Math Problem Statement

Year

Consumer

Credit

Year

Consumer

Credit   20042004

2219.52219.5

20092009

2557.52557.5

20052005

2319.82319.8

20102010

2643.12643.1

20062006

2465.02465.0

20112011

2756.02756.0

20072007

2517.92517.9

20122012

2926.32926.3

20082008

2592.12592.1

20132013

3094.23094.2

Question content area bottom

Part 1

a.

Find an equation for the least squares​ line, letting x equal the number of years since

The equation is

Yequals=enter your response herexplus+enter your response here.

​(Use integers or decimals for any numbers in the equation. Round to two decimal places as​ needed.)

Solution

To find the equation of the least squares line (also known as the line of best fit), we need to determine the relationship between the years (as xx) and the consumer credit (as YY). The least squares line is of the form:

Y=mx+bY = mx + b

where:

  • mm is the slope of the line,
  • bb is the y-intercept.

Step 1: Define the Variables

Let xx be the number of years since 2000. So for each year:

  • 2004: x=4x = 4
  • 2005: x=5x = 5
  • 2006: x=6x = 6
  • 2007: x=7x = 7
  • 2008: x=8x = 8
  • 2009: x=9x = 9
  • 2010: x=10x = 10
  • 2011: x=11x = 11
  • 2012: x=12x = 12
  • 2013: x=13x = 13

The corresponding consumer credit values YY are already given.

Step 2: Calculate the Slope (mm) and Intercept (bb)

The formulas for the slope mm and intercept bb are:

m=n(xy)xynx2(x)2m = \frac{n \sum{(xy)} - \sum{x} \sum{y}}{n \sum{x^2} - (\sum{x})^2} b=ymxnb = \frac{\sum{y} - m \sum{x}}{n}

Where:

  • nn is the number of data points.
  • xx and yy are the individual values of the independent and dependent variables respectively.

Let's calculate these step by step.

Step 3: Perform the Calculations

We have:

  • x={4,5,6,7,8,9,10,11,12,13}x = \{4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}
  • y={2219.5,2319.8,2465.0,2517.9,2592.1,2557.5,2643.1,2756.0,2926.3,3094.2}y = \{2219.5, 2319.8, 2465.0, 2517.9, 2592.1, 2557.5, 2643.1, 2756.0, 2926.3, 3094.2\}

I'll calculate the necessary sums and then determine mm and bb.The equation of the least squares line is:

Y=84.33x+1892.37Y = 84.33x + 1892.37

This equation shows that, on average, consumer credit increases by approximately 84.33 billion dollars per year after 2000, starting from an initial value of 1892.37 billion dollars.

Would you like more details on how this was calculated, or do you have any questions?

Here are some related questions you might find useful:

  1. How do you calculate the correlation coefficient for this data set?
  2. What is the significance of the slope in the context of this data?
  3. How can you use this equation to predict consumer credit in future years?
  4. What are the assumptions behind using a least squares regression line?
  5. How would the line of best fit change if we considered data beyond 2013?

Tip: In regression analysis, the slope of the line indicates the rate of change, while the intercept represents the starting value when the independent variable is zero.

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Math Problem Analysis

Mathematical Concepts

Statistics
Linear Regression
Least Squares Method

Formulas

Least squares line equation: Y = mx + b

Theorems

-

Suitable Grade Level

Advanced High School