Cable TV. since 1990 , when the average monthly price for basic cable TV programming in the Unit

Cable TV. since 1990 , when the average monthly price for...

Key Concepts

Extrapolation

Extrapolation is the process of using a model to predict values outside the range of known data. In many situations, once a valid linear model is established, it is used to forecast future outcomes, assuming that the underlying linear relationship continues. Extrapolation is a key component of predictive analytics, although caution must be exercised as predictions beyond the observed data may carry increased uncertainty.

Interpretation of Intercept

The intercept, typically noted as 'b' in the slope-intercept form, represents the value at which the line crosses the vertical axis (y-axis). It provides the starting value of the dependent variable when the independent variable is zero. This concept is important for establishing baseline conditions in a variety of applications, including cost modeling and other predictive analyses.

Slope-Intercept Form

The slope-intercept form of a linear equation, represented by y = mx + b, is a foundational concept in algebra where 'm' stands for the slope of the line and 'b' is the y-intercept. This form provides a straightforward way to express a linear relationship between two variables and is especially useful for quickly graphing equations and making predictions based on the trend.

Linear Modeling

Linear modeling involves using linear equations to represent relationships between variables. In many real-world applications, such as forecasting costs or trends over time, simplifying complex situations into a linear model allows us to make predictions and understand the underlying pattern. It is an essential technique in statistics, economics, and science for analyzing data and forecasting future outcomes.

Interpretation of Slope

The slope of a line in a linear equation quantifies the rate of change between the two variables. It indicates how much the dependent variable is expected to increase or decrease for a unit change in the independent variable. Understanding the slope is crucial, as it provides insight into the strength and direction of the relationship, and is a key parameter in predictive modeling.

Transcript

00:01 In this problem we are asked to create an equation that represents the number of years since 1990, and that relates the number of years since 1990 and the cost of cable.

00:17 So in this particular case we're asked to use the variable t to represent that, to represent the number of years since 1990, and use the capital letter c here to represent the cost of cable.

00:32 And in this particular case we are given some information.

00:37 We're told that in 1990, so in 1990, let's go ahead and just write this down, so in 1990 the cost of cable was $16.

00:48 So that's about, that's how much a month someone had to pay to get cable in the united states.

00:55 And then they tell us that the cost of cable has risen about $1 .60 per year.

01:05 So every year it's going to go up $1 .60.

01:08 So in 1991, 1991 we're going to go to $17 .60.

01:16 That's going to be $16 plus $1 .60.

01:20 And this is going to continue on, and that's the information that we're given.

01:26 Well, in order for us to write an equation, notice that this is going to be a linear equation.

01:33 And so let's remind ourselves that a linear equation is going to look like y equals mx plus b, where m is the slope of our line and b is our y intercept.

01:45 And in this particular case we are not using the variables y and x.

01:50 We're going to use the variables c and t.

01:52 So essentially what we want is our equation is going to look something like c is equal to the slope m times t plus b.

02:03 And so that's kind of what we're interested in.

02:04 We need to know what this slope is.

02:08 We want to know what the slope is, and we want to know what this intercept is.

02:12 In this particular case this is going to be our c intercept instead of our y intercept.

02:18 This is going to be our c intercept.

02:19 And remember, remind ourselves that the c intercept always happens when our independent variable is equal to zero.

02:31 So in this case when t will be zero, that will give us our intercept, right? because anything times zero is zero, and zero plus b is going to give us that b.

02:41 So we'll keep that in mind because we're going to use that information to help us out.

02:45 So in order to write an equation, in order to write an equation, we need to know some information, and we've given some information.

02:53 But i like to think about these as points on our graph.

02:57 So remind ourselves that t is representing our x.

03:02 So when we have – when we write points, we write them in xy.

03:05 And in this particular case, since our x we're going to use t, and instead of y we're going to use our capital c.

03:12 So our points are going to look like tc as opposed to xy.

03:17 It's the same thing, t and c.

03:20 So let's write down a couple of points.

03:23 Okay, so here we have some data.

03:25 In 1990 it cost $16 per month.

03:29 So again, let's remind ourselves that t is the number of years since – the number of years since 1990.

03:37 So in 1990, how long has it been from 1990? well, to figure that out, we'll just take 1990 minus 1990, and that's going to give us zero.

03:48 So this first point here is going to be zero.

03:54 So we're going to have zero.

03:55 Zero years from 1990.

03:57 And how much did it cost? how much did cable cost? well, that was $16.

04:02 Okay, so that was our first point.

04:05 Then our second point is 1991.

04:10 It cost $17 .60.

04:12 So how many years from 1990 is 1991? well, that's only one year.

04:17 That's only one year from 1990.

04:19 Again, we can take 1991 minus 1990.

04:23 That will give us one.

04:24 And then what was the cost of cable in that year? it was 17.

04:28 It was about $17 .60.

04:30 So about 17 .6.

04:33 And now we've got a couple of points.

04:36 And with those points, now we can write our equation.

04:39 So remember, to write our equation, we want to find our slope, m.

04:44 Remember that m is going to be equal to your rise over your run, rise over run, or your change in y over your change in x.

04:57 How is y changing? how is x changing? so in this particular case, our slope is going to be equal to how does my y change? let's look at my y values.

05:10 17 .6 and the difference between that and 16.

05:13 So i'm going to take 17 .6 minus 16.

05:16 So look at the change in my y over my change in my x.

05:20 So my x changes.

05:22 I went from 0 to 1.

05:23 So it's 1 minus 0 will give me that change.

05:27 And we get 17 .6 minus 16.

05:31 That's going to give us our 1 .6, our 1 .6 over 1, or just 1 .6...