In ongoing economic analyses, the federal government compares per capita incomes not only among different states but also for the same state at different times. Typically, what the federal government finds is that "poor" states tend to stay poor and "wealthy" states tend to stay wealthy. Would we have gotten information about the 1999 per capita income for a state (denoted by y ) from its 1980 per capita income (denoted by x)? The following bivariate data give the per capita income (in thousands of dollars) for a sample of fifteen states in the years 1980 and 1999 (source: U.S. Bureau of Economic Analysis, Survey of Current Business, May 2000). The data are plotted in the scatter plot in Figure 1. Also given is the product of the 1980 per capita income and the 1999 per capita income for each of the fifteen states. (These products, written in the column labelled " xy ", may aid in calculations.) Question table 1980 per capita income, x (in $1000 s) 1999 per capita income, y (in $1000 s) xy Montana 9.1 22.3 202.93 Arizona 9.6 25.3 242.88 Georgia 8.5 27.2 231.2 Virginia 10.2 29.5 300.9 New Jersey 11.8 36.1 425.98 Maryland 11.2 32.2 360.64 Kansas 10.0 26.6 266 Iowa 9.7 25.7 249.29 West Virginia 8.2 20.9 171.38 Illinois 11.1 31.3 347.43 North Carolina 8.2 26.2 214.84 New Hampshire 9.9 30.9 305.91 Louisiana 8.8 22.8 200.64 South Carolina 7.8 23.5 183.3 Ohio 10.1 27.1 273.71 1999 per capita income (in $1000 s) y 20 22 24 26 28 30 32 34 36 38 x 7 8 9 10 11 12 13 1980 per capita income (in $1000 s) Figure 1 What is the sample correlation coefficient for these data? Carry your intermediate computations to at least four decimal places and round your answer to at least three decimal places. (If necessary, consult a list of formulas.
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
In ongoing economic analyses, the federal government compares per capita incomes not only among different states but also for the same state at different times. Typically, what the federal government finds is that "poor" states tend to stay poor and "wealthy" states tend to stay wealthy.
Would we have gotten information about the 1999 per capita income for a state (denoted by y ) from its 1980 per capita income (denoted by x)? The following bivariate data give the per capita income (in thousands of dollars) for a sample of fifteen states in the years 1980 and 1999 (source: U.S. Bureau of Economic Analysis, Survey of Current Business, May 2000). The data are plotted in the
per capita income and the 1999 per capita income for each of the fifteen states. (These products, written in the column labelled " xy ", may aid in calculations.)
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