91Ó°ÊÓ

Consider the following three data sets, in which the variables of interest are \(x=\) commuting distance and \(y=\) commuting time. Based on a scatter plot and the values of \(s\) and \(r^{2}\), in which situation would simple linear regression be most (least) effective, and why? $$ \begin{array}{lllrrrrr} \text { Data Set } & & 1 & & & 2 & & 3 \\ \hline & \boldsymbol{x} & \boldsymbol{y} & \boldsymbol{x} & \boldsymbol{y} & \boldsymbol{x} & \boldsymbol{y} \\ & 15 & 42 & 5 & 16 & 5 & 8 \\ & 16 & 35 & 10 & 32 & 10 & 16 \\ & 17 & 45 & 15 & 44 & 15 & 22 \\ & 18 & 42 & 20 & 45 & 20 & 23 \\ & 19 & 49 & 25 & 63 & 25 & 31 \\ & 20 & 46 & 50 & 115 & 50 & 60 \\ \hline \end{array} $$

Step by step solution

01

Generate Scatter Plots for Each Data Set

For each data set, create a scatter plot with the commuting distance ( x") on the x-axis and commuting time ( "") on the y-axis. Analyze the visual patterns to get an initial understanding of the relationship between the variables. Look for clear linear trends.

02

Calculate the Correlation Coefficient, r

Determine the correlation coefficient ( "r"") for each data set. This value indicates the strength and direction of a linear relationship between x and y. A value of "r"") close to 1 or -1 suggests a strong linear relationship, while a value close to 0 indicates a weak linear relationship.

03

Compute the Coefficient of Determination, r²

Calculate the coefficient of determination ( r²"). This measures how well the data fit a linear regression model, with values closer to 1 indicating a better fit. r²"). It provides insight into the proportion of variance in y hat can be explained by x

04

Determine Standard Error of Estimate, s

Find the standard error of the estimate ( s"). This value helps assess the accuracy of predictions made by the linear regression model. A smaller s"). suggests that the regression line better approximates the actual data points.

05

Compare and Conclude Effectiveness

Based on r²"). and s")., compare the data sets: - The most effective use of linear regression is found in the data set with the highest r²"). and the smallest s")., indicating a good fit with low prediction error. - Conversely, the least effective use of linear regression is found in the data set with the lowest r²"). and highest s")., indicating poor fit and high prediction error.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatter Plot Analysis

Scatter plot analysis is a straightforward way to visually examine the relationship between two variables. In our context, the variables involved are commuting distance ( x ) and commuting time ( y ). By plotting each data point on the graph, with x on the x-axis and y on the y-axis, we can quickly assess whether a linear relationship might exist between these variables. When you look at a scatter plot:

• If the points form a pattern that resembles a straight line, there might be a linear relationship.
• If they are scattered without any apparent form, a linear relationship is likely weak or nonexistent.

Scatter plots are the first step in understanding potential correlations, and they guide us in further quantifying these relationships with numerical measures such as the correlation coefficient.

Correlation Coefficient (r)

The correlation coefficient, denoted as \(r\), is an essential statistic that quantifies the strength and direction of a linear relationship between two variables. It values range from -1 to 1:

• \(r = 1\): perfect positive linear relationship, where both variables increase together.
• \(r = -1\): perfect negative linear relationship, where one variable increases as the other decreases.
• \(r = 0\): no linear relationship.

The closer the value of \(r\) is to 1 or -1, the stronger the linear relationship between the variables. In our data sets, calculating \(r\) gives us an immediate sense of how tightly the data fits a linear trend, but not the full picture, since it doesn't show all variance explained by one variable in terms of another.

Coefficient of Determination (r²)

The coefficient of determination, denoted as \(r^2\), is a key statistic that provides insight into how well a linear regression model explains the variation in the dependent variable (commuting time, \(y\)) based on the independent variable (commuting distance, \(x\)).

• \(r^2\) values range from 0 to 1, where a value closer to 1 suggests that a large proportion of the variation in \(y\) is explained by \(x\).
• A higher \(r^2\) value indicates a better fit of the regression model to the data.

For instance, if \(r^2 = 0.8\), this implies that 80% of the variance in commuting time is explained by the distance. It is a critical metric for determining the effectiveness of a linear regression model. When comparing data sets, the set with the highest \(r^2\) value typically indicates the most effective model for prediction.

Standard Error of Estimate (s)

The standard error of estimate, symbolized by \(s\), gauges the accuracy of the predictions made by a linear regression model. It measures the average distance that data points fall from the regression line. Here’s how it works:

• A small \(s\) value indicates that the predicted \(y\) values are close to the actual data points, reflecting precise predictions.
• A large \(s\) value suggests that predictions are widely spread around the actual data, thereby less reliable.

The \(s\) influences our decision about the model’s effectiveness. When assessing the datasets, the set with the smallest \(s\) typically corresponds to the most accurate predictive model. Hence, in choosing the best linear regression model, one should look for a combination of high \(r^2\) and low \(s\) for optimal results.