Ch. 10 - Correlation and Regression

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 10 - Correlation and Regression

Problem 10.5.18

Chapter 10, Problem 10.5.18

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.
a. Find the sum of squares of the residuals resulting from the linear model.

Verified step by step guidance

Step 1: Understand the concept of residuals. A residual is the difference between the observed value (y) and the predicted value (ŷ) from the model. Mathematically, residuals are calculated as:

e = y - ŷ

Step 2: Refer to the U.S. population data in Table 10-7.a. Identify the observed values (y) and the predicted values (ŷ) from the linear model provided in the table.

Step 3: For each data point, calculate the residual by subtracting the predicted value (ŷ) from the observed value (y). Use the formula:

e = y - ŷ

Step 4: Square each residual to eliminate negative values and emphasize larger deviations. Use the formula:

e^{2}

Step 5: Sum all the squared residuals to find the sum of squares of the residuals. Use the formula:

\sum e^{2}

. This value represents the total deviation of the observed data from the model predictions.

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Key Concepts

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

Residuals

Residuals are the differences between observed values and the values predicted by a statistical model. In regression analysis, they indicate how well the model fits the data; smaller residuals suggest a better fit. Understanding residuals is crucial for evaluating model performance and diagnosing potential issues in the model.

Sum of Squares of Residuals (SSR)

The Sum of Squares of Residuals (SSR) quantifies the total deviation of the observed values from the predicted values in a regression model. It is calculated by squaring each residual and summing these squared values. A lower SSR indicates a better-fitting model, as it reflects less unexplained variability in the data.

Model Quality Assessment

Model quality assessment involves evaluating how well a statistical model represents the data it is intended to explain. This can be done using various metrics, including the sum of squares of residuals, R-squared values, and other diagnostic tools. A thorough assessment helps in selecting the most appropriate model for the data and ensuring reliable predictions.

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