Chapter 14.4, Problem 36E

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Chapter
Section

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Student loans The following table shows the balance of federal direct student loans (in billions of dollars) for selected years from 2011 and projected to 2023.(a) Find the linear regression equation for the federal direct student loan balance as a function of the years past 2010. Report the model with three significant digit coefficients.(b) What does the reported model predict for this balance in 2025?(c) Write a sentence that interprets the slope of your linear regression equation. Years Amount ($billion) 2011 702 2013 940 2015 1220 2017 1500 2019 1775 2021 2000 2023 2250 Source: U.S. Office of Management and Budget (a) To determine To calculate: The linear regression equation for the federal direct student loan balance as a function of the years past 2010. Report the model with 3 significant digit coefficients. The balance of the federal direct student loans (in billions of dollars) for selected is tabulated below.  Years Amount ($ billion) 2011 702 2013 940 2015 1220 2017 1500 2019 1775 2021 2000 2023 2250
Explanation

Given Information:

The balance of the federal direct student loans (in billions of dollars) for selected is tabulated below.

 Years Amount ($billion) 2011 702 2013 940 2015 1220 2017 1500 2019 1775 2021 2000 2023 2250 Formula used: The equation of the best fit line by linear regression that for the given data points (x1,y1),(x2,y2),......,(xn,yn) is y^=a+bx, Where b=âˆ‘i=1nxiâˆ‘i=1nyiâˆ’nâˆ‘i=1nxiyi(âˆ‘i=1nxi)2âˆ’nâˆ‘i=1nxi2, and a=âˆ‘i=1nyiâˆ’bâˆ‘i=1nxin. Calculation: Consider the table,  Years Amount ($ billion) 2011 702 2013 940 2015 1220 2017 1500 2019 1775 2021 2000 2023 2250

Let xi denote the years past 2010 for 1<i<7.

That is, x1=1,x2=3,x3=5,x4=7, x5=9,x6=11,x7=13.

Let yi denote the amount ($billion) for 1<i<7. That is, y1=702,y2=940,y3=1220,y4=1500, y5=1775,y6=2000,y7=2250. Since, the number of data points is 7. Thus, n=7. The value of b for a best fit line from a linear regression is, b=âˆ‘i=1nxiâˆ‘i=1nyiâˆ’nâˆ‘i=1nxiyi(âˆ‘i=1nxi)2âˆ’nâˆ‘i=1nxi2, Substitute 7 for n in b=âˆ‘i=1nxiâˆ‘i=1nyiâˆ’nâˆ‘i=1nxiyi(âˆ‘i=1nxi)2âˆ’nâˆ‘i=1nxi2, (b) To determine To calculate: The balance in 2025 from the reported model. The balance of the federal direct student loans (in billions of dollars) for selected is tabulated below.  Years Amount ($ billion) 2011 702 2013 940 2015 1220 2017 1500 2019 1775 2021 2000 2023 2250

(c)

To determine

The interpretation of the slope of the best fit line for the balance of the federal direct student loans (in billions of dollars) for selected is tabulated below.

 Years Amount (\$ billion) 2011 702 2013 940 2015 1220 2017 1500 2019 1775 2021 2000 2023 2250

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