more consistent with a smaller standard deviation.

Step 1: Understand the problem

Both train lines, Line A and Line B, average a delay of 10 minutes compared to the schedule. The important factor here is the consistency of their delays, which can be measured by the standard deviation: Line A has a standard deviation of 1 minute and Line B has a standard deviation of 5 minutes.

Step 2: Compare Standard Deviations

Standard deviation indicates the variability or spread of a set of values. A smaller standard deviation means the values are closer to the average. Line A, with a standard deviation of 1 minute, is more consistent with its delay time, whereas Line B has a higher variability with a standard deviation of 5 minutes.

Step 3: Choose the more consistent option

Since Mr. Fedora wants to arrive at the same time consistently, he should choose the option with the smaller standard deviation because it's more predictable. Line A has a smaller standard deviation, suggesting that its delays will be closer to the average more consistently than those of Line B.

Key Concepts

Probability and Statistics

Understanding probability and statistics is essential to make informed decisions based on data. In this exercise, Mr. Fedora needs to decide which train line to take to ensure consistent arrival times. Probability and statistics provide tools to analyze and interpret the lateness of trains.

Probability measures the likelihood of events occurring. In the context of train delays, it helps estimate how likely a train is to arrive late. Statistics, on the other hand, is used to collect, analyze, and summarize numerical data.

• Statistics allow us to compute important figures such as averages and standard deviations.
• These figures help us understand the typical behavior and variability of a dataset.

By calculating the standard deviation, Mr. Fedora can evaluate how often each train line deviates from its average delay.

Utilizing these concepts enables him to choose the train line with the most consistent behavior, thereby increasing his predictability in commuting.

Variability

Variability refers to how spread out or dispersed a set of numbers is. In statistics, variability is crucial because it provides insight into how much individual data points in a dataset differ from the average. In Mr. Fedora’s case, understanding variability helps determine which train will typically arrive closer to the expected delay time of 10 minutes.

A lower standard deviation, as seen with Line A at 1 minute, indicates low variability. This means the actual delay times are usually close to 10 minutes. Conversely, a higher standard deviation, like the 5 minutes for Line B, signifies greater variability.

• High variability suggests greater uncertainty in predicting actual arrival times.
• Low variability implies more reliability and consistency in the delay times.

By recognizing these differences in variability, Mr. Fedora can better predict his commute stability, opting for the train line with less fluctuation in its schedule.

Predictability

Predictability is the ability to anticipate future events based on current data. The reliability of data plays a significant role in this aspect. For Mr. Fedora, predictability is crucial because it determines his ability to plan his commute effectively.

Line A, with a standard deviation of 1 minute, offers a predictable schedule, as it shows consistent delay times around the average. This means Mr. Fedora can trust that the train will mostly follow the expected pattern, helping him schedule his morning more accurately.

• Predictability is higher with lower standard deviations.
• This reliability means fewer surprises in actual delay times.

In this scenario, taking Line A would allow Mr. Fedora to rely on established patterns, lessening the likelihood of unexpected tardiness.

Statistical Analysis

Statistical analysis involves the application of statistics to interpret and draw conclusions from data. In choosing a train line, Mr. Fedora applies statistical analysis to determine which option yields more reliable arrival times. This involves comparing two key statistical measures: average delay and standard deviation.

The mean delay provides a central tendency, showing the typical delay expected. However, it’s the standard deviation that highlights the variation from this average, helping identify which line is more dependable.

• Statistical tools are used to measure and compare consistencies in data.
• Statistical analysis simplifies complex decisions by representing data in clear numerical terms.

By employing these statistical methods, Mr. Fedora can make a more informed choice, opting for Line A that presents a smaller variance in delays. This decision demonstrates a practical application of statistical analysis to enhance daily decision-making processes.