The table below shows stopping distances in feet for a car tested 3 times at each of 5 speeds. W

The table below shows stopping distances in feet for a car...

Key Concepts

Extrapolation and Prediction Confidence

Extrapolation involves predicting values outside the range of the observed data and carries inherent risks because the model may not hold under conditions not represented in the sample. Establishing confidence in predictions requires understanding the scope of the data range, the model's fit, and any potential issues with the assumptions made during modeling. Predictions within the observed range are generally more reliable than those made via extrapolation.

Quadratic (or Curvilinear) Relationships

A quadratic or curvilinear relationship occurs when the rate of change in the dependent variable increases or decreases in relation to the independent variable, often modeled as a power function (e.g., the square of a variable). When physical or empirical principles suggest such behavior, incorporating a quadratic term into the model can capture the accelerating or decelerating trend observed in the data, leading to a more accurate representation.

Nonlinearity in Regression

This concept refers to recognizing when a simple linear model does not capture the true relationship between the independent and dependent variables. In many cases, observed data can curve or change rate, indicating that the assumption of a constant rate of change is violated. The identification of nonlinearity guides analysts to consider alternative models that better represent the underlying structure of the data.

Data Transformation

Data transformation involves applying a mathematical change to the variables with the aim of achieving linearity in the relationship. By transforming either the independent variable, the dependent variable, or both, one can often straighten a curvilinear scatterplot, making the relationship more amenable to linear regression techniques. Common transformations include taking logarithms, square roots, or squaring values depending on the shape of the data pattern.

Transcript

00:01 All right.

00:01 So for number 15 were given, um, speeds on stop stopping distances and feet and so we're going to want to.

00:12 First of all, look at what this model looks like.

00:16 So i am a big fan of dust most dot com.

00:19 So that's how i'm going to choose to show you how to do this.

00:22 So it is a website that is great.

00:24 It's free, it's awesome.

00:26 And it will do so much for you to be able to help you be able to get through everything.

00:32 So the first thing you're going to do is go to desk most dot com and you're going to one input your speed as your ex ones and your stop distances as your why one.

00:42 So you go to desk most's, and this is what your screen will look like when you first enter dust most dot com.

00:50 So you're gonna click, start graphing, and this is what a typical starting spot will be.

01:02 So you're gonna want to click ah, this plus sign right here, and this plus sign gives us options to do an expression and note a table full there.

01:12 We want to do a table.

01:13 Since this is what we're given.

01:15 So this is where you start to type in your point.

01:21 So you have 20.

01:25 You have 64.

01:27 Now notice how that was 2064 then there was calm.

01:30 A 62 59.

01:32 That means there's three points there.

01:34 2064 2062 2059.

01:41 Now, for the sake of efficiency in time, i went ahead and and put it all those on a different separate tab.

01:49 So we'll take a look at all those typed up.

01:51 Remember, you can pause at any point in time, so that way you can be able to do this step by step with me.

01:57 So these are all the all the ordered pairs that we need and notice when you do desk most.

02:03 You're not seeing anything yet, and that's because of our scale.

02:07 So i will show you how to adjust your scale.

02:10 Now, you could either zoom out or zoom in, but this little graph settings indicate icon right here.

02:17 We'll let you adjust your scale.

02:19 So we want to adjust the x axis between 20 and 60.

02:22 So my suggestion is to do 10 and 72 so we can see everything.

02:29 And then our y axis ranges from 64 all the way up to to, uh, teoh to to set 3 21 sorry, 21.

02:40 So we're figuring.

02:41 Okay, lets go from zero teoh.

02:46 Well, say 3 50 just so that we again we can see everything's we have to adjust both.

02:50 Now, when we do that, we click out and we can see our graph.

02:56 So with this, um, our image ah, looks like it's a strong, positive, somewhat straight linear situation.

03:08 But when we want to see whether or not there's a residual pattern or not, that's when we're gonna want input, our line of best fit.

03:17 So here's how we go and do that.

03:18 So over here, underneath your table, you're gonna want to start a fresh little box.

03:24 It'll be box, too, and you're always gonna want to do this equation.

03:28 So our table is x one y one.

03:32 So you're gonna type the y equals mx plus b slope intercept form.

03:38 But you gotta do that with terms of what we've already having reference here.

03:42 So i'm gonna do why one you just type, why one and it automatically does that little one for you, and then you're gonna look for that little squiggly button on the upper left corner keyboard going to shift in that button that's called a tilde.

03:56 So you got why one tilled us that's like are equal sign.

03:59 And it's gonna reference now that we need to do em for slow x one for our exes, plus b.

04:06 So you're gonna always type this in whenever you have these types of problems and it gives you so much information, it gives you the line of best fit with the slope in the intercept, and then it gives you the are square to show what kind of confidence level we would want to be at.

04:22 So right now we're at 96 or almost 97% confidence in variability.

04:28 But we definitely want to look at this residuals just to make sure that it's not curve or shows a pattern of some kind.

04:36 So we're gonna go check the residuals now, so i click plot and i noticed we're not gonna see it right away.

04:44 And that's because they're all down here.

04:47 So we're gonna want teoh.

04:49 Either zoom in if we can or re adjuster scale.

04:53 But just looking at this from this perspective, i see a curb.

04:57 So the answer for part a is asking us to create a model and see if free expression is needed.

05:06 And, um, based off of that, um, is this appropriate as is and the answer will be no...

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