Please do the following. (a) Draw a scatter diagram...

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 $\Sigma x y$ 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. The following data are based on information from the Harvard Business Review (Vol. 72, No. 1). Let $x$ be the number of different research programs, and let $y$ be the mean number of patents per program. As in any business, a company can spread itself too thin. For example, too many research programs might lead to a decline in overall research productivity. The following data are for a collection of pharmaceutical companies and their research programs: Complete parts (a) through (e), given $\Sigma x=90, \Sigma y=8.1, \Sigma x^{2}=1420, \Sigma y^{2}=11.83, \Sigma x y=113.8,$ and $r \approx-0.973$. (f) Suppose a pharmaceutical company has 15 different research programs. What does the least-squares equation forecast for $y=$ mean number of patents per program?

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 $\Sigma x y$ 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.
The following data are based on information from the Harvard Business Review (Vol. 72, No. 1). Let $x$ be the number of different research programs, and let $y$ be the mean number of patents per program. As in any business, a company can spread itself too thin. For example, too many research programs might lead to a decline in overall research productivity. The following data are for a collection of pharmaceutical companies and their research programs:
Complete parts (a) through (e), given $\Sigma x=90, \Sigma y=8.1, \Sigma x^{2}=1420, \Sigma y^{2}=11.83, \Sigma x y=113.8,$ and $r \approx-0.973$.
(f) Suppose a pharmaceutical company has 15 different research programs. What does the least-squares equation forecast for $y=$ mean number of patents per program?

Key Concepts

Coefficient of Determination

The coefficient of determination, denoted as r², quantifies the proportion of the variance in the dependent variable that is predictable from the independent variable. It provides a measure of how well the regression line captures the data's variability, with higher values indicating a better fit.

Forecasting with Regression

Forecasting using regression involves using the established least squares equation to predict the value of the dependent variable for a given value of the independent variable. This application of the regression model allows one to estimate outcomes based on historical data and observed trends in the variables involved.

Least Squares Regression Line

The least squares regression line is the line that minimizes the sum of the squared differences between the observed values and the values predicted by the line. It is characterized by a slope and an intercept, and its equation provides the best linear prediction of one variable based on another in a linear regression model.

Summations in Regression Analysis

Summations such as ?x, ?y, ?x², ?y², and ?xy are fundamental components in regression analysis. They are used in calculating key statistics including the sample means, the slope, the intercept of the regression line, and the correlation coefficient. These sums aggregate the data and are essential for applying the formulas that underpin the least squares method.

Scatter Diagram

A scatter diagram is a graphical representation of two quantitative variables where each point represents a pair of values. It is used to visually assess the relationship and potential correlation between the variables, providing initial insights into patterns, trends, and possible outliers in the data.

Sample Correlation Coefficient

The sample correlation coefficient is a numerical measure that quantifies the strength and direction of the linear relationship between two variables. It typically ranges from -1 to 1, where values close to -1 or 1 indicate a strong linear relationship and values near zero suggest a weak or no linear association.

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