1. 3 4 Chapter Describing the Relation between Two Variables
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EXAMPLE Drawing and Interpreting a Scatter Diagram
The data shown to the right are based on a study for drilling rock. The researchers wanted to determine whether the time it takes to dry drill a distance of 5 feet in rock increases with the depth at which the drilling begins. So, depth at which drilling begins is the explanatory variable, x, and time (in minutes) to drill five feet is the response variable, y. Draw a scatter diagram of the data.
Source: Penner, R., and Watts, D.G. “Mining Information.” The American Statistician, Vol. 45, No. 1, Feb. 1991, p. 6.
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EXAMPLE Determining the Linear Correlation Coefficient
Determine the linear correlation coefficient of the drilling data.
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EXAMPLE Does a Linear Relation Exist?
Determine whether a linear relation exists between time to drill five feet and depth at which drilling begins. Comment on the type of relation that appears to exist between time to drill five feet and depth at which drilling begins.
The correlation between drilling depth and time to drill is The critical value for n = 12 observations is Since > 0.576, there is a positive linear relation between time to drill five feet and depth at which drilling begins.
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12. Section 4.2 Least-squares Regression
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EXAMPLE Finding an Equation the Describes Linearly Related Data
Using the following sample data:
(a) Find a linear equation that relates x (the explanatory variable) and y (the response variable) by selecting two points and finding the equation of the line containing the points.
Using (2, 5.7) and (6, 1.9):
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(b) Graph the equation on the scatter diagram.
(c) Use the equation to predict y if x = 3.
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The difference between the observed value of y and the predicted value of y is the error, or residual.
Using the line from the last example, and the predicted value at x = 3:
residual = observed y – predicted y
= 5.2 – 4.75
= 0.45
(3, 5.2) }
residual = observed y – predicted y
= 5.2 – 4.75
= 0.45
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18. residual = observed-predictedx
y observed
y= predicted
residual = observed-predicted
residual squared
5.8
7.6
-1.8
3.24 2
5.7 3
5.2
4.75
0.45
0.2025 5
2.8
2.85
-0.05
0.0025 6
1.9
2.2
0.3
0.09
3.535
y= x predicted
6.55
-0.75
0.5625
5.122
0.578
4.408
0.792
2.98
-0.18
0.0324
2.266
-0.366
-0.066
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EXAMPLE Finding the Least-squares Regression Line
Using the drilling data
Find the least-squares regression line.
Predict the drilling time if drilling starts at 130 feet.
Is the observed drilling time at 130 feet above, or below, average.
Draw the least-squares regression line on the scatter diagram of the data.
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We agree to round the estimates of the slope and intercept to four decimal places.
(b)
(c) The observed drilling time is 6.93 seconds. The predicted drilling time is seconds. The drilling time of 6.93 seconds is below average.
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