Please answer these questions:

$6.$ Given a set of points, the least squares line formed by letting x be the independent variable will not necessarily be the same as the least squares line formed by letting y be the independent.

$9.$ Consider the following table of data.

x $1,1,2,2,9$

y $1,2,1,2,9$

a$.$ Calculate the least squares line and the correlation coefficient.

b$.$ Repeat part $($a$),$ but this time delete the last point.

c$.$ Draw a graph of the data, and use it to explain the dramatic difference between the answers to parts $($a$)$ and $($b$).$

$12.$ Consider the following table of data.

x $0,1,2,3,4$

y $4,1,0,1,4$

a$.$ Calculate the least squares line and the correlation coefficient.

b$.$Sketch a graph of the data.

c$.$Comparing your answers to parts $($a$)$ and $($b$),$does a correlation coefficient of $0$ mean that there is no relationship between the and values? Would some curve other than a line fit the data better? Explain.

$15.$ DECREASE IN BANKS The number of banks in the United States has

been dropping steadily since $1984,$ and the trend in recent years has

been roughly linear. The annual data for the years $2009$ through $2018$

can be summarized as follows, where x represents the years since

$2000$ and y the number of banks, in thousands, in the United States.

n $=10$x $=135$x$^2=1905$

$$y $=57\mathrm{.}232$y$^2=332\mathrm{.}048$xy$=753\mathrm{.}387$

a$.$ Find an equation for the least squares line.

b$.$ Use your result from part $($a$)$ to predict the number of U$.$S$.$ banks in the year $2025.$

c$.$ If this trend continues linearly, in what year will the number of U$.$S$.$ banks drop below $1800?$

d$.$ Find and interpret the correlation coefficient.

$18.$ CONSUMER CREDIT The total amount of consumer credit has been increasing steadily in recent years. The following table gives the total U$.$S$.$ outstanding consumer credit $($in billions of dollars$).$

Year Consumer Credit

$20133078\mathrm{.}7$

$20143287\mathrm{.}7$

$20153527\mathrm{.}0$

$20163632\mathrm{.}3$

$20173814\mathrm{.}9$

$20183979\mathrm{.}2$

a$.$ Find an equation for the least squares line, letting x equal the number of years since $2000.$

b$.$ Based on your answer to part $($a$),$ at approximately what rate is consumer credit growing per year?

c$.$ Use your result from part $($a$)$ to predict the amount of consumer credit in $2030.$

d$.$ If this trend continues linearly, in what year will the total debt first exceed $$7000$ billion?

e$.$ Find and interpret the correlation coefficient.

$21.$ JOBS The following table shows the number of new jobs gained per year in the private sector in the U$.$S$.$ in recent years.

Year $/$New Jobs $($millions$)/$ Year $/$ New Jobs $($millions$)$

$2010/1\mathrm{.}251/2015/2\mathrm{.}574$

$2011/2\mathrm{.}387/2016/2\mathrm{.}116$

$2012/2\mathrm{.}241/2017/2\mathrm{.}068$

$2013/2\mathrm{.}369/2018/2\mathrm{.}583$

$2014/2\mathrm{.}879$

a$.$ Find an equation for the least squares line, letting x be the number of years since $2000.$

b$.$ Find the correlation coefficient for these data.

c$.$ Sketch a scatterplot of these data, and include the least squares line in the graph.

d$.$Based on your answers to parts $($a$)$ through $($c$),$discuss any trends in the number of new jobs over the years between $2010$ and $2018.$

$24.$ SIZE OF HUNTING PARTIES In the $1960$s$,$the famous researcher Jane Goodall observed that chimpanzees hunt and eat meat as part of their regular diet. Sometimes chimpanzees hunt alone, while other times they form hunting parties. The following table summarizes research on chimpanzee hunting parties, giving the size of the hunting party and the percentage of successful hunts.

Number of Chimps in Hunting Party: $1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16$

Percentage of Successful Hunts: $20,30,28,42,40,58,45,62,65,63,75,75,78,75,82$

a$.$ Plot the data. Do the data points lie in a linear pattern?

b$.$ Find the correlation coefficient. Combining this with your answer to part $($a$),$ does the percentage of successful hunts tend to increase with the size of the hunting party?

c$.$ Find the equation of the least squares line, and graph it on yourscatterplot.

$27.$ POVERTY LEVELS The following table lists how poverty level income cutoffs $($in dollars$)$ for a family of four have changed over time.

Year: $1990,1995,2000,2005,2010,2015,2018$

Income: $13,359,15,569,17,604,19,971,22,315,24,257,25,701$

Let x represent the year, with corresponding to $1990,$ and y

represent the income in thousands of dollars.

a$.$ Plot the data. Do the data appear to lie along a straight line?

b$.$ Calculate the correlation coefficient. Does your result agree with your answer to part $($a$)?$

c$.$ Find the equation of the least squares line.

d$.$ Use your answer from part $($c$)$ to predict the poverty level in the year $2030.$