For Problems 7-18, please do the following. (a) Draw a...

For Problems $7-18$, please do the following. (a) Draw a scatter diagram displaying the data. (b) Verify the given sums $\Sigma x, \Sigma y, \Sigma x^{2}, \Sigma y^{2}$, and $\sum x y$ and the value of the sample correlation coefficient $r$. (c) Find $\bar{x}, \bar{y}, a$, and $b$. Then find the equation of the least-squares line $\hat{y}=a+b x$ (d) Graph the least-squares line on your scatter diagram. Be sure to use the point $(\bar{x}, \bar{y})$ as one of the points on the line. (e) Interpretation Find the value of the coefficient of determination $r^{2}$. What percentage of the variation in $y$ can be explained by the corresponding variation in $x$ and the least-squares line? What percentage is unexplained? Answers may vary slightly due to rounding. Basketball: Fouls Data for this problem are based on information from STATS Basketball Scoreboard. It is thought that basketball teams that make too many fouls in a game tend to lose the game even if they otherwise play well. Let $x$ be the number of fouls that were more than (i.e., over and above) the number of fouls made the opposing team made. Let $y$ be the percentage of times the team with the larger number of fouls won the game. $$ \begin{array}{l|rrrr} \hline x & 0 & 2 & 5 & 6 \\ \hline y & 50 & 45 & 33 & 26 \\ \hline \end{array} $$ Complete parts (a) through (e), given $\Sigma x=13, \Sigma y=154, \Sigma x^{2}=65$, $\Sigma y^{2}=6290, \Sigma x y=411$, and $r \approx-0.988$. (f) If a team had $x=4$ fouls over and above the opposing team, what does the least-squares equation forecast for $y$ ?

For Problems $7-18$, please do the following.
(a) Draw a scatter diagram displaying the data.
(b) Verify the given sums $\Sigma x, \Sigma y, \Sigma x^{2}, \Sigma y^{2}$, and $\sum x y$ and the value of the sample correlation coefficient $r$.
(c) Find $\bar{x}, \bar{y}, a$, and $b$. Then find the equation of the least-squares line $\hat{y}=a+b x$
(d) Graph the least-squares line on your scatter diagram. Be sure to use the point $(\bar{x}, \bar{y})$ as one of the points on the line.
(e) Interpretation Find the value of the coefficient of determination $r^{2}$. What percentage of the variation in $y$ can be explained by the corresponding variation in $x$ and the least-squares line? What percentage is unexplained? Answers may vary slightly due to rounding.
Basketball: Fouls Data for this problem are based on information from STATS Basketball Scoreboard. It is thought that basketball teams that make too many fouls in a game tend to lose the game even if they otherwise play well. Let $x$ be the number of fouls that were more than (i.e., over and above) the number of fouls made the opposing team made. Let $y$ be the percentage of times the team with the larger number of fouls won the game.
$$
\begin{array}{l|rrrr}
\hline x & 0 & 2 & 5 & 6 \\
\hline y & 50 & 45 & 33 & 26 \\
\hline
\end{array}
$$
Complete parts (a) through (e), given $\Sigma x=13, \Sigma y=154, \Sigma x^{2}=65$, $\Sigma y^{2}=6290, \Sigma x y=411$, and $r \approx-0.988$.
(f) If a team had $x=4$ fouls over and above the opposing team, what does the least-squares equation forecast for $y$ ?

Key Concepts

Scatter Diagram

A scatter diagram is a graphical representation in which two variables are plotted along two axes, allowing us to visually assess the type and strength of the relationship between them. It is particularly useful for identifying potential trends, patterns, and outliers in data, and is the first step in performing regression analysis.

Summation Statistics in Regression

Summation statistics, including the sums of the x values, y values, squares of x, squares of y, and the products of x and y, are the building blocks in regression analysis. These sums are used to compute key measures such as the means, variances, covariances, and to ultimately derive the formulas for the correlation coefficient and the parameters (slope and intercept) of the least-squares regression line.

Correlation Coefficient

The correlation coefficient quantifies the strength and direction of the linear relationship between two variables. Ranging from -1 to 1, a value close to -1 or 1 indicates a strong linear relationship (negative or positive, respectively), while a value near 0 suggests little or no linear relationship.

Least-Squares Regression Line

The least-squares regression line is a straight line that minimizes the sum of the squared residuals (the differences between the observed values and the values predicted by the line). It is determined by calculating the best estimate for the slope and y-intercept using the averages and summation statistics of the data, providing a model for predicting the dependent variable from the independent variable.

Coefficient of Determination

The coefficient of determination, denoted as r², represents the proportion of the variance in the dependent variable that is predictable from the independent variable. It helps in assessing the explanatory power of the regression model by indicating the percentage of total variation that is accounted for by the model.

Forecasting using the Regression Equation

Forecasting using the regression equation involves inserting a specific value for the independent variable into the least-squares regression equation to estimate the corresponding value of the dependent variable. This predictive capability is a fundamental benefit of regression analysis, allowing for practical decision-making based on historical data.

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