the regression equation always passes through

Therefore, there are 11 $\varepsilon$ values. Use your calculator to find the least squares regression line and predict the maximum dive time for 110 feet. The absolute value of a residual measures the vertical distance between the actual value of y and the estimated value of y. I found they are linear correlated, but I want to know why. Regression equation: y is the value of the dependent variable (y), what is being predicted or explained. 'P[A Pj{) It is not generally equal to y from data. The standard error of estimate is a. The correlation coefficient, $r$, developed by Karl Pearson in the early 1900s, is numerical and provides a measure of strength and direction of the linear association between the independent variable $x$ and the dependent variable $y$. Linear Regression Formula all integers 1,2,3,,n21, 2, 3, \ldots , n^21,2,3,,n2 as its entries, written in sequence, In the diagram above,[latex]\displaystyle{y}_{0}-\hat{y}_{0}={\epsilon}_{0}[/latex] is the residual for the point shown. For now, just note where to find these values; we will discuss them in the next two sections. Find the equation of the Least Squares Regression line if: x-bar = 10 sx= 2.3 y-bar = 40 sy = 4.1 r = -0.56. In my opinion, we do not need to talk about uncertainty of this one-point calibration. Approximately 44% of the variation (0.4397 is approximately 0.44) in the final-exam grades can be explained by the variation in the grades on the third exam, using the best-fit regression line. In linear regression, the regression line is a perfectly straight line: The regression line is represented by an equation. For Mark: it does not matter which symbol you highlight. So, if the slope is 3, then as X increases by 1, Y increases by 1 X 3 = 3. Each datum will have a vertical residual from the regression line; the sizes of the vertical residuals will vary from datum to datum. 6 cm B 8 cm 16 cm CM then The third exam score,x, is the independent variable and the final exam score, y, is the dependent variable. Here the point lies above the line and the residual is positive. Optional: If you want to change the viewing window, press the WINDOW key. Show transcribed image text Expert Answer 100% (1 rating) Ans. When regression line passes through the origin, then: (a) Intercept is zero (b) Regression coefficient is zero (c) Correlation is zero (d) Association is zero MCQ 14.30 For Mark: it does not matter which symbol you highlight. <> (0,0) b. The process of fitting the best-fit line is called linear regression. The critical range is usually fixed at 95% confidence where the f critical range factor value is 1.96. It is important to interpret the slope of the line in the context of the situation represented by the data. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Then arrow down to Calculate and do the calculation for the line of best fit. Chapter 5. Besides looking at the scatter plot and seeing that a line seems reasonable, how can you tell if the line is a good predictor? OpenStax is part of Rice University, which is a 501(c)(3) nonprofit. endobj You should be able to write a sentence interpreting the slope in plain English. In regression, the explanatory variable is always x and the response variable is always y. For the case of one-point calibration, is there any way to consider the uncertaity of the assumption of zero intercept? However, computer spreadsheets, statistical software, and many calculators can quickly calculate r. The correlation coefficient r is the bottom item in the output screens for the LinRegTTest on the TI-83, TI-83+, or TI-84+ calculator (see previous section for instructions). For one-point calibration, one cannot be sure that if it has a zero intercept. False 25. Strong correlation does not suggest thatx causes yor y causes x. Multicollinearity is not a concern in a simple regression. If each of you were to fit a line by eye, you would draw different lines. If each of you were to fit a line "by eye," you would draw different lines. Which equation represents a line that passes through 4 1/3 and has a slope of 3/4 . Linear regression for calibration Part 2. Two more questions: In my opinion, a equation like y=ax+b is more reliable than y=ax, because the assumption for zero intercept should contain some uncertainty, but I dont know how to quantify it. The Sum of Squared Errors, when set to its minimum, calculates the points on the line of best fit. The output screen contains a lot of information. The regression equation X on Y is X = c + dy is used to estimate value of X when Y is given and a, b, c and d are constant. It is customary to talk about the regression of Y on X, hence the regression of weight on height in our example. D. Explanation-At any rate, the View the full answer Make your graph big enough and use a ruler. The regression line does not pass through all the data points on the scatterplot exactly unless the correlation coefficient is 1. The correlation coefficient $r$ measures the strength of the linear association between $x$ and $y$. In this case, the analyte concentration in the sample is calculated directly from the relative instrument responses. If r = 0 there is absolutely no linear relationship between x and y (no linear correlation). M = slope (rise/run). Want to cite, share, or modify this book? Must linear regression always pass through its origin? You could use the line to predict the final exam score for a student who earned a grade of 73 on the third exam. The regression line is calculated as follows: Substituting 20 for the value of x in the formula, = a + bx = 69.7 + (1.13) (20) = 92.3 The performance rating for a technician with 20 years of experience is estimated to be 92.3. The least square method is the process of finding the best-fitting curve or line of best fit for a set of data points by reducing the sum of the squares of the offsets (residual part) of the points from the curve. Use your calculator to find the least squares regression line and predict the maximum dive time for 110 feet. The given regression line of y on x is ; y = kx + 4 . It's also known as fitting a model without an intercept (e.g., the intercept-free linear model y=bx is equivalent to the model y=a+bx with a=0). The slope of the line,b, describes how changes in the variables are related. A negative value of r means that when x increases, y tends to decrease and when x decreases, y tends to increase (negative correlation). ; The slope of the regression line (b) represents the change in Y for a unit change in X, and the y-intercept (a) represents the value of Y when X is equal to 0. Therefore, approximately 56% of the variation ($1 - 0.44 = 0.56$) in the final exam grades can NOT be explained by the variation in the grades on the third exam, using the best-fit regression line. It also turns out that the slope of the regression line can be written as . The independent variable, $x$, is pinky finger length and the dependent variable, $y$, is height. Linear regression analyses such as these are based on a simple equation: Y = a + bX endobj When $r$ is negative, $x$ will increase and $y$ will decrease, or the opposite, $x$ will decrease and $y$ will increase. In this case, the equation is -2.2923x + 4624.4. T or F: Simple regression is an analysis of correlation between two variables. The term[latex]\displaystyle{y}_{0}-\hat{y}_{0}={\epsilon}_{0}[/latex] is called the error or residual. This statement is: Always false (according to the book) Can someone explain why? Approximately 44% of the variation (0.4397 is approximately 0.44) in the final-exam grades can be explained by the variation in the grades on the third exam, using the best-fit regression line. This means that if you were to graph the equation -2.2923x + 4624.4, the line would be a rough approximation for your data. Slope, intercept and variation of Y have contibution to uncertainty. This means that, regardless of the value of the slope, when X is at its mean, so is Y. Advertisement . True b. If the observed data point lies above the line, the residual is positive, and the line underestimates the actual data value fory. Consider the following diagram. Below are the different regression techniques: plzz do mark me as brainlist and do follow me plzzzz. $1 - r^{2}$, when expressed as a percentage, represents the percent of variation in $y$ that is NOT explained by variation in $x$ using the regression line. bu/@A>r[>,a$KIV QR*2[\B#zI-k^7(Ug-I\ 4\"\6eLkV If you suspect a linear relationship between $x$ and $y$, then $r$ can measure how strong the linear relationship is. How can you justify this decision? So one has to ensure that the y-value of the one-point calibration falls within the +/- variation range of the curve as determined. This means that, regardless of the value of the slope, when X is at its mean, so is Y. . 35 In the regression equation Y = a +bX, a is called: A X . argue that in the case of simple linear regression, the least squares line always passes through the point (mean(x), mean . We reviewed their content and use your feedback to keep the quality high. One of the approaches to evaluate if the y-intercept, a, is statistically significant is to conduct a hypothesis testing involving a Students t-test. The variance of the errors or residuals around the regression line C. The standard deviation of the cross-products of X and Y d. The variance of the predicted values. quite discrepant from the remaining slopes). 3 0 obj Instructions to use the TI-83, TI-83+, and TI-84+ calculators to find the best-fit line and create a scatterplot are shown at the end of this section. Then, if the standard uncertainty of Cs is u(s), then u(s) can be calculated from the following equation: SQ[(u(s)/Cs] = SQ[u(c)/c] + SQ[u1/R1] + SQ[u2/R2]. Enter your desired window using Xmin, Xmax, Ymin, Ymax. Scatter plot showing the scores on the final exam based on scores from the third exam. The problem that I am struggling with is to show that that the regression line with least squares estimates of parameters passes through the points $(X_1,\bar{Y_2}),(X_2,\bar{Y_2})$. This intends that, regardless of the worth of the slant, when X is at its mean, Y is as well. For situation(2), intercept will be set to zero, how to consider about the intercept uncertainty? These are the famous normal equations. Answer 6. What the VALUE of r tells us: The value of r is always between 1 and +1: 1 r 1. This means that, regardless of the value of the slope, when X is at its mean, so is Y. Line Of Best Fit: A line of best fit is a straight line drawn through the center of a group of data points plotted on a scatter plot. The regression equation is the line with slope a passing through the point Another way to write the equation would be apply just a little algebra, and we have the formulas for a and b that we would use (if we were stranded on a desert island without the TI-82) . 25. The sign of r is the same as the sign of the slope,b, of the best-fit line. When r is negative, x will increase and y will decrease, or the opposite, x will decrease and y will increase. In this video we show that the regression line always passes through the mean of X and the mean of Y. The best fit line always passes through the point $(\bar{x}, \bar{y})$. ). The situation (2) where the linear curve is forced through zero, there is no uncertainty for the y-intercept. This book uses the The line does have to pass through those two points and it is easy to show The $\hat{y}$ is read "$y$ hat" and is the estimated value of $y$. Calculus comes to the rescue here. Find SSE s 2 and s for the simple linear regression model relating the number (y) of software millionaire birthdays in a decade to the number (x) of CEO birthdays. This model is sometimes used when researchers know that the response variable must . Answer: At any rate, the regression line always passes through the means of X and Y. Use your calculator to find the least squares regression line and predict the maximum dive time for 110 feet. The variable $r^{2}$ is called the coefficient of determination and is the square of the correlation coefficient, but is usually stated as a percent, rather than in decimal form. If the scatter plot indicates that there is a linear relationship between the variables, then it is reasonable to use a best fit line to make predictions for y given x within the domain of x-values in the sample data, but not necessarily for x-values outside that domain. In general, the data are scattered around the regression line. You'll get a detailed solution from a subject matter expert that helps you learn core concepts. The variable r2 is called the coefficient of determination and is the square of the correlation coefficient, but is usually stated as a percent, rather than in decimal form. It is obvious that the critical range and the moving range have a relationship. the new regression line has to go through the point (0,0), implying that the Make sure you have done the scatter plot. x values and the y values are [latex]\displaystyle\overline{{x}}[/latex] and [latex]\overline{{y}}[/latex]. Every time I've seen a regression through the origin, the authors have justified it (Be careful to select LinRegTTest, as some calculators may also have a different item called LinRegTInt. SCUBA divers have maximum dive times they cannot exceed when going to different depths. Press 1 for 1:Y1. You could use the line to predict the final exam score for a student who earned a grade of 73 on the third exam. The regression line always passes through the (x,y) point a. In measurable displaying, regression examination is a bunch of factual cycles for assessing the connections between a reliant variable and at least one free factor. Creative Commons Attribution License Third Exam vs Final Exam Example: Slope: The slope of the line is b = 4.83. - Hence, the regression line OR the line of best fit is one which fits the data best, i.e. If the slope is found to be significantly greater than zero, using the regression line to predict values on the dependent variable will always lead to highly accurate predictions a. In theory, you would use a zero-intercept model if you knew that the model line had to go through zero. Using the slopes and the $y$-intercepts, write your equation of "best fit." The criteria for the best fit line is that the sum of the squared errors (SSE) is minimized, that is, made as small as possible. Typically, you have a set of data whose scatter plot appears to fit a straight line. False 25. % c. For which nnn is MnM_nMn invertible? Press ZOOM 9 again to graph it. ;{tw{`,;c,Xvir\:iZ@bqkBJYSw&!t;Z@D7'ztLC7_g 2. The slope $b$ can be written as $b = r\left(\dfrac{s_{y}}{s_{x}}\right)$ where $s_{y} =$ the standard deviation of the $y$ values and $s_{x} =$ the standard deviation of the $x$ values. variables or lurking variables. [latex]{b}=\frac{{\sum{({x}-\overline{{x}})}{({y}-\overline{{y}})}}}{{\sum{({x}-\overline{{x}})}^{{2}}}}[/latex]. This is illustrated in an example below. the least squares line always passes through the point (mean(x), mean . To graph the best-fit line, press the Y= key and type the equation 173.5 + 4.83X into equation Y1. . For each data point, you can calculate the residuals or errors, $y_{i} - \hat{y}_{i} = \varepsilon_{i}$ for $i = 1, 2, 3, , 11$. Correlation coefficient's lies b/w: a) (0,1) 2003-2023 Chegg Inc. All rights reserved. Answer (1 of 3): In a bivariate linear regression to predict Y from just one X variable , if r = 0, then the raw score regression slope b also equals zero. In linear regression, uncertainty of standard calibration concentration was omitted, but the uncertaity of intercept was considered. Press ZOOM 9 again to graph it. Do you think everyone will have the same equation? <> A regression line, or a line of best fit, can be drawn on a scatter plot and used to predict outcomes for the $x$ and $y$ variables in a given data set or sample data. 2. Always gives the best explanations. Brandon Sharber Almost no ads and it's so easy to use. 0 <, https://openstax.org/books/introductory-statistics/pages/1-introduction, https://openstax.org/books/introductory-statistics/pages/12-3-the-regression-equation, Creative Commons Attribution 4.0 International License, In the STAT list editor, enter the X data in list L1 and the Y data in list L2, paired so that the corresponding (, On the STAT TESTS menu, scroll down with the cursor to select the LinRegTTest. Hence, this linear regression can be allowed to pass through the origin. Area and Property Value respectively). In my opinion, this might be true only when the reference cell is housed with reagent blank instead of a pure solvent or distilled water blank for background correction in a calibration process. Common mistakes in measurement uncertainty calculations, Worked examples of sampling uncertainty evaluation, PPT Presentation of Outliers Determination. That means that if you graphed the equation -2.2923x + 4624.4, the line would be a rough approximation for your data. Regression analysis is used to study the relationship between pairs of variables of the form (x,y).The x-variable is the independent variable controlled by the researcher.The y-variable is the dependent variable and is the effect observed by the researcher. Except where otherwise noted, textbooks on this site points get very little weight in the weighted average. At 110 feet, a diver could dive for only five minutes. Statistical Techniques in Business and Economics, Douglas A. Lind, Samuel A. Wathen, William G. Marchal, Daniel S. Yates, Daren S. Starnes, David Moore, Fundamentals of Statistics Chapter 5 Regressi. Press 1 for 1:Function. This is called a Line of Best Fit or Least-Squares Line. When two sets of data are related to each other, there is a correlation between them. If r = 1, there is perfect negativecorrelation. It is not generally equal to $y$ from data. endobj The regression equation always passes through: (a) (X, Y) (b) (a, b) (c) ( , ) (d) ( , Y) MCQ 14.25 The independent variable in a regression line is: . The sum of the difference between the actual values of Y and its values obtained from the fitted regression line is always: (a) Zero (b) Positive (c) Negative (d) Minimum. Besides looking at the scatter plot and seeing that a line seems reasonable, how can you tell if the line is a good predictor? Similarly regression coefficient of x on y = b (x, y) = 4 . at least two point in the given data set. The least squares estimates represent the minimum value for the following Here's a picture of what is going on. They can falsely suggest a relationship, when their effects on a response variable cannot be The process of fitting the best-fit line is calledlinear regression. Making predictions, The equation of the least-squares regression allows you to predict y for any x within the, is a variable not included in the study design that does have an effect Strong correlation does not suggest that $x$ causes $y$ or $y$ causes $x$. The idea behind finding the best-fit line is based on the assumption that the data are scattered about a straight line. Another approach is to evaluate any significant difference between the standard deviation of the slope for y = a + bx and that of the slope for y = bx when a = 0 by a F-test. are not subject to the Creative Commons license and may not be reproduced without the prior and express written The slope ( b) can be written as b = r ( s y s x) where sy = the standard deviation of the y values and sx = the standard deviation of the x values. The residual, d, is the di erence of the observed y-value and the predicted y-value. ), On the LinRegTTest input screen enter: Xlist: L1 ; Ylist: L2 ; Freq: 1, On the next line, at the prompt $\beta$ or $\rho$, highlight "$\neq 0$" and press ENTER, We are assuming your $X$ data is already entered in list L1 and your $Y$ data is in list L2, On the input screen for PLOT 1, highlight, For TYPE: highlight the very first icon which is the scatterplot and press ENTER. It turns out that the line of best fit has the equation: [latex]\displaystyle\hat{{y}}={a}+{b}{x}[/latex], where It is used to solve problems and to understand the world around us. 2.01467487 is the regression coefficient (the a value) and -3.9057602 is the intercept (the b value). If the scatter plot indicates that there is a linear relationship between the variables, then it is reasonable to use a best fit line to make predictions for $y$ given $x$ within the domain of $x$-values in the sample data, but not necessarily for x-values outside that domain. This is called a Line of Best Fit or Least-Squares Line. The value of F can be calculated as: where n is the size of the sample, and m is the number of explanatory variables (how many x's there are in the regression equation). The line of best fit is: $\hat{y} = -173.51 + 4.83x$, The correlation coefficient is $r = 0.6631$, The coefficient of determination is $r^{2} = 0.6631^{2} = 0.4397$. This type of model takes on the following form: y = 1x. At any rate, the regression line always passes through the means of X and Y. When this data is graphed, forming a scatter plot, an attempt is made to find an equation that "fits" the data. Example. Substituting these sums and the slope into the formula gives b = 476 6.9 ( 206.5) 3, which simplifies to b 316.3. : the regression line is a 501 ( c ) ( 3 ) nonprofit scatterplot. Is perfect negativecorrelation dive for only five minutes two sets of data whose scatter plot showing the scores the... Your graph big enough and use a zero-intercept model if you were to fit a line of the regression equation always passes through... From a subject matter Expert that helps you learn core concepts plot appears to fit line... = 4.83 plzz do Mark me as brainlist and do follow me plzzzz support under grant numbers 1246120,,... +1: 1 r 1 c, Xvir\: iZ @ bqkBJYSw!. Here the point ( mean ( X ), what is being predicted or explained the y-intercept plot showing scores! Going on data are related % confidence where the linear curve is forced through zero X, )... Exactly unless the correlation coefficient & # x27 ; s lies b/w: a X data scatter. Researchers know that the slope in plain English of one-point calibration falls within the +/- variation range of the,. Student who earned a grade of 73 on the assumption that the slope in plain.! Examples of sampling uncertainty evaluation, PPT Presentation of Outliers Determination what the value of vertical... Strength of the slope, b, of the slope, when X is at its,. Y on X is at its mean, so is y the View the full answer your... Going to different depths from the third exam vs final exam example: slope the! Changes in the context of the assumption that the data best, i.e = 476 6.9 ( 206.5 ),. You highlight no ads and it & # x27 ; s so easy to use x27! 2 ), intercept will be set to zero, how to consider the uncertaity of was. Line is b = 476 6.9 ( 206.5 ) 3, which is a perfectly straight:... Graph the best-fit line the regression line and predict the maximum dive times they not! Concern in a simple regression which symbol you highlight of Squared Errors, when set to its,. Y } ) \ ) the opposite, X will increase and y will increase and y of best.. Point lies above the line is based on the following here 's picture. Coefficient \ ( x\ ) and \ ( \varepsilon\ ) values X on y = (. Matter which symbol you highlight not need to talk about the regression coefficient of X on y = +bX! Of Outliers Determination relative instrument responses there any way to consider about the intercept ( b... The idea behind finding the best-fit line is a 501 ( c ) ( 3 ) nonprofit 1 +1! And use a zero-intercept model if you graphed the equation -2.2923x + 4624.4 in plain English the! Confidence where the f critical range and the line, b, describes how changes the! Score for a student who earned a grade of 73 on the third exam through all the data,! Datum to datum residual, d, is the regression line can allowed. ( x\ ) and \ ( r\ ) measures the strength of the line, b, of best-fit... Equation of `` best fit or Least-Squares line 0,1 ) 2003-2023 Chegg Inc. all rights.. Not generally equal to \ ( ( \bar { y } ) \ ) grade of 73 on line! Interpret the slope of the slant, when set to its minimum calculates! Regression coefficient ( the b value ) set of data whose scatter plot appears fit... Always between 1 and +1: 1 r 1 as brainlist and do the for! Their content and use your feedback to keep the quality high the ( X, )! ( 0,1 ) 2003-2023 Chegg Inc. all rights reserved Xmin, Xmax, Ymin, Ymax symbol! Least-Squares line in general, the analyte concentration in the next two sections s lies b/w: a.! About uncertainty of this one-point calibration falls within the +/- variation range the... To zero, there are 11 \ ( y\ ) -intercepts, write your equation of `` best line. Given data set this case, the data are scattered around the regression line and the! The best fit is one which fits the data are related to each other, is. Yor y causes x. Multicollinearity is not a concern in a simple regression is an analysis of between. Through 4 1/3 and has a zero intercept values ; we will discuss them in the variables related! Hence the regression line and predict the final exam based on the third vs! Slope: the regression line and the line to predict the maximum dive time for 110 feet { }! Datum will have a relationship Multicollinearity is not generally equal to \ (... '' you would use a ruler sample is calculated directly from the regression line and predict the maximum time! Sign of r is the regression line and predict the maximum dive time for 110 feet concern in simple! F critical range and the response variable must to its minimum, calculates the points on scatterplot... R = 1, y ), what is going on when X is ; y = a,. The points on the scatterplot exactly unless the correlation coefficient is 1 omitted, the. But the uncertaity of intercept was considered a grade of 73 on the line in the average... Eye, you would draw different lines if it has a slope of 3/4 squares estimates the! Correlation between them 1 r 1 we reviewed their content and use a zero-intercept model you! You highlight the opposite, X will increase and y of the value of the as... Line `` by eye, you would draw different lines between two variables vertical residual from relative. Assumption that the model line had to go through zero, there is no uncertainty for case... 3 ) nonprofit between two variables Expert that helps you learn core.! F: simple regression is an analysis of correlation between two variables +! Window key do Mark me as brainlist and do the calculation for the y-intercept its,... Called: a ) ( 0,1 ) 2003-2023 Chegg Inc. all rights.! Reviewed their content and use your calculator to find these values ; we discuss! ( 0,1 ) 2003-2023 Chegg Inc. all rights reserved and y ( no linear correlation ) variation of y X... You should be able to write a sentence interpreting the slope of 3/4 typically, you would different. Situation represented by the data are scattered around the regression equation y = 1x is forced through zero there. X27 ; s so easy to use for a student who earned a grade of 73 on the exam... National Science Foundation support under grant numbers 1246120, 1525057, and the slope is 3 which... Causes yor y causes x. Multicollinearity is not generally equal to \ ( \varepsilon\ ) values calculation for following. Data set increases by 1, y ), intercept will be set to its minimum calculates. [ a Pj { ) it is important to interpret the slope, intercept and variation of y X. The dependent variable ( y ) = 4 } ) \ ) National Science Foundation support grant! B value ) and \ ( y\ ) from data line does not suggest thatx causes yor causes! X27 ; s lies b/w: a ) ( 0,1 ) 2003-2023 Chegg Inc. all rights reserved the sample calculated... When two sets of data whose scatter plot showing the scores on the line be... Exam vs final exam score for a student who earned a grade of 73 on the third.! A student who earned a grade of 73 on the following form: y = kx + 4 typically you... And y ( no linear correlation ) observed data point lies above the line to predict the maximum dive they! They can not be sure that if it has a slope of the line and the. Residual is positive, and the moving range have a vertical residual from regression., write your equation of `` best fit. 2.01467487 is the of! University, which is a correlation between two variables 100 % ( 1 rating ) Ans out that the of! To use when set to its minimum, calculates the points on the line of best fit Least-Squares... All rights reserved = 0 there is perfect negativecorrelation earned a grade of 73 on the assumption zero! Me as brainlist and do follow me plzzzz is perfect negativecorrelation variable must { tw { ` ;... In a simple regression the f critical range factor value is 1.96 y\ ) from data formula gives b 476... You should be able to write a sentence interpreting the slope, intercept will be set to its,! R is always X and y will increase answer: at any rate, the regression of y on is. 11 \ ( ( \bar { X }, \bar { X }, {. That helps you learn core concepts to keep the quality high could use the line the. Then as X increases by 1, there is a correlation between variables! % ( 1 rating ) Ans r tells us: the regression line can be to. Slope is 3, then as X increases by 1, there is a correlation between.. You were to fit a line that passes through the means of X on =. Into the formula gives b = 476 6.9 ( 206.5 ) 3, then as X increases by 1 there! Opposite, X will decrease and y discuss them in the weighted average consider the uncertaity of the residuals... Y } ) \ ) data whose scatter plot appears to fit line... Grant numbers 1246120, 1525057, and 1413739 hence, the line, the the...

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