Regression and Predictions
Exercises 13–28 use the same ...

Question

Regression and Predictions

Exercises 13–28 use the same data sets as Exercises 13–28 in Section 10-1.

Find the regression equation, letting the first variable be the predictor (x) variable.

Find the indicated predicted value by following the prediction procedure summarized in Figure 10-5.

Taxis Use the distance/fare data from Exercise 15 and find the best predicted fare amount for a distance of 3.10 miles. How does the result compare to the actual fare of $15.30?

Accepted Answer

Welcome back, everyone. In this problem, a car rental company tracks the number of days cars are rented and the corresponding total rental cost. The data below were recorded for the days rented and the total cost in dollars. Find the regression equation letting days rented be the predicted variable. What is the predicted total cost for renting a car for 5 days? Now we'll need to first find the regression equation and then use that to predict the total cost for renting that car for that period of time. So let's first start by letting X, OK. The predictor variable be equal to the days rented, OK. So that X would be 1234, and 6. OK. And why be equal to the total cost, OK? That is 5590, 125, 160, and 230. Now what do we know about the regression equation? Well, recall. That the regression equation is in the form of a linear equation that says Y is going to be equal to A + BX, OK, where A is the intercept and B is the soap. No, we can find a, OK, for a regression equation or intercept because it's gonna be equal to the sum of Y values minus the sum minus B multiplied by the sum of X values divided by the number of values N, OK. And our slope is going to be equal to N multiplied by the sum of Xy values minus the sum of X multiplied by the sum of y values. All divided by n multiplied by the sum of Xsquared values minus the sum of X values all squared. So now from A and B, you realize there are a few things that we'll need to solve for a regression equation. We need the sum of Y values, OK? We need the sum of X values, and we'll also need the sum of Xsquared values and the sum of Xy values. So let's make a table for X versus Y to help us solve for those values, OK? So we have X, it's corresponding Y value, and then X curd value and or Xy values. OK, let me put that table here, OK? And you know what, I think I could even move it up just to make some space. Yes, so I could probably carry this up here. And then moves the word and a bit closer to be. Yeah, so this, this looks a little bit better, I think. Just to help me make some space uh while we work, OK. Let's just move this over a little bit. Yeah, I think this looks good. No, for our X values, we know they were 123. 4 and 6, OK. And for the corresponding Y values, it would have been 55, 90, 125. 160 and 230. Now, to get our X squared values, we'd square the X values, so that would be 149, 16, and 36. And then our XY values would be X multiplied by Y. So that's 1 multiplied by 55, 55, 2 multiplied by 90, 180, 3 multiplied by 125, 375, 4 multiplied by 16, 640, and 6 multiplied by 230 would be 1380. So now when we sum all of these respective values, OK, then we should get the sum of X values to be 16, Y value is 660, uh X2d value is 66, and Xy values 2630. So we have all the sums that we need, OK? Now we can plug them into our formula for the intercept A and the slow B to get our regression equation, OK? So, what this means then. Is that, if we go back to its formula, is going to be the sum of Y values 660 minus and we know that overall, OK, there are, well, if we go back to. There are 5 values, OK. So 660. Well, you know what guys, when we go back, remember that B is here, so we'll need to solve for B first, actually. So we need B first. So if we go to the formula for B, my apologies, it's gonna be N 5 values multiplied by the sum of Xy values, 2630. Minus the sum of X values 16 multiplied by the sum of Y values 660. All divided by n multiplied by the sum of Xquared values 66 minus the sum of x values 16 square. Now when we go ahead and solve here, we should get B. To be equal to 2590 divided by 74, which is approximately equal to 35. I know that we have the B value, we can substitute it to solve for or a value because that means A equals the sum of Y 660 minus B 35 multiplied by the sum of X 16 divided by N5. And we get that to be equal to 100 divided by 5, which is 20. So taking both of these values here for a regression equation, OK. Then let me make some space to the right here. OK. Then we can now say. That the regression equation is Y equals 20 + 35 X. That would be the regression equation for our data. I know that we have this equation, we can predict how much it will cost for 5 days. So now, the prediction for 5 days. That is where X equals 5 because X represents the number of days rented. Would be Y equals 20 + 35 multiplied by 5. That's equal to 20 + 175 and thus. OK. The predicted cost Y would be equal to $195. Therefore, the regression equation is Y E equals 20 + 35 X, and we can predict that the total cost to rent a car for 5 days would be $195. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Regression Analysis

Predictor and Response Variables

Prediction Procedure

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