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A linear regression fits the lineor as Excel prefers to call itto the existing data set. It does so through a technique known as minimizing the sum of the squares of the error terms.
To get the complete result of a regression analysis, select a range 5 rows by 2 columns and array-enter the LINEST function as shown in Figure 4. The first row contains the 2 coefficients a1 and a0 respectively. The rest of the information is important in understanding how well the regression line fits the data, how significant the individual coefficients are, as well as the significance of the regression as a whole.
It also contains key elements needed to build confidence intervals for interpolated or extrapolated estimates, a subject covered in the section titled Confidence Intervals. But, first, we start with some nomenclature.
The number of data points is given by n. The number of independent variables is given by k. If a constant is included in the regression, it increases k by 1.
Each of the recorded observations is denoted by the pair of values. For eachthe value predicted by the regression is given by. The value that one gets from the regression is. For reasons that will soon be apparent, we start with the last row.
There are two values and. These are aggregate measures of something we have already looked at the level of an individual data point. Recall that for each individual data point, the measure of how much the regression explains is and how much remains unexplained is.
The first is the sum of the squared values of how well the regression fits the data or. The second is the sum of the squared values of how much remains unexplained or. Row 4 contains two values: The degrees of freedom is given by the expression n-k, where n and k are explained earlier in this section.
The F statistic, or the observed F-value, is a measure of the significance of the regression as a whole. For the technically minded it tests the null hypothesis that all of the coefficients are insignificant against the alternative hypothesis that at least one of the coefficients is significant.
While Excel provides the value, it can also be computed as. This, the observed F-value, is then compared against a critical F-value, F a, v1, v2where a is 1 - the level of significance we are interested in, and v1 and v2 are as calculated below.
If the observed F-value is greater than the critical F-value, it means the regression as a whole is significant. Row 3 contains the two metrics, R2 and the SEreg. The R2 is measure of how well the regression fits the observed data.
It ranges from 0 to 1 and the closer to 1 the better the fit. Mathematically, it is calculated aswhere each term is explained above. Graphically, in terms of Figure 5, it is a measure of how close the regression line is to all of the observations.
Suppose the regression line were to pass through every observation.
The second item in this row is the Standard Error of the regression, or SEreg. It can also be calculated from what we already know, i.
Keep in mind that n-k is also the df value in row 4 of the result. SSreg will play a role in calculating the confidence intervals later in this chapter. Row 2 provides the standard error of each coefficient, or.
The section Understanding the result addresses how these errors help determine if the coefficients are significant. Used as an array formula in a 5 rows by X columns range, LINEST returns not only the coefficients but also other statistical information about the results.
Some might find it surprising but the Excel documentation for LINEST does a very good functional job of explaining not only contents of all the rows but also the statistical value of that information.
Nonetheless, we will look at one key element of the result that bears repeating — it is overlooked by way too many users of LINEST. Figure 6 When array-entered in 5 rows D2: In row 2 it provides the standard error of each of the coefficients.
Taken together, the two rows contain critical information in estimating whether each of the coefficients is statistically different from zero. Dividing the absolute value of the coefficient value by the standard error yields what is known as the observed t-result, or Comparing the absolute value of this t-result with the corresponding critical t-value lets one decide whether that coefficient should be treated as zero.Slope Intercept Form Showing top 8 worksheets in the category - Slope Intercept Form.
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Definition of a Trend Line. A trend line, often referred to as a line of best fit, is a line that is used to represent the behavior of a set of data to determine if there is a certain pattern.A. This 2-page worksheet gives practice working with equations in Slope Intercept Form (y = mx + b).
Write the linear equation that defines the relationship between the X and Y values in the form y = mx + b. as the Expressions and equations Worksheet shown above. Equation of a line in slope intercept form, as well as how to find equation given slope and one point.
Includes you-tube video Lesson with pictures and many example problems. Introduction. A trendline shows the trend in a data set and is typically associated with regression analysis.
Creating a trendline and calculating its coefficients allows for the quantitative analysis of the underlying data and the ability to both interpolate and extrapolate the data for forecast purposes. This is just a short excerpt for the about page.