Problem 57
Question
Analyzing Real Data For the given data set complete the following. (a) Make a line graph of the data. Let this graph represent a function \(f\) (b) Decide whether \(f\) is linear or nonlinear. Median incomes of full-time female workers $$ \begin{array}{|c|c|c|c|c|}\hline \hline \text { Year } & 1970 & 1980 & 1990 & 2000 \\ \hline \text { Income } & \$ 5,440 & \$ 11,591 & \$ 20,591 & \$ 32,442 \end{array} $$
Step-by-Step Solution
Verified Answer
The function is nonlinear because the changes in income are not consistent.
1Step 1: Identify the Variables
Identify the given data in the problem. We have year as the independent variable, and median income as the dependent variable. The data points are (1970, 5440), (1980, 11591), (1990, 20591), and (2000, 32442).
2Step 2: Plot the Data Points
Create a line graph by plotting the data points on a coordinate plane. The x-axis will represent the years (1970, 1980, 1990, 2000), and the y-axis will represent the median incomes (5440, 11591, 20591, 32442). Connect the plotted points with straight line segments to form the graph.
3Step 3: Analyze the Graph
Look at the plotted line graph to determine whether the line connecting the points is straight or curved. If the line is straight, it represents a linear function, and if it curves, it indicates a nonlinear function.
4Step 4: Check for Linearity
To confirm linearity, calculate the change in income over each decade. From 1970 to 1980, the increase is 11591 - 5440 = 6151. From 1980 to 1990, the increase is 20591 - 11591 = 9000. From 1990 to 2000, the increase is 32442 - 20591 = 11851. Since the increases are different, the function is nonlinear.
Key Concepts
Line GraphLinear and Nonlinear FunctionsIndependent and Dependent Variables
Line Graph
When dealing with real-life data, plotting a line graph is a great way to visually comprehend trends and relationships. A line graph consists of two axes: the x-axis (horizontal) and the y-axis (vertical). This type of graph is particularly suitable for data that changes over a consistent interval, such as time.
In the given exercise, we plot the median incomes over a sequence of years. The x-axis represents the years: 1970, 1980, 1990, and 2000. Meanwhile, the y-axis charts the median incomes corresponding to those years: \(\\)5,440\(, \)\\(11,591\), \(\\)20,591\(, and \)\\(32,442\).
Each year and its corresponding income value form a data pair (e.g., 1970 corresponds with \(\\)5,440$), and these pairs are plotted as points on the graph. After all points are plotted, they are connected with straight line segments to visualize the data trend, making it easier to see how incomes have changed over time.
In the given exercise, we plot the median incomes over a sequence of years. The x-axis represents the years: 1970, 1980, 1990, and 2000. Meanwhile, the y-axis charts the median incomes corresponding to those years: \(\\)5,440\(, \)\\(11,591\), \(\\)20,591\(, and \)\\(32,442\).
Each year and its corresponding income value form a data pair (e.g., 1970 corresponds with \(\\)5,440$), and these pairs are plotted as points on the graph. After all points are plotted, they are connected with straight line segments to visualize the data trend, making it easier to see how incomes have changed over time.
Linear and Nonlinear Functions
Functions can be categorized as either linear or nonlinear based on how the plotted line appears on the graph. This distinction is crucial for understanding the data's underlying trend and growth pattern.
A linear function is characterized by a straight line. This occurs when the rate of change between variables remains constant. In simpler terms, if you move from one point to another on the graph, the addition in the output (dependent variable) remains the same for every equivalent addition in the input (independent variable).
Conversely, if the plotted graph curves, it indicates a nonlinear function. This means that the rate of change is not constant; the difference in the dependent variable varies as the independent variable changes.
For the given dataset, by calculating the income increase between successive decades, we see differences: from 1970 to 1980 the increase is $6,151, from 1980 to 1990 it's $9,000, and from 1990 to 2000 it's $11,851. This variability confirms that the function is nonlinear.
A linear function is characterized by a straight line. This occurs when the rate of change between variables remains constant. In simpler terms, if you move from one point to another on the graph, the addition in the output (dependent variable) remains the same for every equivalent addition in the input (independent variable).
Conversely, if the plotted graph curves, it indicates a nonlinear function. This means that the rate of change is not constant; the difference in the dependent variable varies as the independent variable changes.
For the given dataset, by calculating the income increase between successive decades, we see differences: from 1970 to 1980 the increase is $6,151, from 1980 to 1990 it's $9,000, and from 1990 to 2000 it's $11,851. This variability confirms that the function is nonlinear.
Independent and Dependent Variables
Understanding the roles of independent and dependent variables is fundamental for data analysis. These terms classify quantities that influence each other in an experiment or data set.
The independent variable acts as the input to the situation. It's the variable you can control or change deliberately. In the context of this data set, "year" serves as the independent variable because the progression through time is sequential and predetermined.
The dependent variable is the output that depends on changes in the independent variable. Here, the "median income" is the dependent variable. It changes in response to the progression of years and reflects the trend or outcome we are interested in understanding.
Identifying which variable is independent and which is dependent helps clarify the cause-and-effect relationship in the data, allowing us to make predictions and insights from our analysis.
The independent variable acts as the input to the situation. It's the variable you can control or change deliberately. In the context of this data set, "year" serves as the independent variable because the progression through time is sequential and predetermined.
The dependent variable is the output that depends on changes in the independent variable. Here, the "median income" is the dependent variable. It changes in response to the progression of years and reflects the trend or outcome we are interested in understanding.
Identifying which variable is independent and which is dependent helps clarify the cause-and-effect relationship in the data, allowing us to make predictions and insights from our analysis.
Other exercises in this chapter
Problem 56
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