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Chapter 11: Problem 84

Show that, for the simple linear regression model, the following statements are true: (a) \(\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)=0\) (b) \(\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right) x_{i}=0\) (c) \(\frac{1}{n} \sum_{i=1}^{n} \hat{y}_{i}=\bar{y}\)

Short Answer

Expert verified
(a) Residuals sum to zero. (b) Residuals and predictors are uncorrelated. (c) Mean of predicted values equals mean of observed values.

Step by step solution

01

Understanding the Regression Model

In a simple linear regression model, the observed value \( y_i \) is estimated by \( \hat{y}_i = a + bx_i \), where \( a \) is the y-intercept, \( b \) is the slope of the regression line, and \( x_i \) is the independent variable.
02

Proving (a)

The total sum of residuals, \( \sum_{i=1}^{n} (y_i - \hat{y}_i) \) is zero because the sum of the residuals around the predicted regression line is zero by design. The regression line minimizes the sum of squared residuals, implying that equal positive and negative residuals offset one another.
03

Proving (b)

The statement \( \sum_{i=1}^{n} (y_i - \hat{y}_i) x_i = 0 \) follows because the residuals \( e_i = y_i - \hat{y}_i \) are orthogonal (uncorrelated) to the predictor \( x_i \). In other words, the dot product of the residual vector and the vector of independent variables is zero.
04

Arithmetic Mean of Predicted Values (c)

\( \frac{1}{n} \sum_{i=1}^{n} \hat{y}_i = \bar{y} \) holds because the average value of the predicted values equals the average of the observed values, ensuring that the line of best fit (regression line) passes through the point \((\bar{x}, \bar{y})\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Regression Model
In simple linear regression, our goal is to model the relationship between an independent variable \( x_i \) and a dependent variable \( y_i \). We do this through the regression equation \( \hat{y}_i = a + bx_i \), where \( \hat{y}_i \) is our predicted or estimated value of \( y_i \).
  • \( a \) represents the y-intercept, or the value of \( \hat{y} \) when \( x = 0 \).
  • \( b \) is the slope of the regression line, indicating how much \( \hat{y} \) changes for a one-unit change in \( x \).
  • The equation establishes a straight line that best fits the dataset, balancing the distance of all data points from the line.
The objective is to align the regression line in a way that the disparities, or deviations, between actual and predicted values, known as residuals, are minimized.
Exploring Residuals
Residuals are the differences between observed values \( y_i \) and their corresponding predicted values \( \hat{y}_i \), calculated as \( e_i = y_i - \hat{y}_i \).
Residuals provide crucial insights into how well our model fits the data. When doing regression analysis:
  • The goal is for the sum of these residuals to be zero: \( \sum_{i=1}^{n} (y_i - \hat{y}_i) = 0 \). This condition ensures there is no overall bias in predictions.
  • By minimizing the sum of squared residuals, the regression line is positioned to balance out these deviations effectively.
Residuals are vital diagnostic tools because large residuals can indicate data points that do not fit the model well, which might be outliers or suggest a non-linear relationship.
The Concept of Orthogonality
In regression analysis, orthogonality pertains to the independence, or lack of correlation, between residuals and the predictor variables \( x_i \). What this means is:
  • The dot product, or sum of the element-wise products of two vectors, between residuals \( e_i \) and \( x_i \) equals zero: \( \sum_{i=1}^{n} (y_i - \hat{y}_i)x_i = 0 \).
  • This orthogonality indicates that the residuals contain no systematic information that is related to \( x_i \).
When orthogonality holds, it implies that the linear model captures all the explainable variance in \( y_i \) due to \( x_i \), ensuring that the slope \( b \) is the true best fit for the data.
Understanding Arithmetic Mean in Regression
The arithmetic mean is important in regression for evaluating the central tendency of both observed data and predicted values.
When analyzing a regression model:
  • The average, or mean, of the predicted values \( \hat{y}_i \) aligns with the mean of the observed values \( y_i \), satisfying \( \frac{1}{n}\sum_{i=1}^{n}\hat{y}_i = \bar{y} \).
  • This condition ensures that the regression line passes through the point \((\bar{x}, \bar{y})\), providing an unbiased estimation of the central location of the dataset.
In essence, this property of regression emphasizes balance in data distribution, as it suggests that our model accurately reflects the overall level of the response variable.

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Most popular questions from this chapter

The market research department of a soft drink manufacturer is investigating the effectiveness of a price discount coupon on the purchase of a two-liter beverage product. A sample of 5500 customers was given coupons for varying price discounts between 5 and 25 cents. The response variable was the number of coupons in each price discount category redeemed after one month. The data are shown below. $$\begin{array}{ccc}\hline & & \text { Number } \\\\\text { Discount, } x & \text { Sample Size, } n & \text { Redeemed, } r \\\\\hline 5 & 500 & 100 \\\7 & 500 & 122 \\\9 & 500 & 147 \\\11 & 500 & 176 \\\13 & 500 & 211 \\\15 & 500 & 244 \\\17 & 500 & 277 \\\19 & 500 & 310 \\\21 & 500 & 343 \\\23 & 500 & 372 \\\25 & 500 & 391\end{array}$$ (a) Fit a logistic regression model to the data. Use a simple linear regression model as the structure for the linear predictor. (b) Is the logistic regression model in part (a) adequate? (c) Draw a graph of the data and the fitted logistic regression model. (d) Expand the linear predictor to include a quadratic term. Is there any evidence that this quadratic term is required in the model? (e) Draw a graph of this new model on the same plot that you prepared in part (c). Does the expanded model visually provide a better fit to the data than the original model from part (a)?

Suppose that we are fitting a line and we wish to make the variance of the regression coefficient \(\hat{\beta}_{1}\) as small as possible. Where should the observations \(x_{i}, i=1,2, \ldots, n,\) be taken so as to minimize \(V\left(\hat{\beta}_{1}\right) ?\) Discuss the practical implications of this allocation of the \(x_{i}\).

Suppose that we have \(n\) pairs of observations \(\left(x_{i}, y_{i}\right)\) such that the sample correlation coefficient \(r\) is unity (approximately). Now let \(z_{i}=y_{i}^{2}\) and consider the sample correlation coefficient for the \(n\) -pairs of data \(\left(x_{i}, z_{i}\right) .\) Will this sample correlation coefficient be approximately unity? Explain why or why not.

An article in the Journal of Sound and Vibration (Vol. \(151,1991,\) pp. \(383-394\) ) described a study investigating the relationship between noise exposure and hypertension. The following data are representative of those reported in the article. $$\begin{aligned}&\begin{array}{c|c|c|c|c|c|c|c|c|c|c}y & 1 & 0 & 1 & 2 & 5 & 1 & 4 & 6 & 2 & 3 \\\\\hline x & 60 & 63 & 65 & 70 & 70 & 70 & 80 & 90 & 80 & 80\end{array}\\\&\begin{array}{c|c|c|c|c|c|c|c|c|c|c}y & 5 & 4 & 6 & 8 & 4 & 5 & 7 & 9 & 7 & 6 \\\\\hline x & 85 & 89 & 90 & 90 & 90 & 90 & 94 & 100 & 100 & 100\end{array}\end{aligned}$$ (a) Draw a scatter diagram of \(y\) (blood pressure rise in millimeters of mercury) versus \(x\) (sound pressure level in decibels). Does a simple linear regression model seem reasonable in this situation? (b) Fit the simple linear regression model using least squares. Find an estimate of \(\sigma^{2}\) (c) Find the predicted mean rise in blood pressure level associated with a sound pressure level of 85 decibels.

Suppose that each value of \(x_{i}\) is multiplied by a positive constant \(a\), and each value of \(y_{i}\) is multiplied by another positive constant \(b\). Show that the \(t\) -statistic for testing \(H_{0}: \beta_{1}=0\) versus \(H_{1}: \beta_{1} \neq 0\) is unchanged in value.
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