point of difference

Examples of point of difference in the following topics:

• Introduction to Estimation

  • This value of 0.53 is called a point estimate of the population proportion.
  • It is called a point estimate because the estimate consists of a single value or point.
  • Point estimates are usually supplemented by interval estimates called confidence intervals.
  • Therefore a point estimate of the difference between population means is 30.7.
  • The 95% confidence interval on the difference between means extends from 19.05 to 42.35.

• Summary for inference of the difference of two means

  • When considering the difference of two means, there are two common cases: the two samples are paired or they are independent.
  • When we compute the confidence interval for µ1 −2, the point estimate is the difference in sample means, the value z* corresponds to the confidence level, and the standard error is computed from Equation (5.4) on page 217.
  • This is also true in hypothesis tests for differences of means.
  • In a hypothesis test, we apply the standard framework and use the specific formulas for the point estimate and standard error of a difference in two means.
  • When assessing the difference in two means, the point estimate takes the form $\bar{x}_1-\bar{x}_2$, and the standard error again takes the form of Equation (5.4) on page 217.

• Difference Quotients

  • The difference quotient is used in algebra to calculate the average slope between two points but has broader effects in calculus.
  • The difference between two distinct points, themselves, is known as their Delta (Δ) P, as is the difference in their function result, the particular notation being determined by the direction of formation:
  • The function difference divided by the point difference is known as the difference quotient, attributed to Isaac Newton.
  • The difference quotient is the average slope of a function between two points.
  • If |ΔP| is finite, meaning measurable, then ΔF(P) is known as a finite difference, with specific denotations of DP and DF(P).

• Linear Perspective

  • Any number of vanishing points are possible in a drawing, one for each set of parallel lines that are at an angle relative to the plane of the drawing.
  • Looking up at a tall building is another common example of the third vanishing point.
  • Four-point perspective, also called infinite-point perspective, is the curvilinear variant of two-point perspective.
  • Images of railroad tracks are a common example of one-point perspective.
  • Looking up at a tall building is a common example of the third vanishing point, where the third vanishing point is positioned high in space.

• Maximum and Minimum Values

  • The second partial derivative test is a method used to determine whether a critical point is a local minimum, maximum, or saddle point.
  • The second partial derivative test is a method in multivariable calculus used to determine whether a critical point $(a,b, \cdots )$ of a function $f(x,y, \cdots )$ is a local minimum, maximum, or saddle point.
  • If $M(a,b)>0$ and $f_{xx}(a,b)>0$, then $(a,b)$ is a local minimum of $f$.
  • There are substantial differences between functions of one variable and functions of more than one variable in the identification of global extrema.
  • Apply the second partial derivative test to determine whether a critical point is a local minimum, maximum, or saddle point

• The Distance Formula and Midpoints of Segments

  • In analytic geometry, the distance between two points of the $xy$-plane can be found using the distance formula.  
  • Notice that the length between each point and the triangle's right angle is found by calculating the difference between the $y$-coordinates and $x$-coordinates, respectively.  
  • In geometry, the midpoint is the middle point of a line segment, or the middle point of two points on a line, and thus is equidistant from both end-points.
  • If you have two points, $(x_{1},y_{1})$ and $(x_{2},y_{2})$, the midpoint of the segment connecting the two points can be found with the formula:
  • Calculate the midpoint of a line segment and the distance between two points on a plane

• Introduction to inference for numerical data

  • In this chapter, we encounter several new point estimates and scenarios.
  • Identify an appropriate distribution for the point estimate or test statistic.
  • Each section in Chapter 5 explores a new situation: the difference of two means (5.1, 5.2); a single mean or difference of means where we relax the minimum sample size condition (5.3, 5.4); and the comparison of means across multiple groups (5.5).

• Electric Potential Due to a Point Charge

  • The electric potential of a point charge Q is given by $V=\frac{kQ}{r}$.
  • The potential difference between two points ΔV is often called the voltage and is given by
  • Point charges, such as electrons, are among the fundamental building blocks of matter.
  • The potential of the charged conducting sphere is the same as that of an equal point charge at its center.
  • Express the electric potential generated by a single point charge in a form of equation

• Transitions

  • There are different types of transitions often used in speeches, including:
  • Introductions and summaries are also types of transitions to let listeners know what a person will be speaking about and offering a way to understand the important parts of a speech
  • To move from one point and into the next, you'll want to segue into your new point.
  • If you have multiple pieces of supporting evidence, you may need to transition between examples so that your audience knows you are furthering a point with another example, anecdote or set of researched data.
  • Try to think of transitions as a way to connect the dots of your speech's purpose.

• Point-Slope Equations

  • The point-slope equation is a way of describing the equation of a line.
  • Now choose either of the two points, such as $(-3,6)$.
  • The only difference is the form that they are written in.
  • Graph of the line $y-1=-4(x-2)$, through the point $(2,1)$ with slope of $-4$, as well as the slope-intercept form, $y=-4x+9$.
  • Use point-slope form to find the equation of a line passing through two points and verify that it is equivalent to the slope-intercept form of the equation