11.1 线性关系

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  线线关系

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  Imagine walking through the electronics section of your local department store. On the wall are examples of dozens of television sets, from little 19″ units made to sit on a kitchen counter to 72″+ monsters meant to be the centerpiece of a home theater. Looking at the prices, you note without surprise that the 72″ model is more expensive than the 19″, and a 42″ model is priced in between. It seems rather clear that as the TV gets larger, the price goes up. Does that mean increased screen size causes increased price?
  ::想象一下在您本地百货商店的电子产品区行走。 在墙上,有几十台电视机的例子,从在厨房柜台坐着的19个小单位到72个怪物,这些怪物本来是一家家庭剧院的中心。看看价格,你毫不惊讶地注意到72个小机型比19个小机型更贵,而一个42个小机型在中间定价。似乎相当清楚的是,随着电视越大,价格就越高。这是否意味着屏幕越大导致价格上涨?

  

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  Linear Relationships
  ::线线关系

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  When two quantities are compared, it is not uncommon to note a relationship between them that indicates both quantities increase and decrease at the same time, or that one increases as the other decreases. If both quantities are plotted on coordinate axes, the data points show a general or definite linear trend.
  ::当比较两个数量时,发现它们之间的关系同时表明数量增加和减少,或表明一个数量增加与另一个减少之间的关系并不罕见。 如果两个数量都在坐标轴上绘制,则数据点显示一般或明确的线性趋势。

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  If the points actually form a clearly defined line, the variables may be an example of a deterministic relationship. A deterministic relationship indicates that the value of one variable can be reliably and accurately determined by the manipulation of the other variable. An example might be inches and centimeters: one inch is the same as 2.54 centimeters. If you know how many inches long something is, you can reliably and accurately calculate the number of centimeters long the same item is.
  ::如果点实际上形成一个明确界定的线条,变量可以是确定性关系的一个示例。确定性关系表明,一个变量的价值可以通过对另一个变量的操纵而可靠和准确地确定。一个示例可以是英寸和厘米:一英寸与2.54厘米相同。如果知道有多少英寸长的东西,那么可以可靠和准确地计算出一个变量长的厘米数。

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  As you likely recall from Algebra, the  slope describes the angle of the line created by plotting points from a linear relationship , and the point where the  explanatory variable has a value of zero is called the $\begin{align*}y\end{align*}$ -intercept (commonly denoted $\begin{align*}b\end{align*}$ ).
  ::正如您从代数中可能记得的,斜坡描述线条从线性关系中绘制点所创建的线条角,解释变量零值的点称为y-intercut(通常指b)。

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  Often, particularly in research situations when one or both variables are measured, the plotted values are generally linear, but do not line up precisely. When two variables seem to show a linear relationship, but the values display some amount of randomness, we commonly visually describe the relationship with a scatter plot . As you will see throughout this chapter, the strength of the linear relationship of the variables can be described through mathematics.
  ::通常情况下,特别是在测量一个或两个变量的研究情况下,绘图中的数值一般是线性的,但并不精确地排列。当两个变量似乎显示线性关系,但数值显示的是某种程度的随机性时,我们通常以视觉方式描述与散射图的关系。正如你们在本章中看到的,变量线性关系的强度可以通过数学来描述。

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  Finding the Slope and Y Intercept of an Equation 
  ::查找一个方形的斜坡和 Y 截取器

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  1. Given the equation $\begin{align*}y=2.3x+5\end{align*}$ :
  ::1. 根据y=2.3x+5的方程:

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  a. Create an  $\begin{align*}X-Y\end{align*}$  table to describe the values of at least four points 
  ::a. 创建 X-Y 表格以描述至少四个点的值

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  Pick a value for $\begin{align*}x\end{align*}$ , substitute the chosen value for  $\begin{align*}x\end{align*}$ in the equation, and calculate $\begin{align*}y\end{align*}$ :
  ::选择 x 的值, 替换公式中的 x 的选定值, 计算 y :

  播放段落 $\begin{align*}X\end{align*}$ ::X 十	播放段落 calculation ::计算计算计算	播放段落 $\begin{align*}Y\end{align*}$ ::Y Y Y
  1	$\begin{align*}y=2.3(1)+5\end{align*}$	7.3
  2	$\begin{align*}y=2.3(2)+5\end{align*}$	9.6
  0	$\begin{align*}y=2.3(0)+5\end{align*}$	5
  -1	$\begin{align*}y=2.3(-1)+5\end{align*}$	2.7

  

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  The equation in the problem is in  $\begin{align*}y=mx+b\end{align*}$ form (also known as slope-intercept form ), where  $\begin{align*}b\end{align*}$ is the $\begin{align*}y\end{align*}$ -value when $\begin{align*}x=0\end{align*}$ , and  $\begin{align*}m\end{align*}$ is the slope of the line. Therefore:
  ::问题所在方程式为 y=mx+b 窗体(又称斜度界面), b 是 y 值, x=0, m 是 线的斜度。 因此 :

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  b. What is the slope of the line?
  ::b. 线的斜坡是什么?

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  $\begin{align*}m=2.3\end{align*}$
  ::m=2.3

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  c. What is the  $\begin{align*}y\end{align*}$ -intercept?
  :c) 什么是Y的接口?

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    $\begin{align*}b=5\end{align*}$
  ::b=5 (b=5)

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  2. Given the equation $\begin{align*}y=-3x+3.9\end{align*}$ :
  ::2. 考虑到y3x+3.9的等式:

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  a. Create an  $\begin{align*}X-Y\end{align*}$  table to describe the values of at least four points
  ::a. 创建 X-Y 表格以描述至少四个点的值

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  Pick a value for $\begin{align*}x\end{align*}$ , substitute the chosen value for  $\begin{align*}x\end{align*}$ in the equation, and calculate $\begin{align*}y\end{align*}$ :
  ::选择 x 的值, 替换公式中的 x 的选定值, 计算 y :

  播放段落 $\begin{align*}X\end{align*}$ ::X 十	播放段落 calculation ::计算计算计算	播放段落 $\begin{align*}Y\end{align*}$ ::Y Y Y
  1	$\begin{align*}y=-3(1)+3.9\end{align*}$	6.9
  2	$\begin{align*}y=-3(2)+3.9\end{align*}$	9.9
  0	$\begin{align*}y=-3(0)+3.9\end{align*}$	3.9
  -1	$\begin{align*}y=-3(-1)+3.9\end{align*}$	0.9

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  The equation in the problem is in  $\begin{align*}y=mx+b\end{align*}$ form (also known as slope-intercept form ), where  $\begin{align*}b\end{align*}$ is the $\begin{align*}y\end{align*}$ -value when $\begin{align*}x=0\end{align*}$ , and  $\begin{align*}m\end{align*}$ is the slope of the line. Therefore:
  ::问题所在方程式为 y=mx+b 窗体(又称斜度界面), b 是 y 值, x=0, m 是 线的斜度。 因此 :

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  b. What is the slope of the line?
  ::b. 线的斜坡是什么?

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  $\begin{align*}m=-3\end{align*}$
  ::m3

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  c. What is the  $\begin{align*}y\end{align*}$ -intercept?
  :c) 什么是Y的接口?

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    $\begin{align*}b=3.9\end{align*}$
  ::b=3.9

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  3. Given the equation $\begin{align*}y=-2.8x-9.1\end{align*}$ :
  ::3. 根据y2.8x-9.1.1:

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  a. Create an  $\begin{align*}X-Y\end{align*}$  table to describe the values of at least four points?
  ::a. 创建一个X-Y表,说明至少四个点的值?

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  Pick a value for $\begin{align*}x\end{align*}$ , substitute the chosen value for  $\begin{align*}x\end{align*}$ in the equation, and calculate $\begin{align*}y\end{align*}$ :
  ::选择 x 的值, 替换公式中的 x 的选定值, 计算 y :

  播放段落 $\begin{align*}X\end{align*}$ ::X 十	播放段落 calculation ::计算计算计算	播放段落 $\begin{align*}Y\end{align*}$ ::Y Y Y
  1	$\begin{align*}y=-2.8(1)-9.1\end{align*}$	-11.9
  2	$\begin{align*}y=-2.8(2)-9.1\end{align*}$	-14.7
  0	$\begin{align*}y=-2.8(0)-9.1\end{align*}$	-9.1
  -1	$\begin{align*}y=-2.8(-1)-9.1\end{align*}$	6.3

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  The equation in the problem is in  $\begin{align*}y=mx+b\end{align*}$ form (also known as slope-intercept form ), where  $\begin{align*}b\end{align*}$ is the $\begin{align*}y\end{align*}$ -value when $\begin{align*}x=0\end{align*}$ , and  $\begin{align*}m\end{align*}$ is the slope of the line. Therefore:
  ::问题所在方程式为 y=mx+b 窗体(又称斜度界面), b 是 y 值, x=0, m 是 线的斜度。 因此 :

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  b. What is the slope of the line?
  ::b. 线的斜坡是什么?

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    $\begin{align*}m=-2.8\end{align*}$
  ::2.8 m°2.8

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  c. What is the $\begin{align*}y\end{align*}$ -intecept?
  ::c. 什么是Y概念?

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  $\begin{align*}b=-9.1\end{align*}$
  ::b9.1

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  Earlier Problem Revisited
  ::重审先前的问题

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  Imagine walking through the electronics section of your local department store. On the wall are examples of dozens of television sets, from little 19″ units made to sit on a kitchen counter to 72″+ monsters meant to be the centerpiece of a home theatre. Looking at the prices, you note without surprise that the 72″ model is more expensive than the 19″, and a 42″ model is priced in between. It seems rather clear that as the TV gets larger, the price goes up. Does that mean increased screen size causes increased price ?
  ::想象一下在您本地百货商店的电子路程中行走。 在墙上,有几十台电视机的例子,从在厨房柜台上坐着的19个小机组到72个怪物,这些怪物本来是一家家庭剧院的中心。看看价格,你毫不惊讶地注意到72个机组比19个机组更贵,而一个42个机组则在中间定价。看来相当清楚的是,随着电视机越大,价格就会上涨。 黑屏幕尺寸越大,导致价格上涨吗?

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  No, it does not. This is an example of the difficulty associated with examining linear relationships. Correlation does not imply causation. Just because a pair of variables exhibit a relationship, linear or otherwise, does not mean that one variable causes changes in the other variable.
  ::不,它没有。这是一个与研究线性关系有关的困难的例子。 关联并不意味着因果关系。 仅仅因为一对变量显示出线性关系或其他关系,它并不意味着一个变量引起另一个变量的变化。

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  Examples 
  ::实例

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  Example 1
  ::例1

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  If a linear graph exhibits a positive slope, what can you predict will happen to the response variable as the explanatory variable increases?
  ::如果线性图显示正斜度,您可以预测,随着解释变量的增加,响应变量会发生什么后果?

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  A positive slope indicates that the variables increase and decrease together?
  ::一个积极的斜坡表明变数一起增减?

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  Example 2
  ::例2

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  If a linear graph has no slope, what does that mean?
  ::如果线性图没有斜坡,那是什么意思?

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  A line with no slope is a horizontal line, since the only defined variable is the output. No matter what value is given for the explanatory variable, the response is the same.
  ::没有斜度的直线是水平线, 因为唯一定义的变量是输出。 无论给解释变量给定什么值, 回复都是一样的 。

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  Example 3
  ::例3

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  Given the linear equation $\begin{align*}2y=5.2x+7\end{align*}$ :
  ::根据线性方程式 2y=5.2x+7:

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  a. What is the slope?
  ::a. 什么是斜坡?

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  The slope is 5.2.
  ::斜坡是5.2

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  b. What is the $\begin{align*}y\end{align*}$ -interept?
  ::b. 什么是y -interept?

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  The y-intercept is 7.
  ::y 界面是7。

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  c. What happens to  $\begin{align*}y\end{align*}$ as  $\begin{align*}x\end{align*}$ increases?
  ::c. y作为x增加额会怎样?

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  Since this line has a positive slope, y increases as x increases
  ::由于这条线有正斜坡,y 增加为x增加

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  Example 4
  ::例4

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  Given the equation $\begin{align*}y=2x^2+4\end{align*}$ :
  ::根据y=2x2+4的方程式:

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  a. Is this a linear equation? Why or why not?
  ::a. 这是线性等式吗?

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  No, the explanatory variable is squared, this graph would form a parabola.
  ::否,解释变量为方形,此图将形成抛物线。

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  b. Does this equation represent a relationship?
  ::b. 这个等式是否代表一种关系?

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  Yes! It is just not a linear relationship.
  ::是的,这绝不是线性关系。

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  Review 
  ::回顾

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  For questions 1-8, find the  $\begin{align*}x\end{align*}$ and  $\begin{align*}y\end{align*}$ intercepts of the given equations.
  ::对于问题1-8,请找到给定方程的 x 和 y 截取值 。

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  1. $\begin{align*}-x+4y=8\end{align*}$
  ::1.-x+4y=8

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  2. $\begin{align*}3x+5y=15\end{align*}$
  ::2. 3x+5y=15

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  3. $\begin{align*}-3x+4y=36\end{align*}$
  ::3.-3x+4y=36

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  4. $\begin{align*}-8x+5y=40\end{align*}$
  ::4.-8x+5y=40

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  5. $\begin{align*}5x-6y=-30\end{align*}$
  ::5. 5x-6y30

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  6. $\begin{align*}-9x-3y=-54\end{align*}$
  ::6.-9x-3y54

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  7. $\begin{align*}-x+5y=-10\end{align*}$
  ::7.-x+5y10

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  8. $\begin{align*}-3x+8y=-72\end{align*}$
  ::8. - 3x+8y72

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  For questions 9-15, graph the line.
  ::对于问题9-15,请用线图表示。

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  9. $\begin{align*}x+3y=2\end{align*}$
  ::9. x+3y=2

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  10. $\begin{align*}m=-4\end{align*}$ , $\begin{align*}b=\frac{4}{3}\end{align*}$
  ::10. m4, b=43

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  11. $\begin{align*}x- \text{intercept}= -1\end{align*}$ , $\begin{align*}y- \text{intercept}= 2\end{align*}$
  ::11. x - 截取 #% 1, y - 截取=2

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  12. $\begin{align*}y=-4x+2\end{align*}$
  ::12. y4x+2

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  13. $\begin{align*}m=-1\end{align*}$ , $\begin{align*}b=\frac{1}{2}\end{align*}$
  ::13. m1, b=12

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  14. $\begin{align*}x+2y=5\end{align*}$
  ::14. x+2y=5 14 x+2y=5

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  15. $\begin{align*}-3x+2y=-3\end{align*}$
  ::15. - 3x+2y=3

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  Review (Answers)
  ::回顾(答复)

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。