Class_17 - Linear Regression with Two Variables We return...

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Linear Regression with Two Variables
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We return to the unfinished task of association between two continuous variables . The key to building measures of association for continuous variables is the properties of this type of variable. (1) They have equal intervals throughout their range (this means that the units of measurement are identical); and (2) they have a known and meaningful zero-point (this means that when the variable takes the value of 0.0, the phenomenon that it expresses is absent). It is the first property—equal intervals—that is the key.
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Because of the equal-interval property of continuous variables, one obvious way to describe the relationship between any two such variables (for example, X and Y) is by plotting their association in two-dimensional space. Every observation (data point) is located simultaneously in reference to the values of each variable calibrated along the two axes in Cartesian space (i.e., the x-axis and the y-axis). These axes define four quadrants or sectors where observations lie.
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Here is a simple example to show how this works. Suppose that we are interested in describing the relationship between daily high temperature (Y) and the time (X) when we first saw sunlight in the morning. Our interest is in the month of June in particular because the afternoon temperature can vary widely from day to day. We suspect that the earlier we see the sun in the morning (i.e., the earlier the overcast clouds disappear), the warmer the day will be. For several days we note the time when we first saw sunlight and then record the afternoon high temperature. Here are our data for the first three days:
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———————————————————————— Day Time (X) Temperature (Y) ———————————————————————— 1 10:00 a.m. 76 2 5:30 a.m. 90 3 8:00 a.m. 82 ————————————————————————
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90 __ T * Day 2 e 80 __ * Day 3 m p 70 __ * Day 1 (Y) | | | | | | | 5:00 7:00 9:00 10:00 Time (X)
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This simple device is called a scatterplot , and it tells us two things: (1) the direction of the relationship between X and Y, that is, whether it is DIRECT or INDIRECT ( + or - ); and (2) the strength of the relationship, that is, how strong the association is between X and Y. Strength is indicated by how sharp an angle the plot takes as it falls toward the x-axis. This is an important property called the slope of the line. Here, the angle is not very steep, suggesting that the relationship is not very strong. Clearly, we have an inverse (indirect) relationship, because as the values of the X-variable (time of day) INCREASE, the values of the Y-variable (high temperature) DECREASE.
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This is suggestive but not very precise. Statisticians prefer to describe association NUMERICALLY. This is done by determining how close the data points come to defining a straight line . If it looks as though the data points would like to be a straight line, then we can use an old trick.
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