Problem 60 - Applied Calculus

Question

Read the Application Preview on pages \(3-4\). Evaluate the regression line \(y=-0.356 x+257.44\) at \(x=720\) and at \(x=722\) (corresponding to the years 2620 and 2622). Does the formula give reasonable times for the mile record in these years? [Moral: Linear trends may not continue indefinitely.]

Step-by-Step Solution

Answer

The regression line gives unreasonable mile times for years 2620 and 2622.

Step 1: Evaluate at x=720

To evaluate the regression line at \(x=720\), substitute \(x = 720\) into the equation \(y = -0.356x + 257.44\). \[y = -0.356(720) + 257.44\]Calculate the result:\[y = -256.32 + 257.44 = 1.12\]

Step 2: Evaluate at x=722

To evaluate the regression line at \(x=722\), substitute \(x = 722\) into the equation \(y = -0.356x + 257.44\).\[y = -0.356(722) + 257.44\]Calculate the result:\[y = -257.032 + 257.44 = 0.408\]

Step 3: Analyze the results

The calculated times for \(y\) at \(x = 720\) is \(1.12\) minutes and at \(x = 722\) is \(0.408\) minutes. These times are unrealistically low, suggesting that the linear regression model does not accurately predict the mile record in the given years.

Key Concepts

Regression Line Evaluation

Evaluating a regression line involves substituting specific values of the independent variable into the equation to get predicted values of the dependent variable. In this exercise, the regression line given by the equation \( y = -0.356x + 257.44 \) was evaluated at specific points. These points, \( x = 720 \) and \( x = 722 \), correspond to future years. By plugging these \( x \) values into the equation, we obtained mile times as \( y = 1.12 \) minutes and \( y = 0.408 \) minutes, respectively.
This process is important for assessing whether the linear regression model provides realistic predictions across different scenarios and time points. However, as seen in this example, the predicted times were too low, indicating that the regression line generated does not hold true for these future years. This calls for careful analysis of the assumptions made during model development and whether the relationships hold over extended projections.

Linear Models

Linear models are foundational tools in statistics for predicting and understanding relationships between variables. They assume a linear relationship between the input (independent) variable \( x \) and the output (dependent) variable \( y \). A basic linear model has the form \( y = mx + b \), where \( m \) represents the slope, showing how much \( y \) changes for a unit change in \( x \), and \( b \) is the y-intercept, indicating the value of \( y \) when \( x \) is zero.

• The slope \( (-0.356) \) in our example suggests a decrease in the dependent variable for every unit increase in \( x \).
• The intercept \( 257.44 \) signifies where the line would cross the y-axis.

In this exercise, the linear model was applied to project mile times in the future, but the predictions quickly became unrealistic, showing a limitation of linear models over long-term forecasts. Linear models are powerful, but their simplifying assumptions mean they may not always provide feasible predictions for all types of data or time spans.

Trend Analysis

Trend analysis involves examining data over time to predict future values. Linear regression can be part of trend analysis by extending a discovered linear trend into future observations. In our case, applying the regression line to future \( x \) values aims to continue the trend observed in past data.
However, it's crucial to recognize that real-world phenomena, like mile running records, may not always stay linear over time. Factors such as technological advances, biological limits, and training innovations can alter progress rates.

• The linear trend may suggest consistent improvement, but unforeseen variables often cause deviation from the trend.
• In this exercise, the drastic decrease in mile time to less than half a minute highlights the fallacy of assuming indefinite continuation of past trends.

Effective trend analysis requires regular reevaluation of the model's relevance and assumptions, ensuring that predictions remain in alignment with newer and potentially non-linear influences.