Problem 12 - College Algebra with Corequisite Support

Question

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. $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Population } \\ \hline 1990 & 11,500 \\ \hline 1995 & 12,100 \\ 2000 & 12,700 \\ 2005 & 13,000 \\ \hline 2010 & 13,750 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Extrapolation; estimated year around 2018.

Step 1: Understand the Data

We have data consisting of years and corresponding population values: (1990, 11500), (1995, 12100), (2000, 12700), (2005, 13000), and (2010, 13750). We will plot these points on a scatter plot to visualize the trend in population growth.

Step 2: Draw the Scatter Plot

On graph paper or using graphing software, plot each year on the x-axis and the corresponding population on the y-axis. Place points for (1990, 11500), (1995, 12100), (2000, 12700), (2005, 13000), and (2010, 13750) to form a scatter plot.

Step 3: Assess Trend Line for Estimation

Draw the best-fit line (trend line) through the data points. This line should minimize the distance from all points. Use this line to estimate population values not directly in the data.

Step 4: Determine the Type of Estimation

To find when the population reaches 15,000, check if this value falls within the given range of data years (1990–2010). Since 15,000 is beyond the highest recorded value of 13,750 in 2010, this task involves extrapolation.

Step 5: Estimate Using the Trend Line

Following the trend line, visually extend it to where it meets the population level of 15,000. Estimate the corresponding year along the x-axis. The estimated year is around 2018.

Key Concepts

Understanding Population Growth

Population growth refers to the change in population size over a period of time. By analyzing how a population grows, we can observe patterns and predict future population sizes. In the exercise, we are given specific years with their corresponding population sizes. These data points help us understand the trend over time.

• The increase in population from each recorded year shows a gradual rising trend overall.
• By looking at such data, students can discuss factors that contribute to growth, such as birth rate, death rate, and migration.

The aim is to visually represent these figures to understand the underlying growth pattern, which can often be easier than by just examining the raw numbers.

The Concept of Interpolation

Interpolation involves estimating a value within the range of the given data. For instance, if you have population data for every five years, and there's a missing data point at another year within this range, interpolation can help estimate that year's population.

• You utilize already known data points to estimate a value within these given points.
• Linear interpolation is common, taking the average rate of change between points to find missing values.

In the provided exercise, if a population figure for a year between 1990 and 2010 were required but missing, interpolation would be used to fill that gap.

The Essence of Extrapolation

Extrapolation, unlike interpolation, is used to predict data outside the existing data range. This method is crucial when trying to forecast future trends using current or past data. In the exercise, estimating when the population will hit 15,000 involves looking beyond 2010, the last given data point.

• This approach relies on the assumption that the existing trend will continue beyond the data.
• It generally involves more risk than interpolation because of predictions outside known data.

To predict the population growth reaching 15,000, you extend your trendline and determine at what point this threshold is likely to be crossed.

Creating a Trend Line

A trend line is a straight line that best represents the data points on a scatter plot. It is used to identify the direction and pattern of the data.

• A trend line "smooths" out the data, making it clear at a glance which way things are heading over time.
• The main purpose is to show the overall behavior being exhibited by the data, rather than focus on specific data points.

When drawing a trend line, one aims to reduce the distance of the scatter points from the line itself, indicating how closely data follows a linear trend.

The Importance of Data Visualization

Data visualization is a powerful method used to understand and interpret data clearly and efficiently. It helps to turn raw data into meaningful and digestible information.

• Scatter plots are a key data visualization tool for showing the relationship between two variables, such as time and population in our exercise.
• Through visualization, trends, correlations, and outliers become more evident, aiding in better decision-making and understanding of the presented data.

By visualizing population growth with a scatter plot, we are better equipped to comprehend historical trends and make educated guesses about future outcomes.