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What is the difference between univariate data and bivariate data?

True or False: The least-squares regression line always travels through the point \((\bar{x}, \bar{y})\)

In a recent Harris Poll, a random sample of adult Americans (18 years and older) was asked, "When you see an ad emphasizing that a product is 'Made in America,' are you more likely to buy it, less likely to buy it, or neither more nor less likely to buy it?" The results of the survey, by age group, are presented in the contingency table below. 3 $$\begin{array}{lrrrrr} & 18-34 & 35-44 & 45-54 & 55+ & \text { Total } \\ \hline \text { More likely } & 238 & 329 & 360 & 402 & 1329 \\\\\hline \text { Less likely } & 22 & 6 & 22 & 16 & 66 \\\\\hline \begin{array}{l}\text { Neither more } \\\\\text { nor less likely }\end{array} & 282 & 201 & 164 & 118 & 765 \\\\\hline \text { Total } & 542 & 536 & 546 & 536 & 2160\end{array}$$ (a) How many adult Americans were surveyed? How many were 55 and older? (b) Construct a relative frequency marginal distribution. (c) What proportion of Americans are more likely to buy a product when the ad says "Made in America"? (d) Construct a conditional distribution of likelihood to buy "Made in America" by age. That is, construct a conditional distribution treating age as the explanatory variable. (e) Draw a bar graph of the conditional distribution found in part (d). (f) Write a couple sentences explaining any relation between likelihood to buy and age.

(a) By hand, draw a scatter diagram treating \(x\) as the explanatory variable and y as the response variable. (b) Select two points from the scatter diagram and find the equation of the line containing the points selected. (c) Graph the line found in part (b) on the scatter diagram. (d) By hand, determine the least-squares regression line. (e) Graph the least-squares regression line on the scatter diagram. (f) Compute the sum of the squared residuals for the line found in part (b). (g) Compute the sum of the squared residuals for the leastsquares regression line found in part (d). (h) Comment on the fit of the line found in part (b) versus the least-squares regression line found in part ( \(d\) ). $$ \begin{array}{rrrrrr} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 7 & 6 & 3 & 2 & 0 \\ \hline \end{array} $$

Use the linear correlation coefficient given to determine the coefficient of determination, \(R^{2} .\) Interpret each \(R^{2}\) (a) \(r=-0.32\) (b) \(r=0.13\) (c) \(r=0.40\) (d) \(r=0.93\)

(a) By hand, draw a scatter diagram treating \(x\) as the explanatory variable and y as the response variable. (b) Select two points from the scatter diagram and find the equation of the line containing the points selected. (c) Graph the line found in part (b) on the scatter diagram. (d) By hand, determine the least-squares regression line. (e) Graph the least-squares regression line on the scatter diagram. (f) Compute the sum of the squared residuals for the line found in part (b). (g) Compute the sum of the squared residuals for the leastsquares regression line found in part (d). (h) Comment on the fit of the line found in part (b) versus the least-squares regression line found in part ( \(d\) ). $$ \begin{array}{lrrrrr} \hline x & 20 & 30 & 40 & 50 & 60 \\ \hline y & 100 & 95 & 91 & 83 & 70 \\ \hline \end{array} $$

(a) By hand, draw a scatter diagram treating \(x\) as the explanatory variable and y as the response variable. (b) Select two points from the scatter diagram and find the equation of the line containing the points selected. (c) Graph the line found in part (b) on the scatter diagram. (d) By hand, determine the least-squares regression line. (e) Graph the least-squares regression line on the scatter diagram. (f) Compute the sum of the squared residuals for the line found in part (b). (g) Compute the sum of the squared residuals for the leastsquares regression line found in part (d). (h) Comment on the fit of the line found in part (b) versus the least-squares regression line found in part ( \(d\) ). $$ \begin{array}{llllll} \hline x & 5 & 10 & 15 & 20 & 25 \\ \hline y & 2 & 4 & 7 & 11 & 18 \\ \hline \end{array} $$

You Explain It! Study Time and Exam Scores After the first exam in a statistics course, Professor Katula surveyed 14 randomly selected students to determine the relation between the amount of time they spent studying for the exam and exam score. She found that a linear relation exists between the two variables. The least-squares regression line that describes this relation is \(\hat{y}=6.3333 x+53.0298\). (a) Predict the exam score of a student who studied 2 hours. (b) Interpret the slope. (c) What is the mean score of students who did not study? (d) A student who studied 5 hours for the exam scored 81 on the exam. Is this student's exam score above or below average among all students who studied 5 hours?

You Explain It! \(\mathrm{CO}_{2}\) and Energy Production The leastsquares regression equation \(\hat{y}=0.7676 x-52.6841\) relates the carbon dioxide emissions (in hundred thousands of tons), \(y,\) and energy produced (hundred thousands of megawatts), \(x,\) for all countries in the world. Source: CARMA (www.carma.org) (a) Interpret the slope. (b) Is the \(y\) -intercept of the model reasonable? Why? What would you expect the \(y\) -intercept of the model to equal? Why? (c) The lowest energy-producing country is Rwanda, which produces 0.094 hundred thousand megawatts of energy. The highest energy-producing country is the United States, which produces 4190 hundred thousand megawatts of energy. Would it be reasonable to use this model to predict the \(\mathrm{CO}_{2}\) emissions of a country if it produces 6394 hundred thousand megawatts of energy? Why or why not? (d) China produces 3260 hundred thousand megawatts of energy and emits 3120 hundred thousand tons of carbon dioxide. What is the residual for China? How would you interpret this residual?

In Problems \(17-20,\) (a) draw a scatter diagram of the data, (b) by hand, compute the correlation coefficient, and \((c)\) determine whether there is a linear relation between \(x\) and \(y\). $$ \begin{array}{rrrrrr} \hline x & 2 & 3 & 5 & 6 & 6 \\ \hline y & 10 & 9 & 7 & 4 & 2 \\ \hline \end{array} $$