Question

The value (in 1982 dollars) of each dollar received by producers in each of the years from 1991 to 2005 in the United States is represented by the following ordered pairs. (Source: U.S. Bureau of Labor Statistics) \(\begin{array}{lcl}\text { ordered pairs. } & \text { Source: U.S. Bureau of Labor Statisti } \\ (1991,0.822) & (1992,0.812) & (1993,0.802) \\\ (1994,0.797) & (1995,0.782) & (1996,0.762) \\ (1997,0.759) & (1998,0.765) & (1999,0.752) \\ (2000,0.725) & (2001,0.711) & (2002,0.720) \\ (2003,0.698) & (2004,0.673) & (2005,0.642)\end{array}\) (a) Use a spreadsheet software program to generate a scatter plot of the data. Let \(t=1\) represent 1991. Do the data appear to be linear? (b) Use the regression feature of a spreadsheet software program to find a linear model for the data. (c) Use the model to estimate the value (in 1982 dollars) of 1 dollar received by producers in 2007 and in 2008 . Discuss the reliability of your estimates based on your scatter plot and the graph of your linear model for the data.

Short answer

The scatter plot can give an idea if the data appears to be linear. The linear model will provide a mathematical model of the relationship between years and the dollar value. This model can be used to estimate the dollar value for any year like 2007 and 2008. The reliability of these estimates depends on how well the model represents the data. This can be visualized by comparing the scatter plot and the graph of the linear model.

Step-by-step solution

Data Preparation

Organize the data mentioned in the problem statement in two columns in a spreadsheet software; one for the year (where \(t=1\) represents 1991) and the other for the value of the dollar.

Scatter Plot Generation

Use the charting feature of the spreadsheet software to generate a scatter plot. Carefully inspect the scatter plot to see if a linear relationship can be visualized between the years and the value of dollars. This provides an initial intuitive understanding of the data.

Linear Regression Analysis

Using the regression feature of the spreadsheet software, fit a linear model to the data. The model will provide the relationship as a simple equation, which is a line of best fit. It will represent the relationship between the two variables.

Predict Future Values

Using the generated linear model, predict the value (in 1982 dollars) of 1 dollar received by producers in 2007 and 2008. One only needs to replace the year value in the model's equation and solve for the dollar's value.

Discuss the Reliability

The reliability of these estimates can be evaluated by looking at the scatter plot and the graph of the linear model. If the data points closely follow the line of the best fit, then it can be said that the model is a good fit for the data and thus the predictions are reliable. The further away the data points, the less accurate the model, and thus the less reliable these future predictions are.

Key concepts

Scatter Plot Interpretation

A scatter plot is a graphical representation that displays the values of two variables for a set of data. It's an essential tool in linear regression analysis, where the relationship between the variables is examined. When trying to interpret a scatter plot, one of the first steps is to look for patterns, trends, and correlations.

In our exercise, a scatter plot would visualize how the value of a dollar has changed over the years in a two-dimensional space: the x-axis for time (with each unit increase representing a successive year), and the y-axis for the value of the dollar in 1982 dollars. If the data points form a pattern that suggests a line could reasonably summarize the overall direction, the relationship may be called linear. However, if the points are widely scattered with no discernible pattern, this could indicate a weak or non-linear relationship.

Intuitively, if the plot shows a downward trend, as it likely does give the context, it suggests that the dollar's purchasing power has decreased over the years. A cluster of points close to the resulting line of best fit would confirm a strong linear relationship and a reliable model.

Predictive Modeling

Predictive modeling uses statistics to predict outcomes. In linear regression, it involves creating a model that best fits the data and can then be used to forecast future values. The process usually involves calculating the slope and the y-intercept to develop the linear equation of the form \( y = mx + b \), where \( y \) is the dependent variable, \( x \) is the independent variable, \( m \) is the slope of the line, and \( b \) is the y-intercept.

After establishing a linear model from our data, we can use it for predictive purposes. In the case of this exercise, predictions for the years 2007 and 2008 are obtained by substituting the corresponding time values into the linear equation. This gives an estimate of the dollar's value for those years. It's important to remember that predictions are based on the assumption that the patterns observed in the historical data will continue into the future, which may not always be the case, especially in volatile economic conditions.

Statistical Data Representation

Effective statistical data representation is crucial for analysis and communication in many fields. It involves the visual display of data to highlight its main features and relationships in a clear and concise way. In linear regression analysis, the primary representation of data is through a scatter plot complemented by a line of best fit.

By organizing data into a visually accessible format, such as a graph, chart, or table, complex relationships can be understood and interpreted more straightforwardly. This also helps identify outliers, trends, and patterns that pure numbers might not easily reveal. Ultimately, a well-represented data set not only aids in the understanding of the data itself but also supports the reliability and validity of the predictive models generated from it.

Returning to the exercise, visual representation makes it easier to examine the decline in the dollar's value over time and aids in communicating the findings to others. This can be a valuable skill in many professional contexts, helping to drive informed decision-making.