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 15 using all of the time/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 to find the regression equation, letting the first variable be the predictor X variable, then determine the indicated predicted value. Use the data from 10 hair salon visits listed below. Suppose the appointment time is 75 minutes. How does the predicted tip compare to the actual tip of $12.5? And below the problem, we're given a table, where in the first column, we have the visit, which is numbered 123456789, and 10. The second column is the appointment time in minutes, and it has the values of 40, 45, 50, 60, 65, 70, 75, 80, 85, and 90. And in the last column we have tip in dollars, and here we have 7.50, 8.25, 8.90, 10.50, 11.00, 11.80, 12.25, 13.00, 14.20, and 15.40. We're asking for possible choices as our answers. For choice A, We have $10.06. The predicted value is much lower compared to the actual value. For choice B, we have $24.65. The predicted value is much higher compared to the actual value. For choice C, we have $17.50 and the predicted value is much higher compared to the actual value. And for choice D, we have $12.63. The predicted value is very close to the actual value. So to begin to answer this problem, we first need to find our regression equation. And to do this, we need to first identify the variables. And so for our predictive variable, which is going to be X, we're going to be using using our appointment time. And for the response variable Y, we're going to have our tip. Now, in order to compute our regression, we'll first need to compute the mean and standard deviation for each of our variables, and this is best done in a table. So, on the screen, we have a table. In the first column, we have our X variables. Xay, and it has the values of 40, 45, 50, 60, 65, 70, 75, 80, 85, and 90. And when we sum all of those extra values up, we get 660. And that means that our average, or mean X bar is going to be equal to 660 divided by 10, which is equal to 66. In the next column, we have Y I, so these are our Y values. And we have the values of 7.5, 8.25, 8.9, 10.5, 11, 11.8, 12.25, 13, 14.2, and 15.4. And when we add up all of our Y values, we get 112.8. And so, taking that sum and divided by 10 to get our mean, Y bar is going to be equal to 11.28. So, in order to calculate our standard deviation, we will need to calculate the difference between our data points and the mean. So, in the next column, we'll calculate XI minus X bar, and for each of our values, we get -26, 21, 16, 6149, 1419, and 24. The next column we'll calculate Y I minus Y bar, and we get for the values for each of our points, we get minus 3.78, minus 3.03, minus 2.38, minus 0.78, minus 0.28. 0.52, 0.97, 1.72, 2.92, and 4.12. In the next column, we'll calculate the quantity of Xabi minus X bar squared and do this for each of our data points we get 676, 441, 256, 36, 1, 1681. 196361. And 576. And if we sum all of these values up, we get 2640. Now the next column, we have the quantity of Y minus Y bar in quantity squared. And so calculating these for each of our data points, we get 14.29, 9.18, 5.66, 0.61, 0.08, 0.27, 0.94, 2.96, 8.53, and 16.97. And again, adding all those values up, we get a value of 59.49. So now we have all the information that we need to calculate our standard deviation. So, for our X variable, we'll calculate its standard deviation S of X, and recall that that's going to be equal to the square root. Of the quantity, which of the sum of the quantity of X of I minus X bar in quantity squared, divided by the quantity of N minus 1. And so this is going to be equal to the square root of the quantity, and we calculated. The sum of X5 minus X bar, and that quantity being squared, and that is going to be 2640. Divided by the quantity of 10 minus 1. And when we evaluate this expression, this is going to be equal to. 17 13. We'll do something similar for our Y variables, so we'll calculate the standard deviation as of 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. So this is going to be equal to the square root. Of the quantity of 59.49, divided by the quantity of 10 minus 1. And when we evaluate this expression, we get 2.57. So, now we have all the information that we need to begin calculating quantities for our regression. And so, the first quantity that we're going to calculate is our Correlation coefficient. So we call for a correlation coefficient, we have R. is equal to the sum of quantity of X of I minus X bar. In quantity, multiplied by the quantity of Y I minus Y bar. In quantity, 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 Y minus Y bar in quantity squared in quantity. So we now need to calculate this sum in our numerator, and this is best getting done in our table. And so in the last column of our table, we'll have the quantity of X by minus X bar in quantity multiplied by the quantity of Y minus Y bar in quantity. And we'll calculate this for each of our data points. In doing so, we will get 98.28, 63.63. 38.08, 4.68. 2, 0.28, 2.08. 8.73, 24.08, 55.48, and 98.88. And when we sum all of those values, we get 394.2. So that means that R is going to be equal to. 394.2. Divided by The square root Of the quantity. Of 2640 multiplied by 59.49 in quantity. And evaluating this expression, this is going to be equal to. 0.995. So now we can actually calculate our coefficients that are going to appear in our regression equation. So, the first Coefficient we can calculate is going to be B1, and that's going to be the slope of our line. And we call that B1 is going to be equal to R multiplied by the quantity of Sy divided by S X in quantity. So this is going to be equal to 0.995, multiplied by the quantity of 2.57. Divided by 17.13 in quantity. And when we evaluate this, this is going to be equal to. 0.149. The next value that we need to calculate is going to be B not, and this is going to be the intercept or our regression equation. So this is going to be equal to Y minus B1 multiplied by X bar. And so this is going to be equal to. 11.28. Minus 0.149. Multiplied by 66. And when we evaluate this equation, We get 1.45. So that means that our regression equation. Why hat is going to be equal to. 1.45. Plus 0.149 multiplied by X. So, now that we have found our regression equation, we can use it to make a prediction. So, setting X equals to 75, we get that Y hat is going to be equal to 1.45. Plus 0.149, multiplied by 75. And when we evaluate this expression, this is going to be equal to. 12.63. So this is the answer we were looking for for this problem. And if we compare that, To the actual tip of $12.50 we see that it is actually very close. Our predicted value is very close to the actual value. So, when we look at our answer choices, we see that the correct answer choice actually corresponds to answer choice D. So, I hope that this explanation has been useful. I'll see you in the next video.

Key Concepts

Regression Analysis

Predictor and Response Variables

Data Sets and Sample Size

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