Consumer Credit The total amount of consumer credit has been increasing steadily in recent years

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

Key Concepts

Least Squares Regression

This concept involves finding the best-fitting straight line through a set of data points by minimizing the sum of the squared differences between the observed values and the values predicted by the line. It is a fundamental method in statistics used for modeling the relationship between an independent variable and a dependent variable, especially when assessing trends over time.

Slope Interpretation

In a linear regression equation, the slope represents the rate of change of the dependent variable for every one-unit increase in the independent variable. Interpreting the slope in context allows us to understand how quickly or slowly the dependent variable (such as consumer credit) is increasing or decreasing over time.

Prediction using Regression

Once the regression equation has been determined, it can be used to make predictions about future values of the dependent variable. This involves substituting the desired independent variable value into the equation, which is particularly useful for forecasting trends and making projections based on historical data.

Correlation Coefficient

The correlation coefficient is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, with values close to 1 or -1 indicating a strong linear relationship, and values near 0 indicating a weak linear relationship. This measure helps determine how well the regression line represents the data.

Transcript

00:02 So, in this problem, we are given the total amount of consumer credit in the recent years.

00:16 We are told that it's been increasing steadily and, well, we're giving this table of data, and we are asked first to, we are asked the first to find the the least squares line.

00:38 So to do that, first i organized the data that we are given in these two columns.

00:50 So here x is not the year, but the year since from 2000.

00:59 And consumer credit, which we are told here is measured in million, billions of dollars.

01:06 These are the first two columns and then we have xy the multiplication.

01:14 We have x squared and we have y squared.

01:17 And what we're actually going to use from this table is this row over here.

01:24 That is the sum of each column, color coded.

01:29 And since we have 10 entries and we'll have value 10.

01:39 So this is what i'm going to be referencing when we're doing, we're looking for the least squares line that looks like this, and we want the m and the values for m and b.

01:58 So they're defined like this, where each of these we defined in the slide before.

02:06 So we can go ahead and start plugging in these values.

02:15 So n is, and here is 10.

02:24 The sum of x y is 20, no, 228 ,000, 737.

02:44 And 37 .8.

02:51 Okay, um, so we subtract the sum of the x's that is 85 and the sum of the y's, which is 26, 20 ,000 and 80.

03:23 Point, all of that is over.

03:33 N which is still 10 the sum of the squares which is a hundred and i'm sorry 800 and 5 minus the square of the sum which is um this number 85 squared so we can go ahead and plug in plug all of this into the calculator and we get that am approximately 85 .5.

04:22 So that is the m value or the slope for the least squares line.

04:31 Now to try the y intercept we have on top the sum of the ys which is 26 ,080.

04:45 0 .4 minus this m is the one we just found that is 88 i'm sorry 85 .5 times some of x which is 85 .5.

05:17 I'll add over oops that should be n.

05:37 And in this case it's not 5 it is we go ahead and we flood that in into the calculator and we get that b is approximately 1 ,881 .2.

06:04 So now with these two values we can find the least square's line and that will be y equals this value in.

06:15 Put it in parentheses 85 .5 times x plus b value just this 1 ,8001 .2.

06:40 And here we have the least squares line for the data that we're given here.

06:51 Okay, so next we're asked based on your answer in part a at approximately what rate the consumer credit is a consumer credit growth per year.

07:06 So since, i'm sorry, this rate, this growing rate that they're talking about, it's actually this slope.

07:21 Slope.

07:23 So what this question is asking is the value of m.

07:29 So and we found earlier that m is 85 .5.

07:38 So and because x is measured in billions of dollars, and x is in years, it would be this amount of billions for a year.

08:10 So next question says, use your result from part a, that is a least squares line to predict the amount of consumer credit in 2015.

08:26 So what this number means is since we're measuring x in the number of years from 2000, x will have value 15.

08:38 And this prediction of the amount of consumer credit, it's asking for the corresponding y value.

08:48 And we're going to be using this formula with the values of m and b that we found and this x value.

09:01 So why is the value that we're looking for? m was 85 .5.

09:14 This x we just said it's 15 plus b we have said it is 1 ,885.

09:29 No, sorry, 81 .2.

09:36 So this part you can just go ahead and plug into your calculator.

09:42 Already did and i got that this is approximately 3 ,163 .7 and because um because um oh gram so um this will be the value of this is the value of we are looking for.

10:35 All righty and why is i can't see so since why is measured in billions then memory so then this would be again in billions of dollars next part says if this trend continues linearly as we are assuming in this model in what year will the total debt first succeed 4 ,000 billion.

11:09 So this number here corresponds to y.

11:14 So the value for y would be 4 ,000.

11:23 And we're asked what year it will be.

11:27 So we want to find the corresponding x value...

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