Question

The following data represent the width (in yards) and length (in miles) of various tornadoes. $$ \begin{array}{|cc|} \hline \text { Width (yards), } \boldsymbol{w} & \text { Length (miles), } \boldsymbol{L} \\ \hline 200 & 2.5 \\ \hline 350 & 4.8 \\ \hline 180 & 2.0 \\ \hline 300 & 2.5 \\ \hline 500 & 5.8 \\ \hline 400 & 4.5 \\ \hline 500 & 8.0 \\ \hline 800 & 8.0 \\ \hline 100 & 3.4 \\ \hline 50 & 0.5 \\ \hline 700 & 9.0 \\ \hline 600 & 5.7 \\ \hline \end{array} $$ (a) Draw a scatter plot of the data, treating width as the independent variable. (b) What type of relation appears to exist between the width and the length of tornadoes? (c) Select two points and find a linear model that contains the points. (d) Graph the line on the scatter plot drawn in part (a). (e) Use the linear model to predict the length of a tornado that has a width of 450 yards. (f) Interpret the slope of the line found in part (c).

Short answer

a) Draw the scatter plot. b) The relationship appears linear. c) Linear model: L = 0.011w + 0.3. d) Graph the line on the scatter plot. e) Predicted length: 5.25 miles. f) The slope indicates an increase of 0.011 miles in length per additional yard in width.

Step-by-step solution

Draw the scatter plot

Plot the given data points on a graph with the width (w) on the x-axis and the length (L) on the y-axis. The data points to plot are (200, 2.5), (350, 4.8), (180, 2.0), (300, 2.5), (500, 5.8), (400, 4.5), (500, 8.0), (800, 8.0), (100, 3.4), (50, 0.5), (700, 9.0), and (600, 5.7).

Identify the type of relationship

Observe the scatter plot to determine the type of relationship between width and length. Check if the points form a pattern that suggests a linear, quadratic, or other type of relationship.

Choose two points for the linear model

Select two points from the scatter plot. For example, select (200, 2.5) and (500, 5.8).

Calculate the slope (m) of the line

Use the slope formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\] Where \( (x_1, y_1) = (200, 2.5) \) and \( (x_2, y_2) = (500, 5.8) \). Substituting the values, \[ m = \frac{5.8 - 2.5}{500 - 200} = \frac{3.3}{300} = 0.011 \].

Find the y-intercept (b) of the line

Use the slope-intercept form, \[ L = mw + b \] We use one of the points, for example (200, 2.5), to find b: \[ 2.5 = 0.011(200) + b \] \[ 2.5 = 2.2 + b \] \[ b = 2.5 - 2.2 \] \[ b = 0.3 \] Thus the linear model is \[ L = 0.011w + 0.3 \].

Graph the line on the scatter plot

Draw the line \( L = 0.011w + 0.3 \) on the same scatter plot from Step 1 to visually compare the linear model with the data points.

Predict the length for a width of 450 yards

Substitute \( w = 450 \) into the linear model: \[ L = 0.011(450) + 0.3 \] \[ L = 4.95 + 0.3 \] \[ L = 5.25 \] Therefore, the predicted length is 5.25 miles.

Interpret the slope

The slope of the line, \( m = 0.011 \), means that for every additional yard in width, the length of the tornado increases by 0.011 miles.

Key concepts

linear relationship

In data analysis, recognizing the type of relationship between two variables is crucial. A scatter plot helps visualize these relationships. When analyzing tornado data with width (yards) and length (miles), plotting the width on the x-axis and length on the y-axis can reveal any underlying patterns. In this exercise, the scatter plot suggests a linear relationship. This means as the tornado width increases, its length also tends to increase in a proportional manner, which forms a roughly straight line on the graph.
Linear relationships can vary:
- **Positive linear relationship:** As one variable increases, so does the other.
- **Negative linear relationship:** As one variable increases, the other decreases.
In our tornado example, we observe a positive linear relationship.
Understanding these relationships is key in predicting future values and uncovering deeper insights in your data analysis journey.

slope-intercept form

The slope-intercept form is the standard way to represent a linear equation: ewline `L = mw + b` where `L` is the dependent variable (length), `w` is the independent variable (width), `m` is the slope, and `b` is the y-intercept. ewlineTo find this form, start by identifying the slope (`m`) using the formula: ewline ewline ewline\(m = \frac{(y_2 - y_1)}{(x_2 - x_1)}\) where and ewline `L = mw + b` `y` represents lengths of tornadoes `x` represents widths of tornadoes. ewlineSubstitute the selected points We're using: on tornado dataset: These points yield: ewline\((200, 2.5\)) and \((500, 5.8)\) ewline Our slope `m: ewline = \frac{(5.8 - 2.5)}{(500 - 200)} = 0.011` Next, find the y-intercept (`b`) by rearranging the equation to solve for `b`: ewline \(L = mw + b\) Using the first point (200, 2.5): ewline\(2.5 = 0.011(200) + b\) Simplifies to b = 0.3 So, our linear model \(L = 0.011w + 0.3\) Representing the scatter plot visually highlights the linear relationship.

data analysis

Data analysis is about extracting meaningful insights from data sets. In this exercise, the tornado width and length data help students practice this valuable skill. A scatter plot is a powerful tool for visual analysis. It allows us to see patterns, trends, and potential outliers within paired data points.
Steps in data analysis include:
- **Data Collection:** Gather and compile data points accurately, such as widths and lengths.
- **Data Visualization:** Create visual representations like scatter plots to detect relationships.
- **Statistical Analysis:** Use mathematical models to quantify relationships (e.g., calculating slope and intercept in our linear model).
- **Interpretation:** Draw conclusions about the observed relationships and use them for predictive insights.
By following these steps and using scatter plots, you can make informed predictions and better understand the data. Always remember, the goal is to find and understand patterns that can inform future decisions or insights.

prediction

Prediction is the ultimate goal of many data analysis exercises. Using mathematical models, we can estimate future or unknown values based on the observed data.
In this exercise, we use the linear model to predict the length of a tornado given its width. The established linear model is: ewline ewline\(L = 0.011w + 0.3\) ewline To predict the length for a width of 450 yards, substitute the given width into the model: ewline\(L = 0.011(450)+ 0.3 = 4.95 + 0.3 = 5.25 miles\) ewline Hence, a tornado with a 450-yard width is expected to have a length of 5.25 miles.Ethics is also crucial. Predictions should only guide decision-making, not be taken as guaranteed outcomes. Never overfit using limited data for over-exact predictions.These techniques are invaluable for real-world forecasting beyond just tornado research. By refining our prediction methods, we better navigate uncertainties and improve decision-making.