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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 10.2.33a
Chapter 10, Problem 10.2.33a

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.
a. Identify the nine residuals.

Verified step by step guidance
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Step 1: Recall that a residual is the difference between the observed value (y) and the predicted value (ŷ) from the regression equation. Mathematically, residual = y - ŷ.
Step 2: Use the given regression equation ŷ = -10.9 + 0.174x to calculate the predicted values (ŷ) for each of the nine x-values (ticket sales) provided in Table 10-1.
Step 3: For each x-value, substitute it into the regression equation to compute the corresponding predicted value ŷ. For example, if x = x₁, then ŷ₁ = -10.9 + 0.174(x₁). Repeat this for all nine x-values.
Step 4: Subtract each predicted value (ŷ) from the corresponding observed value (y) to calculate the residual for each data point. For example, residual₁ = y₁ - ŷ₁. Repeat this for all nine data points.
Step 5: List all nine residuals in the form of a table or sequence, showing the observed value (y), predicted value (ŷ), and residual (y - ŷ) for each data point.

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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 regression model. They indicate how far off the predictions are from the actual data points. In the context of regression analysis, calculating residuals helps assess the accuracy of the model, as smaller residuals suggest a better fit.

Least-Squares Method

The least-squares method is a statistical technique used to determine the best-fitting line through a set of data points. It works by minimizing the sum of the squares of the residuals, which are the vertical distances between the observed values and the predicted values on the regression line. This method ensures that the overall error in predictions is as small as possible.
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Regression Equation

A regression equation represents the relationship between independent and dependent variables in a statistical model. In this case, the equation y^ = -10.9 + 0.174x describes how the dependent variable (y) changes with the independent variable (x). Understanding this equation is crucial for predicting outcomes and analyzing the strength of the relationship between the variables.
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Related Practice
Textbook Question

Time and Motion In a physics experiment at Doane College, a soccer ball was thrown upward from the bed of a moving truck. The table below lists the time (sec) that has lapsed from the throw and the corresponding height (m) of the soccer ball.

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a. Find the value of the linear correlation coefficient r.

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Textbook Question

Effects of an Outlier Refer to the Minitab-generated scatterplot given in Exercise 9 of Section 10-1


a. Using the pairs of values for all 10 points, find the equation of the regression line.

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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.

a. What does the symbol y^ represent?

Textbook Question

Time and Motion In a physics experiment at Doane College, a soccer ball was thrown upward from the bed of a moving truck. The table below lists the time (sec) that has lapsed from the throw and the corresponding height (m) of the soccer ball.

[IMAGE]

c. What horrible mistake would be easy to make if the analysis is conducted without a scatterplot?

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Textbook Question

Time and Motion In a physics experiment at Doane College, a soccer ball was thrown upward from the bed of a moving truck. The table below lists the time (sec) that has lapsed from the throw and the corresponding height (m) of the soccer ball.

[IMAGE]

b. Based on the result from part (a), what do you conclude about a linear correlation between time and height?

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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.

a. For this sample of paired data, what does r represent, and what does represent?

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