Consumer Credit The total amount of consumer credit has been...

Consumer Credit The total amount of consumer credit has been increasing steadily in recent years. The following table gives the total U.S. outstanding consumer credit. Source: Federal Reserve. $$\begin{array}{|l|lll|}\hline{\text { Year }} & {2004} & {2005} & {2006} & {2007} & {2008} \\ \hline \text {Consumer credit} \\ \text { (in billions } & {2219.5} & {2319.8} & {2415.0} & {2551.9} & {2592.1} \\ \text {of dollars)} \\ \hline\end{array}$$ a. Find an equation for the least squares line, letting x equal the number of years since 2000. b. Based on your answer to part a, at approximately what rate is the consumer credit growing per year? c. Use your result from part a to predict the amount of consumer credit in the year 2015. d. If this trend continues linearly, in what year will the total debt first exceed $\$ 4000$ billion? e. Find and interpret the correlation coefficient.

Consumer Credit The total amount of consumer credit has been increasing steadily in recent years. The following table gives the total U.S. outstanding consumer credit. Source: Federal Reserve.
$$\begin{array}{|l|lll|}\hline{\text { Year }} & {2004} & {2005} & {2006} & {2007} & {2008} \\ \hline \text {Consumer credit} \\ \text { (in billions } & {2219.5} & {2319.8} & {2415.0} & {2551.9} & {2592.1} \\ \text {of dollars)} \\ \hline\end{array}$$
a. Find an equation for the least squares line, letting x equal the number of years since 2000.
b. Based on your answer to part a, at approximately what rate is the consumer credit growing per year?
c. Use your result from part a to predict the amount of consumer credit in the year 2015.
d. If this trend continues linearly, in what year will the total debt first exceed $\$ 4000$ billion?
e. Find and interpret the correlation coefficient.

Key Concepts

Least Squares Regression

This concept involves finding the line that best fits a set of data points by minimizing the sum of the squared differences between the observed values and those predicted by the line. It is a fundamental statistical method used to model relationships between a dependent and an independent variable, allowing for both prediction and insight into the relationship’s strength and direction.

Slope (Rate of Change)

The slope in a linear regression model represents the rate at which the dependent variable changes with respect to the independent variable. It quantifies the amount of increase or decrease per unit change, helping to interpret how rapidly changes occur over time or across different conditions.

Intercept

In a linear model, the intercept is the value of the dependent variable when the independent variable is zero. It serves as the starting point or baseline level from which changes are measured, and it is critical for understanding the overall equation of the regression line.

Prediction and Extrapolation

Using linear models to predict values involves substituting a specific value of the independent variable into the regression equation to obtain an estimated dependent variable value. Extrapolation extends this idea further by using the model to predict values outside the range of the original data, though caution is advised as the model’s reliability may decrease beyond the observed data set.

Threshold Analysis

This concept involves using the regression equation to determine the point at which the dependent variable exceeds a specified threshold. It requires solving the equation for the independent variable when the dependent variable is set to a particular target value, providing insights into when certain milestones or critical levels might be reached.

Correlation Coefficient

The correlation coefficient is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. Ranging from -1 to 1, it indicates how closely the data points cluster along the regression line, with values near -1 or 1 denoting a strong linear relationship, and values near 0 indicating a weak or no linear correlation.

Transcript

00:01 In question number 13 about a, we want to get the least squares line equation, which is y equals mx plus b, whereas y in this case is a consumer credit in billion dollars, and x is a year since year 2000.

00:27 So we need to get the value of m and the value of b.

00:33 The value of m is equal to n sigma xy minus sigma x by sigma y over n sigma x squared minus sigma x whole squared n b is equal to sigma y minus m sigma x over n.

01:10 So to get the values of these sigmas, i have calculated that.

01:18 So let's see in a tabulated form, we have here each year and the corresponding x values.

01:27 Because year 2004 is four years since year 2000.

01:35 So we have x values, y values, given y values.

01:39 And i have calculated x squared y squared and xy values and at last i have calculated the sum for each element so now we have an n five points one two three four five so now we can calculate the n value which is equal to five multiplied by 73 ,000 567 .1 minus sigma x 30 multiplied by 13 ,098 .3 over n, which is 5, multiplied by 1190 .190.

02:46 0 .3 over n, which is 5, multiplied by 1190, which is sigma x squared, minus sigma x squared.

03:01 So now we have an m value of 97 .73.

03:12 And now we can get the b value, which is equal to 120 ,000, sorry, 0 .3 minus m value, which is 97 .73, multiplied by sigma x.

03:41 Over n which is 5 so we have an b value of 1 ,833 .28 .28.

03:56 So the least squares line equation is y equals 97 .73x plus 1 ,0003.

04:20 Now for part b at which rate is the consumer credit growing.

04:35 We can say that the rate of growing is 97 .7 .7 billion dollars per year, which is the m or the slope value for the least square, these squares line.

05:08 Okay, so this is rate of growth.

05:12 Now is part c.

05:19 We want to know in year 2015 what would be the value of why.

05:33 So we have the year.

05:35 So we can get the x value, the x values equal the year minus the year 2000...

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