The following table shows the number of fiber-optic cable...

The following table shows the number of fiber-optic cable connections to homes in the United States from 2000 to $2004(t=0$ represents 2000$):^{76}$ $$\begin{array}{|r|c|c|c|c|c|}\hline \text { Year } t & 0 & 1 & 2 & 3 & 4 \\\hline \begin{array}{r}\text { Connections } c \\\text { (thousands) }\end{array} & 0 & 10 & 25 & 65 & 150 \\\hline\end{array}$$ a. Use technology to obtain the linear regression line and correlation coefficient $r,$ with all coefficients rounded to two decimal places, and plot the regression line and the given points. b. Does the graph suggest that a nonlinear relationship between $t$ and $c$ would be more appropriate than a linear one? Why or why not? c. Use technology to obtain the residuals. What can you say about the residuals in support of the claim in part (b)?

The following table shows the number of fiber-optic cable connections to homes in the United States from 2000 to $2004(t=0$ represents 2000$):^{76}$ $$\begin{array}{|r|c|c|c|c|c|}\hline \text { Year } t & 0 & 1 & 2 & 3 & 4 \\\hline \begin{array}{r}\text { Connections } c \\\text { (thousands) }\end{array} & 0 & 10 & 25 & 65 & 150 \\\hline\end{array}$$
a. Use technology to obtain the linear regression line and correlation coefficient $r,$ with all coefficients rounded to two decimal places, and plot the regression line and the given points.
b. Does the graph suggest that a nonlinear relationship between $t$ and $c$ would be more appropriate than a linear one? Why or why not?
c. Use technology to obtain the residuals. What can you say about the residuals in support of the claim in part (b)?

Key Concepts

Linear Regression

Linear regression is a statistical method used for modeling the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data. In this context, it involves finding the best-fitting straight line through the given data points, where the slope and intercept minimize the sum of the squared differences between the actual data values and those predicted by the line.

Correlation Coefficient

The correlation coefficient, often denoted as r, quantifies the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, with values close to 1 or -1 indicating strong linear relationships, and values near 0 suggesting a weak or no linear association. It is an important measure to evaluate how well the linear model might represent the data.

Residual Analysis

Residual analysis involves examining the differences between observed values and those predicted by the linear regression model, known as residuals. Analyzing residuals can help diagnose whether the linear model is appropriate. Patterns or systematic errors in residual plots might suggest that the relationship between the variables is not adequately captured by a linear model.

Nonlinear Relationships and Model Fit

When the plot of data or the pattern in the residuals suggests curvature or systematic deviations from linearity, it is an indication that a nonlinear model might be more suitable. Evaluating the fit of a linear model using graphing techniques and residuals helps in determining whether the model properly represents the underlying relationship, or if a more complex model should be considered.

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