$\backslash$table$[[,\backslash$table$[[$Natural$],[$Sciences$]],$Engineering,$\backslash$table$[[$Social$],[$Sciences$]],$Education$],[2010,5,000,7,600,7,900,5,200],[2012,5,400,8,400,8,400,4,800],[2014,5,900,9,600,8,700,4,800],[2016,6,200,9,500,9,000,5,100],[2018,6,300,10,200,8,900,4,800],[2020,6,700,10,800,8,900,4,700]]$

$($a$)$ With $x=$ the number of social science doctorates and $y=$ the number of education doctorates, use technology to obtain the regression equation. $($Round coefficients to three significant digits.$)$

$y(x)=$

Graph the associated points and regression line. $($Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.$)$

Update Graph

Student Response Graph

Student Response

Graph Description

$\backslash$table$[[5500,,,],[5400,,,],[5300,,,],[5200,,,],[5100,,,],[5000,,,],[4900,,,],[4800,,,],[4700,,,],[\backslash$table$[[4600],[7500]],8000,8500,9000]]$

$($b$)$ What does the slope tell you about the relationship between the number of social science doctorates and the number of education doctorates?

Since the slope is negative, its absolute value tells us the decrease in the number of education doctorates for each additional social science doctorate.

Since the slope is positive, its absolute value tells us the increase in the number of social science doctorates for each additional education doctorate.

Since the slope is positive, its absolute value tells us the increase in the number of education doctorates for each additional social science doctorate.

Since the slope is negative, its absolute value tells us the decrease in the number of social science doctorates for each additional education doctorate.

$($c$)$ Use technology to obtain the coefficient of correlation $r.($Round your answer to three decimal places.$)$

$r=$

Does the value of $r$ suggest a strong correlation between $x$ and $y?$

Since $|r|$ is not close to $1,$ the correlation between $x$ and $y$ is a weak one.

Since $|r|$ is close to $1,$ the correlation between $x$ and $y$ is a strong one.

Since $|r|$ is not close to $1,$ the correlation between $x$ and $y$ is a strong one.

Since $|r|$ is close to $1,$ the correlation between $x$ and $y$ is a weak one.