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.

Oscars Listed below are ages of recent Oscar winners matched by the years in which the awards were won (from Data Set 21 “Oscar Winner Age” in Appendix B). Find the best predicted age of an Oscar-winning actress given that the Oscar winner for best actor is 59 years of age. How does the result compare to the actual actress age of 60 years?

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Accepted Answer

Hello everyone. Let's take a look at this question together. Listed below are marathon completion times in minutes for elite male and female runners in recent international marathons matched by the same event year. Find the best predicted completion time for an elite female runner, given that an elite male runner completes a marathon in 140 minutes. How does the predicted time compare to the actual female runner's time of 155 minutes? Is it answer choice A, 159 minutes, and the predicted time is slightly higher than the actual time? Answer choice B, 150 minutes, and the predicted time is slightly lower than the actual time. Answer choice C 155 minutes, and the predicted time is exactly equal to the actual time, or answer choice D 169 minutes, and the predicted time is higher than the actual time. So in order to solve this question, we must first find the best predicted completion time for an elite female runner, given that an elite male runner completes a marathon in 14. 40 minutes, as well as how does the predicted time compare to the actual female runner's time of 155 minutes. So the first step in solving this problem is identifying both the predictor and response variables, which we know that the completion time of elite male runners is the predictor variable or X, and the completion time of elite female runners is the variable or why. Well, from the information in the question, we are given the completion time of an elite male runner, which was 140 minutes, so our predictor variable X is equal to 140, and then we can compute the regression using the least squares regression equation, which is Y is equal to B0 plus B1 multiplied by X, and using the given. Data we can compute the following values, and using these values, we can then calculate the standard deviation of X and Y using the formulas for the standard deviation of X and the standard deviation of Y, which plugging in the values from our data for the standard deviation of X, we calculate 10.80 minutes, and for the standard deviation of Y, we calculate 11. 87 minutes. And then we can also calculate the correlation coefficient, which is R, using the formula for the correlation coefficient and plugging in the values from our data, we calculate 0.991398, which is approximately 0.9914. So then we use the regression formula, which is Y equals B0 plus B1X, but we Need to calculate the values for B0 and B1, which the values for B1 and B0 can be calculated using their formulas, where B1 is calculated as R multiplied by the standard deviation of Y divided by the standard deviation of X, which plugging in those values that we already calculated, we get 1.09 for B1, and then for B 0, it can be calculated. By taking Y minus B1 multiplied by X bar, which those values are from the data that we calculated, giving us 5.956 as the value for B0. So we have B1, which is 1.09, and B0, which is 5.956, as well as X is equal to 140. So plugging in those values into our regression equation, we Get Y hat equals 5.956 plus 1.09 multiplied by X. And so using the regression formula to get our predicted female completion time, we note that Y hat, which is equal to 5.956 plus 1.09 X, subbing in the value of 144 X, we can calculate 158.556 or approximate. 159 minutes as the predicted female completion time if the elite male runner has a completion time of 140 minutes. And using the predicted time of 159 minutes, we can compare it to the actual time, which was 155 minutes for the female runner, we can identify that the predicted value is 4 minutes higher than the time. And so looking at our answer choices, based on our calculations, we can identify that answer choice A is the correct answer, as the predicted time was 159 minutes, which is slightly higher than the actual time of 155 minutes. So answer choice A is the only correct answer choice. I hope you found this video to be helpful. Thank you and goodbye.

Key Concepts

Regression Analysis

Predictor and Response Variables

Prediction Procedure

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