Question

the total amount of consumer credit has been increasing steadily in recent years. the following table gives the total outstanding consumer credit (in billions of dollars) answer parts (a) through (a) below.

year	consumer credit	year	consumer credit
2004	2350.8	2009	2643.1
2005	2437.0	2010	2756.0
2006	2540.9	2011	2926.3
2007	2595.1	2012	3094.2

a. find an equation for the least squares line, letting x equal the number of years since 2001.
the equation is ( y = square x + square ).
(use integers or decimals for any numbers in the equation. round to two decimal places as needed.)

Explanation:

Step1: Define variables

Let $x$ = years since 2001, so:
2003: $x=2$, 2004: $x=3$, 2005: $x=4$, 2006: $x=5$, 2007: $x=6$,
2008: $x=7$, 2009: $x=8$, 2010: $x=9$, 2011: $x=10$, 2012: $x=11$
Let $Y$ = consumer credit (billions of dollars)

Step2: Calculate sums ($n=10$)

$\sum x = 2+3+4+5+6+7+8+9+10+11 = 65$
$\sum Y = 2210.5+2350.8+2437.0+2540.9+2595.1+2551.5+2643.1+2756.0+2926.3+3094.2 = 26105.4$
$\sum xY = (2×2210.5)+(3×2350.8)+(4×2437.0)+(5×2540.9)+(6×2595.1)+(7×2551.5)+(8×2643.1)+(9×2756.0)+(10×2926.3)+(11×3094.2) = 175112.3$
$\sum x^2 = 2^2+3^2+4^2+5^2+6^2+7^2+8^2+9^2+10^2+11^2 = 505$

Step3: Calculate slope $m$

Slope formula: $m = \frac{n\sum xY - \sum x \sum Y}{n\sum x^2 - (\sum x)^2}$
$m = \frac{10×175112.3 - 65×26105.4}{10×505 - 65^2} = \frac{1751123 - 1696851}{5050 - 4225} = \frac{54272}{825} ≈ 65.78$

Step4: Calculate y-intercept $b$

Intercept formula: $b = \frac{\sum Y - m\sum x}{n}$
$b = \frac{26105.4 - 65.78×65}{10} = \frac{26105.4 - 4275.7}{10} = \frac{21829.7}{10} ≈ 2182.97$

Answer:

The equation is $Y = 65.78x + 2182.97$