Question

The number of miles of U.S. urban roadways (millions of miles) for the years \(2000-2015\) is reported below. \({ }^{6}\) The years are simplified as years 0 through \(15 .\) $$ \begin{array}{l|cccccccc} \text { Miles of Road- } & & & & & & & & \\ \text { ways (millions) } & 0.85 & 0.88 & 0.89 & 0.94 & 0.98 & 1.01 & 1.03 & 1.04 \\ \hline \text { Year }-2000 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \end{array} $$ $$ \begin{array}{l|cccccccc} \begin{array}{l} \text { Miles of Road- } \\ \text { ways (millions) } \end{array} & 1.07 & 1.08 & 1.09 & 1.10 & 1.11 & 1.18 & 1.20 & 1.21 \\ \hline \text { Year }-2000 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \end{array} $$ a. Draw a scatterplot of the number of miles of roadways in the U.S. over time. Describe the pattern that you see. b. Find the least-squares line describing these data. Do the data indicate that there is a linear relationship between the number of miles of roadways and the year? Test using a \(t\) statistic with \(\alpha=.05\). c. Construct the ANOVA table and use the \(F\) statistic to answer the question in part b. Verify that the square of the \(t\) statistic in part \(\mathrm{b}\) is equal to \(F\). d. Calculate \(r^{2}\). What does this value tell you about the effectiveness of the linear regression analysis?

Short answer

#tag_title# Question: #tag_content# Based on the analysis of the U.S. urban roadways data from 2000 to 2015, describe the pattern observed in the scatterplot and explain the effectiveness of the linear regression analysis using the calculated r^2 value.

Step-by-step solution

Prepare the data

Organize the data into a table with two columns: Year and Miles of Roadways (millions). Year | Miles of Roadways (millions) ---- | --------------------------- 0 | 0.85 1 | 0.88 2 | 0.89 3 | 0.94 4 | 0.98 5 | 1.01 6 | 1.03 7 | 1.04 8 | 1.07 9 | 1.08 10 | 1.09 11 | 1.10 12 | 1.11 13 | 1.18 14 | 1.20 15 | 1.21 Next, plot this data in a scatterplot.

Describe the pattern

From the scatterplot, it appears that there is a positive linear relationship between the number of miles of roadways and the year. The number of miles of roadways seems to increase as the years progress. **Step 2: Find the least-squares line and test for a linear relationship**

Compute the least-squares line

Use the given data to calculate the least-squares regression line, which will have the form y = a + bx, where a is the y-intercept and b is the slope.

Test for a linear relationship using the t-statistic

Perform a t-test with a significance level of 0.05(α = 0.05) to determine if there is a linear relationship between the number of miles of roadways and the year. **Step 3: Construct an ANOVA table**

Create the ANOVA table

Use the provided data and the least-squares regression line to construct the ANOVA table, which will include the following columns: Source, Degrees of Freedom (df), Sum of Squares (SS), Mean Squares (MS), F-Statistic, and p-value.

Verify the relationship with the F-statistic

Compare the t-statistic squared obtained in step 2 with the F-statistic obtained from the ANOVA table and make sure they are equal in value. **Step 4: Calculate r^2**

Calculate r^2

Compute the Coefficient of Determination (r^2) using the ANOVA table.

Interpret r^2

The r^2 value represents the proportion of variability in the dependent variable (miles of roadways) that can be explained by the independent variable (year). A higher r^2 value indicates a better fit between the data and the linear regression line.

Key concepts

Scatterplot

A scatterplot is a graphical representation used in statistics to show the relationship between two quantitative variables. It's a valuable initial step in identifying the kind of association that might exist between the variables.

In the given exercise, the scatterplot would display 'Year' on the horizontal axis and 'Miles of Roadways' on the vertical axis. Each point on the plot represents a year and the corresponding miles of roadways. From the scatterplot, one can visually inspect the pattern or trend of the data. A positive upward trend, as seen in the exercise, indicates that as the years increase, so do the miles of roadways, suggesting a possible linear relationship.

Least-Squares Regression

The least-squares regression method is used to find the line that best fits the data points in a scatterplot. This line, often called the least-squares regression line or simply the regression line, minimizes the sum of the squared differences between the observed values and the values predicted by the line.

Mathematically, this line has the equation \( y = a + bx \) where \( y \) is the predicted value, \( a \) is the y-intercept, \( b \) is the slope of the line, and \( x \) is the independent variable. To assess the linear relationship, one can perform hypothesis testing on the slope coefficient \( b \) using the \( t \) statistic. If the \( t \) statistic is significant, it suggests a linear relationship exists between the variables.

ANOVA Table

The Analysis of Variance (ANOVA) table is used in regression analysis to partition the total variability of the dependent variable into components associated with the regression and the residuals.

The ANOVA table breaks down and displays the components of variation including the sums of squares for the regression (SSR), sums of squares for error (SSE), total sums of squares (SST), degrees of freedom (df), mean squares for regression (MSR), mean squares for error (MSE), the F-statistic, and the significance level (p-value). The F-statistic tests the overall significance of the model, and if it's sufficiently large, we reject the null hypothesis that there is no linear relationship.

Coefficient of Determination

The Coefficient of Determination, denoted as \( r^2 \) in statistics, quantifies the proportion of the variance in the dependent variable that is predictable from the independent variable(s) in a regression model.

An \( r^2 \) value ranges from 0 to 1 and can be interpreted as a percentage. For example, an \( r^2 \) of 0.90 would suggest that 90% of the variability in the dependent variable can be explained by the model. A higher \( r^2 \) indicates a better fit of the regression line to the data, meaning the independent variable is a good predictor of the dependent variable.

t-statistic

The \( t \) statistic is a ratio used in hypothesis testing to determine whether to reject the null hypothesis. It is used in the context of regression to test if the slope of the regression line is significantly different from zero, which would indicate a linear relationship between the dependent and independent variables.

In this exercise, after finding the least-squares regression line, the \( t \) statistic is calculated to decide if the slope coefficient is significant. With a significance level \( \alpha = 0.05 \) being typical, a \( t \) statistic beyond the critical value from \( t \) distribution would lead to rejecting the null hypothesis, suggesting evidence of a linear trend in the data.