Problem 76 - Brief Applied Calculus

Question

GENERAL: Seat Belt Use Because of driver education programs and stricter laws, seat belt use has increased steadily over recent decades. The following table gives the percentage of automobile occupants using seat belts in selected years. $$ \begin{array}{lcccc} \hline \text { Year } & 1995 & 2000 & 2005 & 2010 \\ \hline \text { Seat Belt Use (\%) } & 60 & 71 & 81 & 86 \\ \hline \end{array} $$ a. Number the data columns with \(x\) -values \(1-4\) and use linear regression to fit a line to the data. State the regression formula. [Hint: See Example 8.] b. Interpret the slope of the line. From your answer, what is the yearly increase? c. Use the regression line to predict seat belt use in \(2015 .\) d. Would it make sense to use the regression line to predict seat belt use in 2025 ? What percentage would you get?

Step-by-Step Solution

Answer

The regression formula is \( y = 8.67x + 56.5 \). Seat belt use increases by ~1.734% yearly. In 2015, predicted use is 100.85%, and predicting 2025 (117.19%) is impractical.

Step 1: Assign x-values to Years

Assign the x-values as follows: for 1995 assign x = 1, for 2000 assign x = 2, for 2005 assign x = 3, and for 2010 assign x = 4. The corresponding y-values (Seat Belt Use) are 60, 71, 81, and 86.

Step 2: Use Linear Regression Formula

To find the linear regression line in the form \( y = mx + b \), calculate the slope \( m \) and y-intercept \( b \). Using the data points (1, 60), (2, 71), (3, 81), (4, 86), calculate \( m = \frac{\Sigma(xy) - \frac{\Sigma x \Sigma y}{n}}{\Sigma(x^2) - \frac{(\Sigma x)^2}{n}} \) and \( b = \frac{\Sigma y - m \Sigma x}{n} \). Compute \( m \approx 8.67 \) and \( b \approx 56.5 \), so the formula is \( y = 8.67x + 56.5 \).

Step 3: Interpret the Slope

The slope \( m \approx 8.67 \) represents the average increase in seat belt use percentage per the x-unit (each 5-year period). Thus, the yearly increase is \( 8.67 / 5 = 1.734 \% \) per year.

Step 4: Predict Seat Belt Use in 2015

Substitute \( x = 5 \) for the year 2015 into the regression equation \( y = 8.67(5) + 56.5 \). Calculate \( y = 100.85 \). Thus, the predicted seat belt use in 2015 is 100.85%, which theoretically means complete compliance, though not practical.

Step 5: Predict Seat Belt Use in 2025

Substitute \( x = 7 \) for the year 2025 into the regression equation \( y = 8.67(7) + 56.5 \). Calculate \( y = 117.19 \). Predicting values beyond full compliance is not sensible, indicated by more than 100% seat belt use.

Key Concepts

Seat Belt Use Over Time

Seat belt usage has significantly increased over the years, influenced by numerous factors like public policies and education programs. These changes are apparent through historical data representation. For example:
- In 1995, seat belt use was recorded at 60%.
- By 2000, it rose to 71%.
- In 2005, the number further increased to 81%.
- In 2010, it reached 86%.
This steady growth reflects positive outcomes from initiatives promoting road safety.
Understanding seat belt use changes involves statistical methods like linear regression to analyze trends, providing a numerical picture of how seat belt use has evolved. Data helps indicate how often people follow safety measures and can guide future improvements in public safety strategies.

Understanding the Slope in Linear Regression

In linear regression, the slope of a line is vital as it describes the rate of change between variables. Here, we investigate how seat belt use changes over defined periods.

The slope, derived as \( m \approx 8.67 \), tells us that seat belt use percentage increases by about 8.67% every five years. This increment indicates the average change in seat belt use per time unit in the study. To find the yearly increase, we divided by five years, resulting in approximately 1.734% per year.

Such interpretation helps understand the impact of policies and education on increasing seat belt adherence. It guides predictions and informs authorities about necessary measures to continue improving public safety.

Predicting Future Data

Predicting future values based on past data is an essential part of the analysis. Linear regression aids in this by providing a formula that estimates future behavior. For the given data:
- Using the equation \( y = 8.67x + 56.5 \), where \( x \) is a time unit, we predict future seat belt use.
Substituting \( x = 5 \) (for 2015), we get a prediction of 100.85%, indicating every driver would use seat belts. Exceeding 100% signals limitations in real-life applicability.
- Projecting seat belt use for 2025 using \( x = 7 \) yields 117.19%. This value surpasses practical expectations, showing prediction challenges beyond established data ranges.

Thus, while regression predicts trends effectively, out-of-range forecasts often require caution. They may not account for realistic constraints or changes in influencing factors.