In Exercises 39-45, the correlation coefficient , r, is given for a sample of n data points. Use the α = 0.05 column in Table 12.19 on page 828 to determine whether or not we may conclude that a correlation does exist in the population. (Using the α = 0.05 column, there w a probability of 0.05 that the variables are not really correlated in the population and our results could be attributed to chance. Ignore this possibility when concluding whether or not there is a correlation in the population.) n = 12 , r = 0.5
In Exercises 39-45, the correlation coefficient , r, is given for a sample of n data points. Use the α = 0.05 column in Table 12.19 on page 828 to determine whether or not we may conclude that a correlation does exist in the population. (Using the α = 0.05 column, there w a probability of 0.05 that the variables are not really correlated in the population and our results could be attributed to chance. Ignore this possibility when concluding whether or not there is a correlation in the population.) n = 12 , r = 0.5
Solution Summary: The author explains that a correlation does not exist in the population.
In Exercises 39-45, the correlation coefficient, r, is given for a sample of n data points. Use the
α
=
0.05
column inTable 12.19on page 828 to determine whether or not we may conclude that a correlation does exist in the population. (Using the
α
=
0.05
column, there w a probability of 0.05 that the variables are not really correlated in the population and our results could be attributed to chance. Ignore this possibility when concluding whether or not there is a correlation in the population.)
n
=
12
,
r
=
0.5
Definition Definition Statistical measure used to assess the strength and direction of relationships between two variables. Correlation coefficients range between -1 and 1. A coefficient value of 0 indicates that there is no relationship between the variables, whereas a -1 or 1 indicates that there is a perfect negative or positive correlation.
The Martin-Beck Company operates a plant in St. Louis with an annual capacity of 30,000 units. Product is shipped to regional distribution centers located in Boston, Atlanta, and Houston. Because of an anticipated increase in demand, Martin-Beck plans to increase capacity by constructing a new plant in one or more of the following cities: Detroit, Toledo, Denver, or Kansas. The following is a linear program used to
determine which cities Martin-Beck should construct a plant in.
Let
y₁ = 1 if a plant is constructed in Detroit; 0 if not
y₂ = 1 if a plant is constructed in Toledo; 0 if not
y₂ = 1 if a plant is constructed in Denver; 0 if not
y = 1 if a plant is constructed in Kansas City; 0 if not.
The variables representing the amount shipped from each plant site to each distribution center are defined just as for a transportation problem.
*,, = the units shipped in thousands from plant i to distribution center j
i = 1 (Detroit), 2 (Toledo), 3 (Denver), 4 (Kansas City), 5 (St.Louis) and…
Consider the following mixed-integer linear program.
Max
3x1
+
4x2
s.t.
4x1
+
7x2
≤
28
8x1
+
5x2
≤
40
x1, x2 ≥ and x1 integer
(a)
Graph the constraints for this problem. Indicate on your graph all feasible mixed-integer solutions.
On the coordinate plane the horizontal axis is labeled x1 and the vertical axis is labeled x2. A region bounded by a series of connected line segments, and several horizontal lines are on the graph.
The series of line segments connect the approximate points (0, 4), (3.889, 1.778), and (5, 0).
The region is above the horizontal axis, to the right of the vertical axis, and below the line segments.
At each integer value between 0 and 4 on the vertical axis, a horizontal line extends out from the vertical axis to the series of connect line segments.
On the coordinate plane the horizontal axis is labeled x1 and the vertical axis is labeled x2. A region bounded by a series of connected line segments, and several…
Consider the nonlinear optimization model stated below.
Min
s.t.
2x²-18x + 2XY + y² - 14Y + 53
x + 4Y ≤ 8
(a) Find the minimum solution to this problem.
|at (X, Y) =
(b) If the right-hand side of the constraint is increased from 8 to 9, how much do you expect the objective function to change?
Based on the dual value on the constraint X + 4Y ≤ 8, we expect the optimal objective function value to decrease by
(c) Resolve the problem with a new right-hand side of the constraint of 9. How does the actual change compare with your estimate?
If we resolve the problem with a new right-hand-side of 9 the new optimal objective function value is|
, so the actual change is a decrease of
rather than what we expected in part (b).
Chapter 12 Solutions
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