Question

According to the National Association of Home Builders, the average size of a home in 1950 was \(983 \mathrm{ft}^{2}\). The average size increased to \(1500 \mathrm{ft}^{2}\) in \(1970,2080 \mathrm{ft}^{2}\) in 1990 ; and \(2330 \mathrm{ft}^{2}\) in 2003 (San Luis Obispo Tribune, October 16,2005\()\). a. Construct a time-series plot that shows how the average size of a home has changed over time. b. If the trend of the time-series plot were to continue, what would you predict the average home size to be in \(2010 ?\)

Short answer

Based on the trend of increasing average home size over time, a prediction can be made that the average home size in 2010 would be larger than the average home size in 2003. The exact value would depend on the trend line that is drawn based on the provided data.

Step-by-step solution

Construct Time-Series Plot

The first step is to create a time-series plot. Plotting time on the x-axis and the average size of the house on the y-axis. The given points are (1950, 983), (1970, 1500), (1990, 2080), and (2003, 2330). Connect the dots to create a line that represents the trend over time.

Observe The Trend

The next step is to observe the trend. Looking at the plot you have made, you should note that the average size of a house is increasing over time.

Predict the Average Home Size for 2010

To predict the average home size for 2010, extend the trend line you have drawn until it reaches the 2010 mark on the x-axis. Note down the y-value at this point, which is the estimated average home size for 2010.

Key concepts

Trend Prediction

Trend prediction in time series analysis involves identifying the direction in which data is moving over a period of time. For instance, when analyzing the average home size from 1950 to 2003 as provided in the exercise, a clear upward trend is observed. This trend can be extrapolated to predict future values.

When predicting the average home size for 2010, we must account for the rate of change observed in the historical data. By connecting the data points and extending the line graph, the estimated size for the year 2010 can be determined by where the line intersects the 2010 mark on the x-axis. This method, while simplistic, assumes that the historical rate of change will continue without interruption or variation, which may not always be the case.

Statistical Visualization

Statistical visualization is an essential tool in data analysis and time series analysis is no exception. Creating a time-series plot as was done in step 1 of the exercise solution allows individuals to visually understand trends, patterns, and outliers within a dataset over time. The x-axis typically represents time, while the y-axis represents the variable being measured—in this case, the average size of a home.

The straightforwardness of a time-series plot helps to quickly convey the direction and magnitude of trends. By visualizing data, it becomes easier to synthesize information, communicate findings, and support decision-making processes. Effective visualization requires the use of proper scales, labels, and may include additional information, such as confidence intervals or prediction bands, to provide context to the trend line.

Data Analysis

Data analysis is the process of examining, cleaning, transforming, and modeling data to discover useful information, draw conclusions, and support decision-making. In the context of the provided exercise, data analysis involves observing the historical trend in home sizes and using that information to make a future prediction.

The initial analysis includes descriptive statistics like finding the mean or median. Then, data visualization techniques, like the time-series plot constructed in step 1, are used. Next, reviewing the trend observed in step 2 assists in understanding the direction and strength of a trend over time. Lastly, inferential statistics or prediction methods are applied, which may include linear regression or other models to forecast future values based on historical data points.

While the exercise solution provides a simple linear prediction, more advanced techniques could incorporate factors like economic conditions or changes in housing market preferences to refine the prediction accuracy.