The data below represent commute times (in minutes) and scores on a well-being survey. Complete parts (a) through (d) below. Commute Time (minutes), x Well-Being Index Score, y 15 25 35 60 84 105 O 69.0 67.7 66.5 65.6 63.4 62.0 59.4 (a) Find the least-squares regression line treating the commute time, x, as the explanatory variable and the index score, y, as the response variable. ý = - 0.090 x+ ( 69.045) (Round to three decimal places as needed.) (b) Interpret the slope and y-intercept, if appropriate. Interpret the slope. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. For every unit increase in commute time, the index score falls by 0.090 , on average. (Round to three decimal places as needed.) O B. For an index score of zero, the commute time is predicted to be minutes. (Round to three decimal places as needed.) O C. For a commute time of zero minutes, the index score is predicted to be (Round to three decimal places as needed.) O D. For every unit increase in index score, the commute time falls by , on average. (Round to three decimal places as needed.) O E. It is not appropriate to interpret the slope. Interpret the y-intercept. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. For every unit increase in index score, the commute time falls by , on average. (Round to three decimal places as needed.) O B. For a commute time of zero minutes, the index score is predicted to be 69.045 . (Round to three decimal places as needed.) O C. For an index score of zero, the commute time is predicted to be minutes. (Round to three decimal places as needed.) O D. For every unit increase in commute time, the index score falls by , on average. (Round to three decimal places as needed.) O E. It is not appropriate to interpret the y-intercept because a commute time of zero minutes does not make sense and the value of zero minutes is much smaller than those observed in the data set. (c) Predict the well-being index of a person whose commute is 30 minutes. The predicted index score is 66.3. (Round to one decimal place as needed.) (d) Suppose Barbara has a 20-minute commute and scores 66.5 on the survey. Is Barbara more "well-off" than the typical individual who has a 20-minute commute? Select the correct choice below and fill in the answer box to complete your choice. (Round to one decimal place as needed.) O A. Yes, Barbara is more well-off because the typical individual who has a 20-minute commute scores O B. No, Barbara is less well-off because the typical individual who has a 20-minute commute scores

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Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.