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Chapter 2: Problem 222

For the following data, draw a scatter plot. If we wanted to know when the population would reach 15,000, would the answer involve interpolation or extrapolation? Eyeball the line, and estimate the answer. $$\begin{array}{|c|c|c|c|c|c|}\hline \text { Year } & {1990} & {1995} & {2000} & {2005} & {2010} \\ \hline \text { Population } & {11,500} & {12,100} & {12,700} & {13,000} & {13,750} \\ \hline\end{array}$$

Short Answer

Expert verified
Extrapolation is required, estimating around the year 2020.

Step by step solution

01

Plot the Data Points

Create a scatter plot by plotting the given data points on a graph with 'Year' on the x-axis and 'Population' on the y-axis. The points to plot are (1990, 11500), (1995, 12100), (2000, 12700), (2005, 13000), and (2010, 13750).
02

Draw a Best-Fit Line

After plotting the scatter points, draw a line that best fits or approximates the general direction of the points. This line may not pass through all the points but should represent the trend of the data.
03

Identify Nature of the Prediction

Determine whether to use interpolation or extrapolation to predict the year when the population reaches 15,000. Interpolation applies within the range of existing data, while extrapolation extends beyond it.
04

Estimate Using Extrapolation

Since 15,000 is beyond the last known data point (2010, 13750), use extrapolation by extending the best-fit line. Estimate where this line would hit 15,000 on the y-axis, and project that back to the x-axis to find the corresponding year.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interpolation
Interpolation is a method used to estimate unknown values within the range of a set of known data points. Imagine you have a series of data points forming a line or a curve. Interpolation allows you to predict values that fall inside the endpoints of this line.

For example, if you have population data from the years 1990 to 2010, interpolation would help you estimate the population for any year between 1990 and 2010, such as 2002 or 2008. The main idea is that we assume a continuous trend between the data points.

Here are some key points to remember:
  • Interpolation is only applicable within the data range.
  • It relies on existing data trends to predict unknown points.
  • It is often more accurate than extrapolation since it uses closely spaced data points.
Extrapolation
Extrapolation is quite similar to interpolation, but instead of estimating values within the range of known data points, it is used for predicting values outside this range. It requires extending the existing trend observed in the data.

In the context of the given data, you would use extrapolation to predict the population for a year beyond 2010, like 2015 or 2020. Our exercise uses extrapolation to estimate when the population might reach 15,000, a number beyond our data scope.

Key points to consider about extrapolation:
  • Extrapolation is less reliable than interpolation because it assumes the current trend will continue indefinitely.
  • It carries a higher risk of error as it ventures beyond the observed range.
  • Can be helpful for long-term predictions but should be used with caution.
Best-Fit Line
The best-fit line serves as a visual representation of the data's trend on a scatter plot. It is the line that minimizes the distance from all the data points collectively, not necessarily passing through any of them. This line is crucial for both interpolation and extrapolation.

Creating a best-fit line involves placing it such that it reflects the central tendency of the scatter points. This helps in making more accurate predictions.
  • The slope of the line indicates how the dependent variable (e.g., population) changes with the independent variable (e.g., year).
  • It can be estimated by "eyeballing," which is a simple visual technique, or calculated precisely with statistical software.
  • A good best-fit line can simplify the complexity and noise of data, providing a clearer view of trends.
Population Prediction
Population prediction uses statistical methods to estimate future populations based on current data trends. This can involve both interpolation and extrapolation to render these forecasts. Accurate population predictions can have significant implications for planning in areas like urban development, health care, and education.

To predict when a population will reach a specific number, like 15,000 in our exercise, a best-fit line on a scatter plot is essential. By extending this line to reach the desired population value, extrapolation is used.

Important aspects of population prediction:
  • Helps in understanding demographic shifts and planning resource allocation.
  • Requires careful analysis given uncertainties in long-range predictions.
  • Can inform policies to accommodate future growth or decline.

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