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Ch. 10 - Correlation and Regression
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 10 - Correlation and RegressionProblem 10.2.33b
Chapter 10, Problem 10.2.33b

Least-Squares Property According to the least-squares property, the regression line minimizes the sum of the squares of the residuals. Refer to the jackpot/tickets data in Table 10-1 and use the regression equation y^ = -10.9 + 0.174x that was found in Examples 1 and 2 of this section.
b. Find the sum of the squares of the residuals.

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1
Step 1: Understand the problem. The goal is to calculate the sum of the squares of the residuals for the given regression equation y^ = -10.9 + 0.174x. Residuals are the differences between the observed values (y) and the predicted values (y^) from the regression equation.
Step 2: For each data point in the jackpot/tickets data (Table 10-1), calculate the predicted value y^ using the regression equation y^ = -10.9 + 0.174x, where x is the independent variable (tickets).
Step 3: Compute the residual for each data point by subtracting the predicted value y^ from the observed value y. Mathematically, residual = y - y^.
Step 4: Square each residual to eliminate negative values and emphasize larger deviations. This is done by calculating (y - y^)² for each data point.
Step 5: Sum all the squared residuals to find the total sum of the squares of the residuals. This value represents how well the regression line fits the data, with smaller values indicating a better fit.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Least-Squares Property

The least-squares property is a fundamental principle in regression analysis that states the best-fitting line minimizes the sum of the squares of the vertical distances (residuals) between the observed data points and the predicted values on the line. This method ensures that the overall error in predictions is as small as possible, leading to a more accurate model.
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Residuals

Residuals are the differences between the observed values and the values predicted by a regression model. They indicate how far off the predictions are from the actual data points. In the context of the least-squares method, the goal is to minimize the sum of the squares of these residuals to achieve the best fit for the regression line.

Sum of Squares of Residuals

The sum of squares of residuals (SSR) is a key metric in regression analysis that quantifies the total deviation of the predicted values from the actual values. It is calculated by squaring each residual and then summing these squared values. A lower SSR indicates a better fit of the regression model to the data, as it reflects less overall prediction error.
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Related Practice
Textbook Question

Notation Using the weights (lb) and highway fuel consumption amounts (mi/gal) of the 48 cars listed in Data Set 35 “Car Data” of Appendix B, we get this regression equation:

y^ = 58.9 - 0.00749x, where x represents weight.

c. What is the predictor variable?

Textbook Question

Notation The author conducted an experiment in which the height of each student was measured in centimeters and those heights were matched with the same students’ scores on the first statistics test.

c. Does r change if the heights are converted from centimeters to inches?

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Textbook Question

Variation and Prediction Intervals

In Exercises 17–20, find the (a) explained variation, (b) unexplained variation, and (c) indicated prediction interval. In each case, there is sufficient evidence to support a claim of a linear correlation, so it is reasonable to use the regression equation when making predictions.

Altitude and Temperature Listed below are altitudes (thousands of feet) and outside air temperatures (°F) recorded by the author during Delta Flight 1053 from New Orleans to Atlanta. For the prediction interval, use a 95% confidence level with the altitude of 6327 ft (or 6.327 thousand feet).

Textbook Question

Sum of Squares Criterion In addition to the value of another measurement used to assess the quality of a model is the sum of squares of the residuals. Recall from Section 10-2 that a residual is (the difference between an observed y value and the value predicted from the model). Better models have smaller sums of squares. Refer to the U.S. population data in Table 10-7.

c. Verify that according to the sum of squares criterion, the quadratic model is better than the linear model.

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Textbook Question

Outlier Refer to the accompanying Minitab-generated scatterplot.

b. After identifying the 10 pairs of coordinates corresponding to the 10 points, find the value of the correlation coefficient r and determine whether there is a linear correlation.

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Textbook Question

Notation The author conducted an experiment in which the height of each student was measured in centimeters and those heights were matched with the same students’ scores on the first statistics test.

b. Without doing any research or calculations, estimate the value of r.

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