Large Data Sets
Exercises 29–32 use the same Appendix B ...

Question

Large Data Sets

Exercises 29–32 use the same Appendix B data sets as Exercises 29–32 in Section 10-1. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted values following the prediction procedure summarized in Figure 10-5.

Taxis Repeat Exercise 16 using all of the distance/tip data from the 703 taxi rides listed in Data Set 32 “Taxis” from Appendix B.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says the table below lists the dinner durations in minutes and the corresponding tips in dollars received by a restaurant over 10 evenings. Find the best predicted tip when the dinner duration is 82 minutes. And below the text we're given a table that has 3 columns. In the first column, we have the visit, and here we just label the visits with the values from 1 to 10, starting from 1 and going to 10. The next column is the dinner duration in minutes, and we have the values in this column of 50, 55, 60, 65, 70, 75, 80, 85, 90, and 95. And in the last column, we have the tip in dollars, and we have the values of 6.00, 6.80, 7.25, 7.80, 8.30, 8.75, 9.25, 9.80, 10.25, and 10.70. We're also given 4 possible choices as our answers. For choice A, we have $1.85. For choice B, we have $9.46. For choice C, we have $6.54 and for choice D we have $4.70. So, the first thing we need to do is actually find our least squares fit regression equation. And to do this, we'll need to actually calculate our mean and standard deviations for both our X data and our Y data, and this is best done in a table. So, here on the screen, we have created a table in the first column. We have our X values X to buy, and in this column, we have the values of 50, 55, 60, 65, 70, 75, 80, 85, 90, and 95. And if we sum all those X values, we get 725. So that means that our mean X bar is going to be equal to 725 divided by the number of visits, which is 10. And so this is going to be equal to 72.5. Our next column table contains our Y values Y to I, and here we have the values of 6, 6.8, 7.25, 7.8, 8.3, 8.75, 9.25, 9.8, 10.25, 10.7. And if we add up all these values, We get 84.9. So that means that our mean, Y bar gonna be equal to 84.9 divided by 10, which is going to be equal to 8.49. Now, in order to calculate our standard deviations, we'll need to calculate. The quantity of Xi minus X bar, as well as the quantity of Y minus Y. So in the next column in our table, we have X by minus X bar, and we calculate the values of -22.5, minus 17.5, minus 12.5, minus 7.5, minus 2.5, 2.5, 7.5, 12.5, 17.5, and 22.5. And the next column, we have the quantity of Y minus Y bar. And we, when we calculate these values, we get. -2.49, 1.69, 1.24, minus 0.69, minus 0.19. 0.26, 0.76, 1.31, 1.76, and 2.21. And so to calculate the standard deviation. In the next column, we're going to calculate the quantity of X by minus X bar in quantity squad. And doing so, we just square our column of X by minus X bar. In doing that, we get the values of 506.25, 306.25, 156.25, 56.25, 6.25, 6.25, 56.25, 156.25, 306.25, and 506.25. And when we add that column, Up, we get a value of 2,062.5. The next column has the quantity of Y minus Y bar in quantity squared. And Here we have the values of 6.2. 2.86 1.54 0.48, 0.04 0.07 0.58, 1.72, 3.1, and 4.88. And when we add that column together. The sum is equal to 21.47. So calculating the standard deviation for both our X and Y variable. Or standard deviation of X, that's gonna be S of X, and that's going to be equal to, if you recall, square root. Of the quantity of the sum of the quantity of X of I minus X bar in quantity squared, divided by the quantity of N minus 1. So this is going to be equal to the square root. Of the quantity of 2,062.5. That it's going to be divided by the quantity of 10 minus 1 in quantity. And when we evaluate this expression, this is going to be equal to. 15.1. And so we'll do the same thing for our Y variable. So we'll have S Y, and that's going to be equal to the square root of the quantity of the sum of the quantity of Y I minus Y bar in quantity squared divided by the quantity of N minus 1, and that's going to be equal to the square root of the quantity. Of 21.47. Divided by the quantity of 10 minus 1. And when we evaluate this expression, this is going to be equal to. 1.54. Now there is one last column that we need to calculate in our table here. And in our final column, we have the quantity of X of I minus X bar in quantity, multiplied by the quantity of Y minus Y bar in quantity. And we wouldn't need this for our correlation coefficient. And when we calculate this, we get the values of 56.03, 29.58, 15.5, 5.18, 0.47, 0.65, 5.7, 16.38, 30.8, and 49.73. And when we sum up that column, we get 210. So we're going to use that sum to calculate our correlation coefficient R. So recall that R. is equal to the sum of the quantity of X I minus X bar in quantity, multiplied by the quantity of YI minus Y bar. In quantity, and that is divided by The square root Of the quantity of the sum of the quantity of X of I minus X bar in quantity squared multiplied by the sum. Of the quantity of wise of I. Minus y bar in quantity squared in quantity. And so this is going to be equal to. 210, divided by D square root Of the quantity of 2,062.5. Multiplied by 21.47 in quantity. And when we evaluate this expression, this is going to be equal to. 0.998. So now we can calculate the coefficients of our regression equation. So, the first coefficient that we're going to calculate is B1, which is going to be our slope of our linear line. I recall that this is going to be equal to R multiplied by S Y divided by S X. So this is going to be equal to 0.998. Multiplied by the quantity of 1.5 or. Divided by 15.1. In quantity And when we evaluate this expression, this is going to be equal to. 0.102. And the next coefficient that we need to calculate is the knot, which is going to be your slope. I recall that this is going to be equal to. Why bar minus. B1 multiplied by X4. So this is going to be equal to. 8.49. Minus 0.102, multiplied by 72.5. And when we evaluate this expression, this is going to be equal to. 1.10. So now we can finally write down our regression equation, and so we will have that. Why hat? is equal to, and here we have B knot, which is 1.10. Plus, B1, which is 0.102, multiplied by X. And so now we just need to predict our Y value when X is equal to 82. So we will have Y hat. is equal to 1.10 plus 0.102, multiplied by 82. And when we evaluate this expression, this is going to be equal to 9.46. So this here is the answer that we're looking for for this problem. And when we compare that to our answer choices, we see that the correct answer choice corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

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Key Concepts

Regression Analysis

Predictor and Response Variables

Prediction Procedure

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