Question

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

year	consumer credit	year	consumer credit
2004	2303.8	2009	2645.1
2005	2465.0	2010	2759.0
2006	2517.9	2011	2928.3
2007	2597.1	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.)
b. based on the answer to part (a), at approximately what rate is consumer credit growing per year?
the consumer credit is growing about $\square$ billion per year.
(round to two decimal places as needed.)
c. use the result from part (a) to predict the amount of consumer credit in the year 2016.
the amount of consumer credit in the year 2016 will be $\square$ billion.

Explanation:

Step1: Calculate x values (years since 2001)

x for 2003=2, 2004=3,...,2012=11
List (x,y): (2,2250.5),(3,2303.8),(4,2465.0),(5,2517.9),(6,2597.1),(7,2553.5),(8,2645.1),(9,2759.0),(10,2928.3),(11,3094.2)

Step2: Compute sums for least squares

n=10
$\sum x=2+3+4+5+6+7+8+9+10+11=65$
$\sum y=2250.5+2303.8+2465.0+2517.9+2597.1+2553.5+2645.1+2759.0+2928.3+3094.2=26114.4$
$\sum xy=2×2250.5+3×2303.8+...+11×3094.2=177246.3$
$\sum x^2=2²+3²+...+11²=485$

Step3: Calculate slope m

$m=\frac{n\sum xy - \sum x\sum y}{n\sum x² - (\sum x)²}=\frac{10×177246.3 - 65×26114.4}{10×485 - 65²}=\frac{1772463 - 1697436}{4850 - 4225}=\frac{75027}{625}=120.0432≈120.04$

Step4: Calculate intercept b

$b=\frac{\sum y - m\sum x}{n}=\frac{26114.4 - 120.0432×65}{10}=\frac{26114.4 - 7802.808}{10}=\frac{18311.592}{10}=1831.1592≈1831.16$

Step5: Find growth rate (slope)

Rate = m ≈120.04

Step6: Predict 2016 (x=15)

$Y=120.04×15 + 1831.16=1800.6 + 1831.16=3631.76$

Answer:

a. $Y = 120.04x + 1831.16$
b. 120.04
c. 3631.76