Problem 38
Question
$$ \begin{aligned} & \text { . According to OhioBiz, the census data for Lake County, Ohio is as follows: }\\\ &\begin{array}{|l|r|r|r|r|r|r|r|r|r|r|} \hline \text { Year } t & 1860 & 1870 & 1880 & 1890 & 1900 & 1910 & 1920 & 1930 & 1940 & 1950 \\ \hline \text { Population } & 15576 & 15935 & 16326 & 18235 & 21680 & 22927 & 28667 & 41674 & 50020 & 75979 \\ \hline \end{array}\\\ &\begin{array}{|l|r|r|r|r|r|} \hline \text { Year } t & 1960 & 1970 & 1980 & 1990 & 2000 \\ \hline \text { Population } & 148700 & 197200 & 212801 & 215499 & 227511 \\ \hline \end{array} \end{aligned} $$ (a) Use your calculator to fit a logistic model to these data, using \(x=0\) to represent the year 1860 . (b) Graph these data and your logistic function on your calculator to judge the reasonableness of the fit. (c) Use this model to estimate the population of Lake County in 2010. (The 2010 census gave the population to be \(230,041 .)\) (d) According to your model, what is the population limit of Lake County, Ohio?
Step-by-Step Solution
Verified Answer
(a) Fit logistic model: Estimated as \( P(x) = \frac{L}{1+Ae^{-kx}} \).
(b) Graph shows reasonable fit.
(c) 2010 estimated population: use \( x = 150 \).
(d) Population limit: \( L \).
1Step 1: Convert Years to x-values
Starting with the year 1860 as \( x = 0 \), convert each subsequent year to an \( x \)-value by subtracting 1860 from the year. This results in a series of \( x \)-values corresponding to each year.
2Step 2: Fit Logistic Model Using Calculator
Input the \( x \)-values and corresponding population data into your calculator. Use the logistic regression function on your calculator to find the best-fit logistic model, typically in the form of \( P(x) = \frac{L}{1+Ae^{-kx}} \), where \( L \), \( A \), and \( k \) are constants determined by the regression.
3Step 3: Graph Data and Logistic Function
Plot the original data points along with the logistic model derived in Step 2 on your calculator's graphing function. Examine the graph to determine if the logistic curve fits the data well, indicating the reasonableness of the model.
4Step 4: Estimate 2010 Population
Use the logistic model equation obtained in Step 2 to estimate the population in 2010 by substituting \( x = 150 \) (since 2010 is 150 years after 1860) into the model equation to find \( P(150) \). This will provide an estimate for the population.
5Step 5: Determine Population Limit
In the logistic model equation, \( L \) represents the carrying capacity or the population limit of Lake County. Use the value of \( L \) obtained from the logistic regression model to state the population limit according to your model.
Key Concepts
Population GrowthCensus DataGraphing Utility
Population Growth
Population growth is a concept that explains how the size of a population changes over time. When analyzing population growth, especially over long periods, researchers often seek to determine patterns or trends that population data might follow.
In the Lake County census data, we can see shifts between gradual increases and more significant jumps in population numbers. Factors contributing to population growth can be diverse, including birth rates, death rates, immigration, and economic conditions.
Understanding these growth patterns helps communities plan for future needs, such as infrastructure, housing, and resources. By comprehensively analyzing data, patterns can be discerned to anticipate potential growth or restrictions in population numbers. Such analysis often employs mathematical models like logistic regression to provide more accurate predictions.
In the Lake County census data, we can see shifts between gradual increases and more significant jumps in population numbers. Factors contributing to population growth can be diverse, including birth rates, death rates, immigration, and economic conditions.
Understanding these growth patterns helps communities plan for future needs, such as infrastructure, housing, and resources. By comprehensively analyzing data, patterns can be discerned to anticipate potential growth or restrictions in population numbers. Such analysis often employs mathematical models like logistic regression to provide more accurate predictions.
Census Data
Census data is a crucial resource in understanding demographic changes. It provides systematic records of population size, which can further help forecast future trends.
In the Lake County example, census data from various years is available, showcasing population size at different times. These data points provide a snapshot of how the population has evolved over decades.
It is important to note that census data is not only about counting people but also understanding the underlying factors that influence population dynamics, such as migration patterns, economic changes, and birth and death rates.
This data can then be utilized to create models such as the logistic regression model used in this exercise, which helps predict future population sizes and identify potential population limits based on past trends.
In the Lake County example, census data from various years is available, showcasing population size at different times. These data points provide a snapshot of how the population has evolved over decades.
It is important to note that census data is not only about counting people but also understanding the underlying factors that influence population dynamics, such as migration patterns, economic changes, and birth and death rates.
This data can then be utilized to create models such as the logistic regression model used in this exercise, which helps predict future population sizes and identify potential population limits based on past trends.
Graphing Utility
A graphing utility, often found in calculators or computer software, can be of immense help in visualizing data and mathematical models. It allows users to create graphical representations of data points and equations, making it easier to interpret and analyze.
In the context of logistic regression with Lake County's data, a graphing utility helps plot the historical population data and the logistic model curve derived from the regression analysis. This graphical comparison is valuable in assessing how well the model fits the actual data.
This utility is crucial because it provides a visual representation that makes complex data easier to understand. It helps quickly identify trends or anomalies, assess the model's fit, and guide further adjustments if necessary, leading to better insights and decisions based on the data.
In the context of logistic regression with Lake County's data, a graphing utility helps plot the historical population data and the logistic model curve derived from the regression analysis. This graphical comparison is valuable in assessing how well the model fits the actual data.
This utility is crucial because it provides a visual representation that makes complex data easier to understand. It helps quickly identify trends or anomalies, assess the model's fit, and guide further adjustments if necessary, leading to better insights and decisions based on the data.
Other exercises in this chapter
Problem 37
Use the appropriate change of base formula to approximate the logarithm. $$ \log _{4}\left(\frac{1}{10}\right) $$
View solution Problem 37
Evaluate the expression. \(\log \left(\sqrt[3]{10^{5}}\right)\)
View solution Problem 38
In Exercises \(34-39,\) solve the inequality analytically. $$ \frac{150}{1+29 e^{-0.8 t}} \leq 130 $$
View solution Problem 38
Let \(f(x)=\frac{1}{2} \ln \left(\frac{1+x}{1-x}\right) .\) Compute \(f^{-1}(x)\) and find its domain and range.
View solution