1. Intro to Stats and Collecting Data
Intro to Stats
2:29 minutesProblem 10.2.33a
Textbook Question
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 guidance1
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 yÌ‚. For example, if x = xâ‚, then yÌ‚â‚ = -10.9 + 0.174(xâ‚). Repeat this for all nine x-values.
Step 4: Subtract each predicted value (yÌ‚) from the corresponding observed value (y) to calculate the residual for each data point. For example, residualâ‚ = yâ‚ - yÌ‚â‚. 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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