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Chapter 12: Problem 26

Show that the "point of averages" \((\bar{x}, \bar{y})\) lies on the estimated regression line.

Short Answer

Expert verified
The point of averages \((\bar{x}, \bar{y})\) lies on the regression line because substituting \(\bar{x}\) into the line equation verifies \(\bar{y} = a\bar{x} + b\).

Step by step solution

01

Understand the Regression Line Equation

The equation for the regression line, also known as the line of best fit, can be given by \[ Y = aX + b \]where \( a \) is the slope of the line and \( b \) is the y-intercept. The regression line represents the line that minimizes the sum of squared differences between observed values and those predicted by the line.
02

Define the "Point of Averages"

The point \((\bar{x}, \bar{y})\) represents the average of all the x-values and y-values. It is calculated as:- \( \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \)- \( \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i \) where \( n \) is the number of data points.
03

Substitute Averages into Regression Line

To check if the point of averages \((\bar{x}, \bar{y})\) lies on the regression line, substitute \( \bar{x} \) into the regression line equation:\[ \bar{y} = a\bar{x} + b \]
04

Solve for b Using Means and Regression Slope

Using statistical formulas, we can find that for the line of best fit:\[ b = \bar{y} - a\bar{x} \]which is obtained by rearranging the regression line formula to solve for the y-intercept when \( x = \bar{x} \).
05

Verify Point of Averages on the Line

To show that \((\bar{x}, \bar{y})\) lies on the regression line, substitute \( \bar{x} \) and \( b \) back into the line equation:\[ \bar{y} = a\bar{x} + (\bar{y} - a\bar{x}) \]Simplifying provides:\[ \bar{y} = \bar{y} \]This confirms that the point of averages \((\bar{x}, \bar{y})\) lies on the estimated regression line since both sides of the equation are equal.

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Key Concepts

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

Point of Averages
The 'point of averages' is an important concept in regression analysis that helps in understanding statistical relationships. For any given dataset, the point of averages
  • (\(\bar{x}\), \(\bar{y}\))
represents the average or mean of all observations.
To calculate these averages, simply sum up all the x-values and divide by the number of x-values to get \(\bar{x}\).
Do the same with all the y-values to find \(\bar{y}\).
This point is crucial because it helps in establishing the central tendency of the data, acting as a balancing point. By understanding where the averages lie, one can better comprehend the distribution of the data, providing useful insights into the general behavior of the dataset.
Regression Line Equation
The regression line equation is fundamental in regression analysis. It's often referred to as the line of best fit and is denoted as:
  • \( Y = aX + b \)
Here, \( a \) is the slope, indicating how much \( Y \) will change for a one-unit change in \( X \).
Meanwhile, \( b \) is the y-intercept, representing the point where the line crosses the Y-axis.
This line minimizes the sum of squared differences between observed and predicted values, thus providing the best linear representation of the relationship between X and Y. By fitting a regression line, one can make informed predictions about future outcomes based on past observations.
Slope and Intercept
In the context of the regression line, understanding the slope and intercept is key to interpreting the model. The slope \( a \)
  • reflects the steepness or incline of the line, providing insight into the strength and direction of the relationship between two variables.
  • A positive slope indicates that as X increases, Y also increases. Conversely, a negative slope signals that Y decreases as X increases.
Meanwhile, the intercept \( b \)
  • is the value of Y when X is zero. It signifies where the line meets the Y-axis.
Both the slope and intercept together define the positioning and orientation of the regression line, thus allowing the visualization of data trends and patterns effectively.
Statistical Formulas
Statistical formulas underpin regression analysis, allowing us to calculate critical elements such as the slope and intercept accurately. For the regression line, the slope \( a \) can be calculated using specific formulas based on the covariance and variance, and is typically expressed as:
  • \( a = \frac{\sum{(X_i - \bar{X})(Y_i - \bar{Y})}}{\sum{(X_i - \bar{X})^2}} \)
The intercept \( b \) follows as:
  • \( b = \bar{Y} - a \bar{X} \)
These formulas ensure that the regression line reflects the closest association between X and Y by minimizing error terms. Grasping these calculations helps in ensuring the regression model accurately represents the underlying data patterns, empowering the researcher to derive valid insights.

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Most popular questions from this chapter

The article "Characterization of Highway Runoff in Austin, Texas, Area" (J. of Envir: Engr., 1998: 131-137) gave a scatter plot, along with the least squares line, of \(x=\) rainfall volume \(\left(\mathrm{m}^{3}\right)\) and \(y=\) runoff volume \(\left(\mathrm{m}^{3}\right)\) for a particular location. The accompanying values were read from the plot. $$ \begin{aligned} &\begin{array}{l|llllllll} x & 5 & 12 & 14 & 17 & 23 & 30 & 40 & 47 \\ \hline y & 4 & 10 & 13 & 15 & 15 & 25 & 27 & 46 \end{array}\\\ &\begin{array}{l|lllllll} x & 55 & 67 & 72 & 81 & 96 & 112 & 127 \\ \hline y & 38 & 46 & 53 & 70 & 82 & 99 & 100 \end{array} \end{aligned} $$ a. Does a scatter plot of the data support the use of the simple linear regression model? b. Calculate point estimates of the slope and intercept of the population regression line. c. Calculate a point estimate of the true average runoff volume when rainfall volume is 50 . d. Calculate a point estimate of the standard deviation \(\sigma\). e. What proportion of the observed variation in runoff volume can be attributed to the simple linear regression relationship between runoff and rainfall?

A study to assess the capability of subsurface flow wetland systems to remove biochemical oxygen demand (BOD) and various other chemical constituents resulted in the accompanying data on \(x=\mathrm{BOD}\) mass loading \((\mathrm{kg} / \mathrm{ha} / \mathrm{d})\) and \(y=\) BOD mass removal (kg/ha/d) ("Subsurface Flow WetlandsA Performance Evaluation," Water Envir: Res., 1995: 244-247). $$ \begin{array}{l|cccccccccccccc} x & 3 & 8 & 10 & 11 & 13 & 16 & 27 & 30 & 35 & 37 & 38 & 44 & 103 & 142 \\ \hline y & 4 & 7 & 8 & 8 & 10 & 11 & 16 & 26 & 21 & 9 & 31 & 30 & 75 & 90 \end{array} $$ a. Construct boxplots of both mass loading and mass removal, and comment on any interesting features. b. Construct a scatter plot of the data, and comment on any interesting features.

The article "Objective Measurement of the Stretchability of Mozzarella Cheese" (J. of Texture Studies, 1992: 185-194) reported on an experiment to investigate how the behavior of mozzarella cheese varied with temperature. Consider the accompanying data on \(x=\) temperature and \(y=\) elongation \((\%)\) at failure of the cheese. [Note: The researchers were Italian and used real mozzarella cheese, not the poor cousin widely available in the United States.] $$ \begin{array}{l|rrrrrrr} x & 59 & 63 & 68 & 72 & 74 & 78 & 83 \\ \hline y & 118 & 182 & 247 & 208 & 197 & 135 & 132 \end{array} $$ a. Construct a scatter plot in which the axes intersect at \((0,0)\). Mark \(0,20,40,60,80\), and 100 on the horizontal axis and \(0,50,100,150,200\), and 250 on the vertical axis. b. Construct a scatter plot in which the axes intersect at ( 55 , 100 ), as was done in the cited article. Does this plot seem preferable to the one in part (a)? Explain your reasoning. c. What do the plots of parts (a) and (b) suggest about the nature of the relationship between the two variables?

The Turbine Oil Oxidation Test (TOST) and the Rotating Bomb Oxidation Test (RBOT) are two different procedures for evaluating the oxidation stability of steam turbine oils. The article "Dependence of Oxidation Stability of Steam Turbine Oil on Base Oil Composition" (J. of the Society of Tribologists and Lubrication Engrs., Oct. 1997: 19-24) reported the accompanying observations on \(x=\) TOST time (hr) and \(y=\) RBOT time (min) for 12 oil specimens. $$ \begin{array}{lrrrrrr} \text { TOST } & 4200 & 3600 & 3750 & 3675 & 4050 & 2770 \\ \text { RBOT } & 370 & 340 & 375 & 310 & 350 & 200 \\ \text { TOST } & 4870 & 4500 & 3450 & 2700 & 3750 & 3300 \\ \text { RBOT } & 400 & 375 & 285 & 225 & 345 & 285 \end{array} $$ a. Calculate and interpret the value of the sample correlation coefficient (as did the article's authors). b. How would the value of \(r\) be affected if we had let \(x=\) RBOT time and \(y=\) TOST time? c. How would the value of \(r\) be affected if RBOT time were expressed in hours? d. Construct normal probability plots and comment. e. Carry out a test of hypotheses to decide whether RBOT time and TOST time are linearly related.

The following summary statistics were obtained from study that used regression analysis to investigate the relationship between pavement deflection and surface temperature of the pavement at various locations on a state highway. Here \(x=\) temperature \(\left({ }^{\circ} \mathrm{F}\right)\) and \(y=\) deflection adjustment factor \((y \geq 0)\) : $$ \begin{aligned} &n=15 \quad \sum x_{i}=1425 \quad \sum y_{i}=10.68 \\ &\sum x_{i}^{2}=139,037.25 \quad \sum x_{i} y_{i}=987.645 \\ &\sum y_{i}^{2}=7.8518 \end{aligned} $$ (Many more than 15 observations were made in the study; the reference is "Flexible Pavement Evaluation and Rehabilitation," Transportation Eng. J., 1977: 75-85.) a. Compute \(\hat{\beta}_{1}, \hat{\beta}_{0}\), and the equation of the estimated regression line. Graph the estimated line. b. What is the estimate of expected change in the deflection adjustment factor when temperature is increased by \(1^{\circ} \mathrm{F}\) ? c. Suppose temperature were measured in \({ }^{\circ} \mathrm{C}\) rather than in \({ }^{\circ} \mathrm{F}\). What would be the estimated regression line? Answer part (b) for an increase of \(1^{\circ} \mathrm{C}\). [Hint: \({ }^{\circ} \mathrm{F}=(9 / 5)^{\circ} \mathrm{C}+32\); now substitute for the "old \(x\) " in terms of the "new \(x\)."] d. If a \(200^{\circ} \mathrm{F}\) surface temperature were within the realm of possibility, would you use the estimated line of part (a) to predict deflection factor for this temperature? Why or why not?
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