Calculus: Graphical, Numerical, Algebraic
Calculus: Graphical, Numerical, Algebraic
3rd Edition
ISBN: 9780133688399
Author: Finney, Ross L.
Publisher: Pearson/Prentice Hall
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Correlation
Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.
Linear Correlation
A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.
Regression Analysis
Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as p…
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Chapter 5, Problem 41RE

a.

To determine

To find: the logistic regression for the data.

a.

Expert Solution
Check Mark

Answer to Problem 41RE

  P=12655.17381+12.87142e−0.03250t .

Explanation of Solution

Given information: Table shows the growth of the U.S. population from 1790 to 2000 in 30- year’s increments and indicates the growth in 2010.

    Year Growth
    17903929
    18209638
    185023,192
    188050189
    191092,228
    1940132,165
    1970203,302
    2000281,422
    2010309,447

Calculation:

The logistic regression curve, for population P, t years after 1820 as produced by graphing utility:

  P=1633001.591+17.471e−0.06378t .

b.

To determine

To graph: the data in a scatter plot and super impose the regression curve.

b.

Expert Solution
Check Mark

Answer to Problem 41RE

Explanation of Solution

Given information: P=1633001.591+17.471e−0.06378t

Calculation: Let t = 0 for 1970 then the above table would be written as.

    Year ( t)Growth ( P)
    03929
    309638
    6023,192
    9050189
    12092,228
    150132,165
    180203,302
    210281,422
    220309,447

Scatter plot of the table data shown below.

  Calculus: Graphical, Numerical, Algebraic, Chapter 5, Problem 41RE , additional homework tip 1

c.

To determine

To predict: the U.S. population in the 2050.

c.

Expert Solution
Check Mark

Answer to Problem 41RE

  P(260)≈1632999.79 thousands.

Explanation of Solution

Given information: P=1633001.591+17.471e−0.06378t

Calculation:

  P=1633001.591+17.471e−0.06378t .

2050 is 260 years after 1970, so t = 260 plug that into the regression equation to find the U.S. population in the 2050.

  P(260)=1633001.591+17.471e−0.06378×260≈1632999.79.P(260)≈1632999.79 thousands.

d.

To determine

To find: in what year was the U.S. population growing the fastest and what significant behavior does the graph of the regression equation exhibit at that point

d.

Expert Solution
Check Mark

Answer to Problem 41RE

About year 2000.

Explanation of Solution

Given information: P=1633001.591+17.471e−0.06378t .

Calculation:

Scatter plot of the table data shown below.

  Calculus: Graphical, Numerical, Algebraic, Chapter 5, Problem 41RE , additional homework tip 2

From the graph, the population is increasing the fastest around t =210 .or the year 2000.

The slope of the tangent line at that point is greatest there, in other words, the change in population over the change in time is largest.

The concavity also changes at that point, so it is a point of inflection.

e.

To determine

To find: what does the regression equation indicate about the population of Pennsylvania in the long run.

e.

Expert Solution
Check Mark

Answer to Problem 41RE

In the long term, the equation says the population will stabilize at around 1633001.59 .

Explanation of Solution

Given information: P=1633001.591+17.471e−0.06378t

Calculation:

  limt→∞P=1633001.591+17.471e−0.06378t=1633001.59 .

So, in the long term, the equation says the population will stabilize at around 1633001.59

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Calculus: Graphical, Numerical, Algebraic

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