Show that: $\sum {x}_{i}{e}_{i}=0$ and also show that $\sum {\hat{y}}_{i}{e}_{i}=0$. Now I do believe that being able to solve the first sum will make the solution to the second sum more clear. So far I have proved that $\sum {e}_{i}=0$.

Mariah Sparks
2022-07-22
Answered

Show that: $\sum {x}_{i}{e}_{i}=0$ and also show that $\sum {\hat{y}}_{i}{e}_{i}=0$. Now I do believe that being able to solve the first sum will make the solution to the second sum more clear. So far I have proved that $\sum {e}_{i}=0$.

You can still ask an expert for help

Kitamiliseakekw

Answered 2022-07-23
Author has **23** answers

Here's one way of viewing it. We want to write

$\left[\begin{array}{c}{y}_{1}\\ \vdots \\ {y}_{n}\end{array}\right]=\hat{\alpha}\left[\begin{array}{c}1\\ \vdots \\ 1\end{array}\right]+\hat{\beta}\left[\begin{array}{c}{x}_{1}\\ \vdots \\ {x}_{n}\end{array}\right]+\left[\begin{array}{c}{\hat{\epsilon}}_{1}\\ \vdots \\ {\epsilon}_{n}\end{array}\right]$

and choose the values of $\hat{\alpha}$ and $\hat{\beta}$ that minimze ${\hat{\epsilon}}_{1}^{2}+\cdots +{\hat{\epsilon}}_{n}^{2}$. The sum of the first two terms on the right is $[{\hat{y}}_{1},\dots ,{\hat{y}}_{n}{]}^{T}$.

That means the point $\left[\begin{array}{c}{\hat{y}}_{1}\\ \vdots \\ {y}_{n}\end{array}\right]=\hat{\alpha}\left[\begin{array}{c}1\\ \vdots \\ 1\end{array}\right]+\hat{\beta}\left[\begin{array}{c}{x}_{1}\\ \vdots \\ {x}_{n}\end{array}\right]$ is closer to $\left[\begin{array}{c}{y}_{1}\\ \vdots \\ {y}_{n}\end{array}\right]$ in ordinary Euclidean distance than is any other point in the plane spanned by $\left[\begin{array}{c}1\\ \vdots \\ 1\end{array}\right]$ and $\left[\begin{array}{c}{x}_{1}\\ \vdots \\ {x}_{n}\end{array}\right]$. The point in a plane that is closet to $\mathbf{y}$ is the point you get by dropping a perpendicular from $\mathbf{y}$ to the plane. That means $\hat{\epsilon}=\hat{\mathbf{y}}-\mathbf{y}$ is perpendicular to the two columns that span the plane, and thus perpendicular to every linear combination of them, such as $\hat{\mathbf{y}}$. "Perpendicular" means the dot-product is zero. Q.E.D.

The vector of fitted values $\hat{\mathbf{y}}=\left[\begin{array}{c}{\hat{y}}_{1}\\ \vdots \\ {\hat{y}}_{n}\end{array}\right]$ is the orthogonal projection of the vector $\mathbf{y}=\left[\begin{array}{c}{y}_{1}\\ \vdots \\ {y}_{n}\end{array}\right]$ onto the column space of the design matrix $X=\left[\begin{array}{cc}1& {x}_{1}\\ \vdots & \vdots \\ 1& {x}_{n}\end{array}\right]$.

The orthogonal projection is a linear transformation whose matrix is the "hat matrix" is $H=X({X}^{T}X{)}^{-1}{X}^{T}$, an $n\times n$ matrix of rank $2$. Observe that if $\mathbf{w}$ is orthogonal to that column space then $X\mathbf{w}=0$ so $H\mathbf{w}=0$, and if $\mathbf{w}$ is in the column space, then $\mathbf{w}=Xu$ for some $u\in {\mathbb{R}}^{2}$, and so $H\mathbf{w}=\mathbf{w}$.

It follows that $\hat{\epsilon}=\mathbf{y}-\hat{\mathbf{y}}=(I-H)\mathbf{y}$ is orthogonal to the column space. Since $\hat{\mathbf{y}}$ is in the column space, $\hat{\epsilon}$ is orthogonal to $\hat{\mathbf{y}}$.

$\left[\begin{array}{c}{y}_{1}\\ \vdots \\ {y}_{n}\end{array}\right]=\hat{\alpha}\left[\begin{array}{c}1\\ \vdots \\ 1\end{array}\right]+\hat{\beta}\left[\begin{array}{c}{x}_{1}\\ \vdots \\ {x}_{n}\end{array}\right]+\left[\begin{array}{c}{\hat{\epsilon}}_{1}\\ \vdots \\ {\epsilon}_{n}\end{array}\right]$

and choose the values of $\hat{\alpha}$ and $\hat{\beta}$ that minimze ${\hat{\epsilon}}_{1}^{2}+\cdots +{\hat{\epsilon}}_{n}^{2}$. The sum of the first two terms on the right is $[{\hat{y}}_{1},\dots ,{\hat{y}}_{n}{]}^{T}$.

That means the point $\left[\begin{array}{c}{\hat{y}}_{1}\\ \vdots \\ {y}_{n}\end{array}\right]=\hat{\alpha}\left[\begin{array}{c}1\\ \vdots \\ 1\end{array}\right]+\hat{\beta}\left[\begin{array}{c}{x}_{1}\\ \vdots \\ {x}_{n}\end{array}\right]$ is closer to $\left[\begin{array}{c}{y}_{1}\\ \vdots \\ {y}_{n}\end{array}\right]$ in ordinary Euclidean distance than is any other point in the plane spanned by $\left[\begin{array}{c}1\\ \vdots \\ 1\end{array}\right]$ and $\left[\begin{array}{c}{x}_{1}\\ \vdots \\ {x}_{n}\end{array}\right]$. The point in a plane that is closet to $\mathbf{y}$ is the point you get by dropping a perpendicular from $\mathbf{y}$ to the plane. That means $\hat{\epsilon}=\hat{\mathbf{y}}-\mathbf{y}$ is perpendicular to the two columns that span the plane, and thus perpendicular to every linear combination of them, such as $\hat{\mathbf{y}}$. "Perpendicular" means the dot-product is zero. Q.E.D.

The vector of fitted values $\hat{\mathbf{y}}=\left[\begin{array}{c}{\hat{y}}_{1}\\ \vdots \\ {\hat{y}}_{n}\end{array}\right]$ is the orthogonal projection of the vector $\mathbf{y}=\left[\begin{array}{c}{y}_{1}\\ \vdots \\ {y}_{n}\end{array}\right]$ onto the column space of the design matrix $X=\left[\begin{array}{cc}1& {x}_{1}\\ \vdots & \vdots \\ 1& {x}_{n}\end{array}\right]$.

The orthogonal projection is a linear transformation whose matrix is the "hat matrix" is $H=X({X}^{T}X{)}^{-1}{X}^{T}$, an $n\times n$ matrix of rank $2$. Observe that if $\mathbf{w}$ is orthogonal to that column space then $X\mathbf{w}=0$ so $H\mathbf{w}=0$, and if $\mathbf{w}$ is in the column space, then $\mathbf{w}=Xu$ for some $u\in {\mathbb{R}}^{2}$, and so $H\mathbf{w}=\mathbf{w}$.

It follows that $\hat{\epsilon}=\mathbf{y}-\hat{\mathbf{y}}=(I-H)\mathbf{y}$ is orthogonal to the column space. Since $\hat{\mathbf{y}}$ is in the column space, $\hat{\epsilon}$ is orthogonal to $\hat{\mathbf{y}}$.

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Case: Dr. Jung’s Diamonds Selection

With Christmas coming, Dr. Jung became interested in buying diamonds for his wife. After perusing the Web, he learned about the “4Cs” of diamonds: cut, color, clarity, and carat. He knew his wife wanted round-cut earrings mounted in white gold settings, so he immediately narrowed his focus to evaluating color, clarity, and carat for that style earring.

After a bit of searching, Dr. Jung located a number of earring sets that he would consider purchasing. But he knew the pricing of diamonds varied considerably. To assist in his decision making, Dr. Jung decided to use regression analysis to develop a model to predict the retail price of different sets of round-cut earrings based on their color, clarity, and carat scores. He assembled the data in the file Diamonds.xls for this purpose. Use this data to answer the following questions for Dr. Jung.

1) Prepare scatter plots showing the relationship between the earring prices (Y) and each of the potential independent variables. What sort of relationship does each plot suggest?

2) Let X1, X2, and X3 represent diamond color, clarity, and carats, respectively. If Dr. Jung wanted to build a linear regression model to estimate earring prices using these variables, which variables would you recommend that he use? Why?

3) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?

4) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?

5) Dr. Jung now remembers that it sometimes helps to perform a square root transformation on the dependent variable in a regression problem. Modify your spreadsheet to include a new dependent variable that is the square root on the earring prices (use Excel’s SQRT( ) function). If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?

1

6) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?

7) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must actually square the model’s estimates to convert them to price estimates.) Which sets of earring appears to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?

8) Dr. Jung now also remembers that it sometimes helps to include interaction terms in a regression model—where you create a new independent variable as the product of two of the original variables. Modify your spreadsheet to include three new independent variables, X4, X5, and X6, representing interaction terms where: X4 = X1 × X2, X5 = X1 × X3, and X6 = X2 × X3. There are now six potential independent variables. If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?

9) Suppose Dr. Jung decides to use color (X1), carats (X3) and the interaction terms X4 (color * clarity) and X5 (color * carats) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?

10) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must square the model’s estimates to convert them to actual price estimates.) Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?

With Christmas coming, Dr. Jung became interested in buying diamonds for his wife. After perusing the Web, he learned about the “4Cs” of diamonds: cut, color, clarity, and carat. He knew his wife wanted round-cut earrings mounted in white gold settings, so he immediately narrowed his focus to evaluating color, clarity, and carat for that style earring.

After a bit of searching, Dr. Jung located a number of earring sets that he would consider purchasing. But he knew the pricing of diamonds varied considerably. To assist in his decision making, Dr. Jung decided to use regression analysis to develop a model to predict the retail price of different sets of round-cut earrings based on their color, clarity, and carat scores. He assembled the data in the file Diamonds.xls for this purpose. Use this data to answer the following questions for Dr. Jung.

1) Prepare scatter plots showing the relationship between the earring prices (Y) and each of the potential independent variables. What sort of relationship does each plot suggest?

2) Let X1, X2, and X3 represent diamond color, clarity, and carats, respectively. If Dr. Jung wanted to build a linear regression model to estimate earring prices using these variables, which variables would you recommend that he use? Why?

3) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?

4) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?

5) Dr. Jung now remembers that it sometimes helps to perform a square root transformation on the dependent variable in a regression problem. Modify your spreadsheet to include a new dependent variable that is the square root on the earring prices (use Excel’s SQRT( ) function). If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?

1

6) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?

7) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must actually square the model’s estimates to convert them to price estimates.) Which sets of earring appears to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?

8) Dr. Jung now also remembers that it sometimes helps to include interaction terms in a regression model—where you create a new independent variable as the product of two of the original variables. Modify your spreadsheet to include three new independent variables, X4, X5, and X6, representing interaction terms where: X4 = X1 × X2, X5 = X1 × X3, and X6 = X2 × X3. There are now six potential independent variables. If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?

9) Suppose Dr. Jung decides to use color (X1), carats (X3) and the interaction terms X4 (color * clarity) and X5 (color * carats) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?

10) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must square the model’s estimates to convert them to actual price estimates.) Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?

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LightCastle, a Bangladeshi business management consultant, employs a large number of data analysts to analyze the data in survey project. The time it takes for new data analysts to learn the analysis procedure is known to have a normal distribution with a mean of 101 minutes and a standard.

a. What is the probability that the new data analysts will take more than 123 minutes to learn the analysis procedure?

b. What is the probability that the new data analysts will take between 97 to 141 minutes to learn the analysis procedure?

c. To maintain employee standars, LightCastle does not hire any analyst who fails to learn the process within a specified time. Therefore, what should be the specified time limit if they wish to retain at most 95.59% of new analysts?

a. What is the probability that the new data analysts will take more than 123 minutes to learn the analysis procedure?

b. What is the probability that the new data analysts will take between 97 to 141 minutes to learn the analysis procedure?

c. To maintain employee standars, LightCastle does not hire any analyst who fails to learn the process within a specified time. Therefore, what should be the specified time limit if they wish to retain at most 95.59% of new analysts?