slope

Examples of slope in the following topics:

• Slope and Y-Intercept of a Linear Equation

  • For the linear equation y = a + bx, b = slope and a = y-intercept.
  • What is the y-intercept and what is the slope?
  • The slope is 15 (b = 15).
  • (a) If b > 0, the line slopes upward to the right.
  • (c) If b < 0, the line slopes downward to the right.

• Slope and Intercept

  • The concepts of slope and intercept are essential to understand in the context of graphing data.
  • The slope or gradient of a line describes its steepness, incline, or grade.
  • A higher slope value indicates a steeper incline.
  • Slope is normally described by the ratio of the "rise" divided by the "run" between two points on a line.
  • It also acts as a reference point for slopes and some graphs.

• Slope and Intercept

  • In the regression line equation the constant $m$ is the slope of the line and $b$ is the $y$-intercept.
  • The constant $$$m$ is slope of the line and $b$ is the $y$-intercept -- the value where the line cross the $y$ axis.
  • An equation where y is the dependent variable, x is the independent variable, m is the slope, and b is the intercept.

• Making Inferences About the Slope

  • The slope of the regression line describes how changes in the variables are related.
  • It is important to interpret the slope of the line in the context of the situation represented by the data.
  • You should be able to write a sentence interpreting the slope in plain English.
  • where $y$ is the dependent variable, $x$ is the independent variable, $m$ is the slope, and $b$ is the intercept.
  • Infer how variables are related based on the slope of a regression line

• The Equation of a Line

  • The slope of the fitted line is equal to the correlation between $y$ and $x$ corrected by the ratio of standard deviations of these variables.
  • Where $m$ (slope) and $b$ (intercept) designate constants.
  • In this particular equation, the constant $m$ determines the slope or gradient of that line, and the constant term $b$ determines the point at which the line crosses the $y$-axis, otherwise known as the $y$-intercept.
  • Three lines — the red and blue lines have the same slope, while the red and green ones have same y-intercept.

• Inferential Statistics for b and r

  • This section shows how to conduct significance tests and compute confidence intervals for the regression slope and Pearson's correlation.
  • As you will see, if the regression slope is significantly different from zero, then the correlation coefficient is also significantly different from zero.
  • As applied here, the statistic is the sample value of the slope (b) and the hypothesized value is 0.
  • Therefore, the slope is not significantly different from 0.
  • The method for computing a confidence interval for the population slope is very similar to methods for computing other confidence intervals.

• Exercises

  • b. the slope of the regression line and test if it differs significantly from zero.
  • If the standard error of b is .4, is the slope statistically significant at the .05 level?
  • True/false: If the slope of a simple linear regression line is statistically significant, then the correlation will also always be significant.
  • Test to see if the slope is significantly different from 0.
  • (f) Comment on possible assumption violations for the test of the slope.

• Finding the least squares line

  • The slope of the least squares line can be estimated by
  • You might recall the point-slope form of a line from math class (another common form is slope-intercept).
  • Given the slope of a line and a point on the line, (x0,y0), the equation for the line can be written as: y−y0 = slope×(x−x0)
  • Noting that the point ($y-\bar{y}=b_1(x-\bar{x})$) is on the least squares line, use x0 = $y-\bar{y}=b_1(x-\bar{x})$ and y0 = $y-\bar{y}=b_1(x-\bar{x})$ along with the slope b1 in the point-slope equation: $y-\bar{y}=b_1(x-\bar{x})$
  • Apply the point-slope equation using (101.8,19.94) and the slope b1 = −0.0431:

• Estimating and Making Inferences About the Slope

  • $b_1$ is the estimated slope of a regression of $Y$ on $X_1$, if all of the other $X$ variables could be kept constant.
  • When the purpose of multiple regression is prediction, the important result is an equation containing partial regression coefficients (slopes).
  • Discuss how partial regression coefficients (slopes) allow us to predict the value of $Y$ given measured $X$ values.

• Midterm elections and unemployment

  • There is a negative slope in the line shown in Figure 7.20.
  • However, this slope (and the y-intercept) are only estimates of the parameter values.
  • We might wonder, is this convincing evidence that the "true" linear model has a negative slope?
  • The true linear model has slope zero.
  • The true linear model has a slope less than zero.