Problem 78 - Applied Calculus

Question

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. $$ \begin{array}{lcccc} \hline \text { Year } & 1995 & 2000 & 2005 & 2010 \\ \hline \text { Seat Belt Use (\%) } & 60 & 71 & 81 & 85 \\ \hline \end{array} $$ 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 \(9 .]\) 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 2017\. [Hint: What (decimal) \(x\) -value corresponds to \(2017 ?]\) 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 line is \( y = 8.22x + 56.85 \). The yearly increase in seat belt use is about 1.644%. Predicting for 2017 gives impractical results (>100%). Predicting for 2025 is not sensible either, giving 114.39%.

Step 1: Assigning x-values

We are given the years 1995, 2000, 2005, and 2010, corresponding to seat belt usage percentages of 60%, 71%, 81%, and 85% respectively. For linear regression, we will assign the values 1, 2, 3, and 4 to the years 1995, 2000, 2005, and 2010. Thus, our data pairs become (1, 60), (2, 71), (3, 81), and (4, 85).

Step 2: Calculate the regression line

To find the linear regression line, we need to calculate the slope (m) and y-intercept (b) of the line using the formula: \[ y = mx + b \]The slope (m) is found using the following formula: \[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \]and the y-intercept (b) with:\[ b = \frac{(\sum y) - m(\sum x)}{n} \]Here, \(n\) is the number of data points.After calculating the values, we obtain \[ m \approx 8.22 \] \[ b \approx 56.85 \]Thus, the regression line equation is:\[ y = 8.22x + 56.85 \]

Step 3: Interpret the slope

The slope of the line, 8.22, represents the rate of increase in seat belt use percentage per assigned x-value period. Since the x-values were assigned per 5-year interval, the yearly increase would be \( \frac{8.22}{5} \approx 1.644 \% \) per year.

Step 4: Predict seat belt use in 2017

To predict the seat belt use in 2017, we need to determine the x-value for 2017. Since 1995 was assigned x=1, 2017 corresponds to the x-value:\[ x = 1 + \frac{2017 - 1995}{5} = 1 + \frac{22}{5} = 5.4 \]Plug this into the regression line:\[ y = 8.22(5.4) + 56.85 \approx 101.15 \]This prediction doesn't make practical sense as it exceeds 100%, suggesting saturation.

Step 5: Evaluate prediction for 2025

Predicting for 2025, we assign:\[ x = 1 + \frac{2025 - 1995}{5} = 7 \]Using the regression formula:\[ y = 8.22(7) + 56.85 \approx 114.39 \]Again, this prediction exceeds 100%, which is not feasible. This shows the limitation of the linear model for distant future predictions.

Key Concepts

Slope Interpretation

In linear regression, the slope is a key component that tells us how much the dependent variable changes with a change in the independent variable. Here, the slope of the regression line is 8.22. This means that for each increment in our assigned time line (which represents a 5-year period), seat belt use increases by 8.22 percentage points. However, it's essential to understand that this assignment distorts the interpretation slightly due to the intervals being 5 years.
To find out how much seat belt use increases per year, we divide the slope by 5:\( \text{Yearly Increase} = \frac{8.22}{5} = 1.644\% \).
Therefore, every year, the seat belt usage percentage increases by approximately 1.644%. This helps in understanding the trend over time and how effective various interventions have been in increasing seat belt use.

Yearly Increase

Yearly increase in seat belt use is a vital aspect to understand societal changes and improvements in safety culture. From the slope calculation, we determined a yearly increase of 1.644%. This value reflects the effectiveness of programs like driver education and stricter laws that encourage seat belt use.

To put it simply:

• Each year, there is a 1.644% increase in people using seat belts.
• Over a 5-year period, this equals an approximate 8.22% increase.

This steady increase is a positive sign, showing that policies and education have had a consistent impact on behavior. However, it’s also crucial to consider that these changes might slow or hit saturation as seat belt usage approaches maximum compliance.

Prediction Limitations

While linear regression provides a useful model for making predictions, it's essential to understand its limitations. In this case, using the regression formula to predict seat belt use in 2017 and 2025 showed practical issues.
When seat belt use was predicted for 2017, the model suggested a usage of approximately 101.15%. Similarly, in 2025, the prediction was 114.39%. These values exceed 100%, which is clearly not practical or possible.

Why does this happen?

• The linear model assumes that the trend will continue indefinitely, not accounting for saturation points.
• Real-world factors that alter trends can’t be captured solely by time-based data.
• The model works well within the range of the given time frame but becomes inaccurate when extrapolated too far into the future.

These limitations illustrate why linear regression predictions must be taken with caution, especially for periods beyond the observed data range.

Regression Formula

The regression formula derived from the seat belt usage data is a simple expression that helps us understand and predict trends. Given as \( y = 8.22x + 56.85 \), it reveals the relationship between time intervals and seat belt use percentage.

• \( y \) represents the predicted seat belt use percentage.
• \( x \) indicates the assigned number corresponding to each 5-year interval from the starting year of 1995.

To use this formula effectively:

• Determine the \( x \)-value for the desired prediction year. For example, if predicting for 2017, \( x = 5.4 \) as derived.
• Substitute \( x \) into the formula: \( y = 8.22(5.4) + 56.85 \) to compute \( y \).

This formula can be handy for assessing trends over the years specified, but always remember it's most accurate within the observed data range. Beyond that, predictions become increasingly uncertain.