Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)
Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)
4th Edition
ISBN: 9780133178579
Author: Ross L. Finney
Publisher: PEARSON
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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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Question
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Chapter 5.3, Problem 31E

a.

To determine

To find: the logistic regression for the data.

a.

Expert Solution
Check Mark

Answer to Problem 31E

  P=12655.17381+12.87142e−0.03250t .

Explanation of Solution

Given information: Table shows the population of Pennsylvania in each 10 years

between 1830 and 1950.

    Year since 1820Population in Thousands
    101348
    201724
    302312
    402906
    503522
    604283
    705258
    806302
    907665
    1008720
    1109631
    1209900
    13010,498

Calculation:

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

  P=12655.17381+12.87142e−0.03250t .

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 31E

Explanation of Solution

Given information: P=12655.17381+12.87142e−0.03250t

Calculation:

Scatter plot of the table data with P=12655.17381+12.87142e−0.03250t shown below.

  Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy), Chapter 5.3, Problem 31E , additional homework tip 1

c.

To determine

To predict: the Pennsylvania population in the 2000 census.

c.

Expert Solution
Check Mark

Answer to Problem 31E

  P(180)≈12209.87 thousands.

Explanation of Solution

Given information: P=12655.17381+12.87142e−0.03250t

Calculation:

  P=12655.17381+12.87142e−0.03250t .

2000 is 180 years after 1820, so t = 180 plug that into the regression equation to find the Pennsylvania population in the 2000 census.

  P(180)=12655.17381+12.87142e−.03250(180)≈12209.87.P(180)≈12209.87 thousands.

d.

To determine

To find: in what year was the Pennsylvania 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 31E

About year 1900.

Explanation of Solution

Given information: P=12655.17381+12.87142e−0.03250t .

Calculation:

Scatter plot of the table data with P=12655.17381+12.87142e−0.03250t shown below.

  Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy), Chapter 5.3, Problem 31E , additional homework tip 2

From the graph, the population is increasing the fastest around t =80 .or the year 1900.

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 31E

In the long term, the equation says the population will stabilize at around 12.66 million.

Explanation of Solution

Given information: P=12655.17381+12.87142e−0.03250t

Calculation:

  limt→∞12655.17381+12.87142e−.03250t=12655.1738 .

So, in the long term, the equation says the population will stabilize at around 12.66 million.

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Chapter 5.3, Problem 30E
Next
Chapter 5.3, Problem 32E

Chapter 5 Solutions

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

Ch. 5.1 - Prob. 1QRCh. 5.1 - Prob. 2QRCh. 5.1 - Prob. 3QRCh. 5.1 - Prob. 4QRCh. 5.1 - Prob. 5QRCh. 5.1 - Prob. 6QRCh. 5.1 - Prob. 7QRCh. 5.1 - Prob. 8QRCh. 5.1 - Prob. 9QRCh. 5.1 - Prob. 10QR
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