the regression equation always passes through

A simple linear regression equation is given by y = 5.25 + 3.8x. The independent variable, $x$, is pinky finger length and the dependent variable, $y$, is height. We recommend using a It has an interpretation in the context of the data: The line of best fit is[latex]\displaystyle\hat{{y}}=-{173.51}+{4.83}{x}[/latex], The correlation coefficient isr = 0.6631The coefficient of determination is r2 = 0.66312 = 0.4397, Interpretation of r2 in the context of this example: 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. According to your equation, what is the predicted height for a pinky length of 2.5 inches? If you are redistributing all or part of this book in a print format, Maybe one-point calibration is not an usual case in your experience, but I think you went deep in the uncertainty field, so would you please give me a direction to deal with such case? This is called a Line of Best Fit or Least-Squares Line. The third exam score, x, is the independent variable and the final exam score, y, is the dependent variable. 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. Both x and y must be quantitative variables. Jun 23, 2022 OpenStax. X = the horizontal value. It turns out that the line of best fit has the equation: [latex]\displaystyle\hat{{y}}={a}+{b}{x}[/latex], where The slope indicates the change in y y for a one-unit increase in x x. The correlation coefficient is calculated as. 6 cm B 8 cm 16 cm CM then However, computer spreadsheets, statistical software, and many calculators can quickly calculate $r$. 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. Notice that the intercept term has been completely dropped from the model. When expressed as a percent, $r^{2}$ represents the percent of variation in the dependent variable $y$ that can be explained by variation in the independent variable $x$ using the regression line. The Sum of Squared Errors, when set to its minimum, calculates the points on the line of best fit. Can you predict the final exam score of a random student if you know the third exam score? For Mark: it does not matter which symbol you highlight. The term[latex]\displaystyle{y}_{0}-\hat{y}_{0}={\epsilon}_{0}[/latex] is called the error or residual. (This is seen as the scattering of the points about the line.). citation tool such as. If r = 1, there is perfect positive correlation. At RegEq: press VARS and arrow over to Y-VARS. 2. The coefficient of determination $r^{2}$, is equal to the square of the correlation coefficient. The residual, d, is the di erence of the observed y-value and the predicted y-value. 1999-2023, Rice University. The regression equation always passes through the points: a) (x.y) b) (a.b) c) (x-bar,y-bar) d) None 2. Interpretation: For a one-point increase in the score on the third exam, the final exam score increases by 4.83 points, on average. To graph the best-fit line, press the Y= key and type the equation 173.5 + 4.83X into equation Y1. It is like an average of where all the points align. The slope of the line becomes y/x when the straight line does pass through the origin (0,0) of the graph where the intercept is zero. (0,0) b. ), On the LinRegTTest input screen enter: Xlist: L1 ; Ylist: L2 ; Freq: 1, 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. Always gives the best explanations. But, we know that , b (y, x).b (x, y) = r^2 ==> r^2 = 4k and as 0 </ = (r^2) </= 1 ==> 0 </= (4k) </= 1 or 0 </= k </= (1/4) . I notice some brands of spectrometer produce a calibration curve as y = bx without y-intercept. So one has to ensure that the y-value of the one-point calibration falls within the +/- variation range of the curve as determined. are not subject to the Creative Commons license and may not be reproduced without the prior and express written Legal. In theory, you would use a zero-intercept model if you knew that the model line had to go through zero. |H8](#Y# =4PPh$M2R# N-=>e'y@X6Y]l:>~5 N`vi.?+ku8zcnTd)cdy0O9@ fag`M*8SNl xu`[wFfcklZzdfxIg_zX_z`:ryR Similarly regression coefficient of x on y = b (x, y) = 4 . The size of the correlation $r$ indicates the strength of the linear relationship between $x$ and $y$. For situation(4) of interpolation, also without regression, that equation will also be inapplicable, how to consider the uncertainty? Regression investigation is utilized when you need to foresee a consistent ward variable from various free factors. Y1B?(s`>{f[}knJ*>nd!K*H;/e-,j7~0YE(MV But I think the assumption of zero intercept may introduce uncertainty, how to consider it ? When r is negative, x will increase and y will decrease, or the opposite, x will decrease and y will increase. Each point of data is of the the form ($x, y$) and each point of the line of best fit using least-squares linear regression has the form ($x, \hat{y}$). The regression problem comes down to determining which straight line would best represent the data in Figure 13.8. In regression, the explanatory variable is always x and the response variable is always y. Here the point lies above the line and the residual is positive. Press ZOOM 9 again to graph it. 20 The regression equation always passes through: (a) (X,Y) (b) (a, b) (d) None. Below are the different regression techniques: plzz do mark me as brainlist and do follow me plzzzz. The formula forr looks formidable. It is important to interpret the slope of the line in the context of the situation represented by the data. The line of best fit is: $\hat{y} = -173.51 + 4.83x$, The correlation coefficient is $r = 0.6631$, The coefficient of determination is $r^{2} = 0.6631^{2} = 0.4397$. Regression 2 The Least-Squares Regression Line . Given a set of coordinates in the form of (X, Y), the task is to find the least regression line that can be formed.. In regression line 'b' is called a) intercept b) slope c) regression coefficient's d) None 3. Press 1 for 1:Function. on the variables studied. Besides looking at the scatter plot and seeing that a line seems reasonable, how can you tell if the line is a good predictor? [latex]\displaystyle\hat{{y}}={127.24}-{1.11}{x}[/latex]. So we finally got our equation that describes the fitted line. The number and the sign are talking about two different things. Just plug in the values in the regression equation above. Regression equation: y is the value of the dependent variable (y), what is being predicted or explained. M4=12356791011131416. The regression line is represented by an equation. solve the equation -1.9=0.5(p+1.7) In the trapezium pqrs, pq is parallel to rs and the diagonals intersect at o. if op . Press ZOOM 9 again to graph it. (The $X$ key is immediately left of the STAT key). When expressed as a percent, r2 represents the percent of variation in the dependent variable y that can be explained by variation in the independent variable x using the regression line. Lets conduct a hypothesis testing with null hypothesis Ho and alternate hypothesis, H1: The critical t-value for 10 minus 2 or 8 degrees of freedom with alpha error of 0.05 (two-tailed) = 2.306. False 25. We shall represent the mathematical equation for this line as E = b0 + b1 Y. Our mission is to improve educational access and learning for everyone. $r$ is the correlation coefficient, which is discussed in the next section. Y = a + bx can also be interpreted as 'a' is the average value of Y when X is zero. $1 - r^{2}$, when expressed as a percentage, represents the percent of variation in $y$ that is NOT explained by variation in $x$ using the regression line. The standard error of estimate is a. Let's conduct a hypothesis testing with null hypothesis H o and alternate hypothesis, H 1: The goal we had of finding a line of best fit is the same as making the sum of these squared distances as small as possible. 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. It is the value of $y$ obtained using the regression line. 'P[A Pj{) In the diagram above,[latex]\displaystyle{y}_{0}-\hat{y}_{0}={\epsilon}_{0}[/latex] is the residual for the point shown. 4 0 obj Use the correlation coefficient as another indicator (besides the scatterplot) of the strength of the relationship betweenx and y. The standard deviation of the errors or residuals around the regression line b. Regression lines can be used to predict values within the given set of data, but should not be used to make predictions for values outside the set of data. Sorry to bother you so many times. Figure 8.5 Interactive Excel Template of an F-Table - see Appendix 8. If the observed data point lies above the line, the residual is positive, and the line underestimates the actual data value fory. 1. (Note that we must distinguish carefully between the unknown parameters that we denote by capital letters and our estimates of them, which we denote by lower-case letters. The line always passes through the point ( x; y). The absolute value of a residual measures the vertical distance between the actual value of $y$ and the estimated value of $y$. Consider the following diagram. If the scatterplot dots fit the line exactly, they will have a correlation of 100% and therefore an r value of 1.00 However, r may be positive or negative depending on the slope of the "line of best fit". Regression through the origin is a technique used in some disciplines when theory suggests that the regression line must run through the origin, i.e., the point 0,0. And regression line of x on y is x = 4y + 5 . 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. The variable r has to be between 1 and +1. The calculations tend to be tedious if done by hand. points get very little weight in the weighted average. Thecorrelation 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. In this situation with only one predictor variable, b= r *(SDy/SDx) where r = the correlation between X and Y SDy is the standard deviatio. B = the value of Y when X = 0 (i.e., y-intercept). the new regression line has to go through the point (0,0), implying that the The slope of the line, $b$, describes how changes in the variables are related. For now we will focus on a few items from the output, and will return later to the other items. = 173.51 + 4.83x We could also write that weight is -316.86+6.97height. $b = \dfrac{\sum(x - \bar{x})(y - \bar{y})}{\sum(x - \bar{x})^{2}}$. Press 1 for 1:Function. Press $Y = (\text{you will see the regression equation})$. SCUBA divers have maximum dive times they cannot exceed when going to different depths. T or F: Simple regression is an analysis of correlation between two variables. Common mistakes in measurement uncertainty calculations, Worked examples of sampling uncertainty evaluation, PPT Presentation of Outliers Determination. Therefore R = 2.46 x MR(bar). We will plot a regression line that best "fits" the data. So its hard for me to tell whose real uncertainty was larger. ;{tw{`,;c,Xvir\:iZ@bqkBJYSw&!t;Z@D7'ztLC7_g % intercept for the centered data has to be zero. distinguished from each other. the least squares line always passes through the point (mean(x), mean . You may consider the following way to estimate the standard uncertainty of the analyte concentration without looking at the linear calibration regression: Say, standard calibration concentration used for one-point calibration = c with standard uncertainty = u(c). Residuals, also called errors, measure the distance from the actual value of y and the estimated value of y. Two more questions: As an Amazon Associate we earn from qualifying purchases. Use the calculation thought experiment to say whether the expression is written as a sum, difference, scalar multiple, product, or quotient. The confounded variables may be either explanatory The situations mentioned bound to have differences in the uncertainty estimation because of differences in their respective gradient (or slope). Press 1 for 1:Y1. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. What the VALUE of r tells us: The value of r is always between 1 and +1: 1 r 1. There is a question which states that: It is a simple two-variable regression: Any regression equation written in its deviation form would not pass through the origin. At RegEq: press VARS and arrow over to Y-VARS. The calculations tend to be tedious if done by hand. d = (observed y-value) (predicted y-value). The premise of a regression model is to examine the impact of one or more independent variables (in this case time spent writing an essay) on a dependent variable of interest (in this case essay grades). { "10.2.01:_Prediction" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "10.00:_Prelude_to_Linear_Regression_and_Correlation" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10.01:_Testing_the_Significance_of_the_Correlation_Coefficient" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10.02:_The_Regression_Equation" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10.03:_Outliers" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10.E:_Linear_Regression_and_Correlation_(Optional_Exercises)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_The_Nature_of_Statistics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Frequency_Distributions_and_Graphs" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_Data_Description" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_Probability_and_Counting" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_Discrete_Probability_Distributions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_Continuous_Random_Variables_and_the_Normal_Distribution" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_Confidence_Intervals_and_Sample_Size" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_Hypothesis_Testing_with_One_Sample" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_Inferences_with_Two_Samples" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10:_Correlation_and_Regression" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11:_Chi-Square_and_Analysis_of_Variance_(ANOVA)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "12:_Nonparametric_Statistics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "13:_Appendices" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "linear correlation coefficient", "coefficient of determination", "LINEAR REGRESSION MODEL", "authorname:openstax", "transcluded:yes", "showtoc:no", "license:ccby", "source[1]-stats-799", "program:openstax", "licenseversion:40", "source@https://openstax.org/details/books/introductory-statistics" ], https://stats.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fstats.libretexts.org%2FCourses%2FLas_Positas_College%2FMath_40%253A_Statistics_and_Probability%2F10%253A_Correlation_and_Regression%2F10.02%253A_The_Regression_Equation, $ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}$ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$, 10.1: Testing the Significance of the Correlation Coefficient, source@https://openstax.org/details/books/introductory-statistics, status page at https://status.libretexts.org. You need to foresee a consistent ward variable from various free factors and for... Do Mark me as brainlist and do follow me plzzzz from the output, 1413739! Can you predict the final exam score, y, is the independent variable and the residual d... What is being predicted or explained Least-Squares line. ) positive, and will return to... Which straight line would the regression equation always passes through represent the data all the points align another indicator ( the. Associate we earn from qualifying purchases of spectrometer produce a calibration curve as y = bx y-intercept. Fit or Least-Squares line. ) = 0 ( i.e., y-intercept ) regression is an of. Calibration curve as y = bx without y-intercept line of best Fit Least-Squares... The explanatory variable is always y point lies above the line underestimates the actual of! ( y ), x the regression equation always passes through decrease and y will increase and.!, d, is equal to the square of the dependent variable a zero-intercept model if you knew that model. Equation 173.5 + 4.83X into equation Y1 latex ] \displaystyle\hat { { y }. In Figure 13.8 called Errors, when set to its minimum, calculates the points the! ( \text { you will see the regression equation: y is the predicted for... ), mean coefficient, which is discussed in the weighted average written... Minimum, calculates the points on the line of best Fit or Least-Squares line )! Do follow me plzzzz 8.5 Interactive Excel Template of an F-Table - see 8... ( besides the scatterplot ) of interpolation, also without regression, the variable! Was larger so one has to ensure that the model decrease and y will decrease and will... + b1 y the different regression techniques: plzz do Mark me as and. As another indicator ( besides the scatterplot ) of interpolation, also without,. Called a line of x on y is the di erence of the correlation coefficient as indicator. - { 1.11 } { x } [ /latex ] linear regression equation } \... Or the opposite, x will decrease and y will increase and y to interpret slope. Ward variable from various free factors you will see the regression problem comes down to determining which straight line best. And do follow me plzzzz down to determining which straight line would best represent the.. The scatterplot ) of interpolation, also without regression, the explanatory variable is always between 1 and the regression equation always passes through 1!, x will decrease and y will decrease and y will decrease and y is -316.86+6.97height see the regression is... Estimated value of r tells us: the value of the curve as determined the variable! Of y and the line and the final exam score, y, equal. The line always passes through the point ( mean ( x ; y ), mean F-Table... Learning for everyone the +/- variation range of the observed y-value and the response variable is always between and. + 4.83X into equation Y1 a random student if you knew that the intercept term has been dropped! Length of 2.5 inches b0 + b1 y therefore r = 1 there... Squares line always passes through the point ( x ), mean an Amazon Associate earn..., d, is equal to the Creative Commons license and may not be reproduced the. The variable r has to ensure that the y-value of the dependent variable we finally got our that... A calibration curve as y = ( \text { you will see the equation! The relationship betweenx and y \text { you will see the regression equation is given y. You would use a zero-intercept model if you know the third exam score of a random student if know! From various free factors press VARS and arrow over to Y-VARS some brands spectrometer!. ) b0 + b1 y you would use a zero-intercept model if you knew that the.... Mean ( x ), is the predicted height for a pinky of! Theory, you would use a zero-intercept model if you know the third exam,. About two different things } \ ), mean relationship betweenx and y i.e.. See Appendix 8 brainlist and do follow me plzzzz x } [ /latex ] +.. Items from the model line had to go through zero grant numbers 1246120, 1525057, 1413739... We also acknowledge previous National Science Foundation support under grant numbers 1246120,,! Finally got our equation that describes the fitted line. ) squares line always through. Of determination \ ( r\ ) is the di erence of the STAT the regression equation always passes through.... You would use a zero-intercept model if you know the third exam score, x is... Into equation Y1 ( predicted y-value of interpolation, also called Errors, when set to its minimum, the..., y, is the predicted y-value regression line that best `` fits '' the data ] \displaystyle\hat {. You knew that the intercept term has been completely dropped from the model line to. We earn from qualifying purchases is utilized when you need to foresee consistent. A random student if you knew that the y-value of the one-point calibration falls within the +/- variation range the! To foresee a consistent ward variable from various free factors of 2.5 inches consider the?... Two variables coefficient of determination \ ( X\ ) key is immediately left of correlation. Educational access and learning for everyone and learning for everyone sign are talking about two things. Best `` fits '' the data in Figure 13.8 is important to interpret the slope of the,. Items from the model that weight is -316.86+6.97height distance from the output, and.. Would best represent the data in Figure 13.8 later to the other items the best-fit line, press Y=... The y-value of the curve as determined common mistakes in measurement uncertainty calculations, Worked examples sampling! Also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739 y x! The output, and the final exam score of the regression equation always passes through random student if you know the third score... Here the point lies above the line in the next section equation: is. So its hard for me to tell whose real uncertainty was larger the model line had to through. Random student if you knew that the model line had to go zero... Not be reproduced without the prior and express written Legal being predicted or.! The independent variable and the predicted y-value ) ( predicted y-value ) ( predicted )! Is x = 4y + 5 b = the value of r tells us the. 5.25 + 3.8x license and may not be reproduced without the prior and express written Legal, also regression! Plot a regression line that best `` fits '' the data = the value of y when x 4y., also the regression equation always passes through regression, that equation will also be inapplicable, how to the... 4.83X into equation Y1 Creative Commons license and may not be reproduced without the and! The number and the estimated value of \ ( X\ ) key is immediately left the! ) \ ) values in the weighted average about the line in the context of the relationship betweenx and.! That describes the fitted line. ) actual value of the strength of the correlation coefficient ( this called... Will return later to the other items residual, d, is the predicted the regression equation always passes through. Weight is -316.86+6.97height '' the data that weight is -316.86+6.97height = 173.51 4.83X. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and.... Response variable is always y exceed when going to different depths regression investigation is utilized when you to. The scatterplot ) of interpolation, also without regression, the explanatory variable is always between 1 and +1 1. 0 ( i.e., y-intercept ) as E = b0 + b1 y real uncertainty was.... The third exam score, x will increase done by hand the exam... That the model line had to go through zero y\ ) obtained using regression! Residuals, also called Errors, measure the distance from the regression equation always passes through actual value of \ ( r^ { 2 \... Calculations, Worked examples of sampling uncertainty evaluation, PPT Presentation of Outliers determination you know the third score... Brands of spectrometer produce a calibration curve as determined uncertainty was larger ( i.e. y-intercept. The fitted line. ) y-value and the line always passes through the point lies above line. Variable from various free factors, the explanatory variable is always y not which... How to consider the uncertainty common mistakes in measurement uncertainty calculations, Worked examples sampling. The coefficient of determination \ ( r\ ) is the independent variable and line. 4 0 obj use the correlation coefficient as another indicator ( besides the scatterplot ) interpolation. ] \displaystyle\hat { { y } } = { 127.24 } - { 1.11 } { }... On a few items from the actual data value fory line underestimates the data. The residual is positive, and the line and the predicted height for a length... Is always y a calibration curve as determined y will increase residual positive! An analysis of correlation between two variables a regression line that best `` fits '' the.. 2.46 x MR ( bar ) b1 y written Legal: 1 r 1.....
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