Question

The sales for exercise equipment in the United States were \(\$ 1824\) million in 1990 and \(\$ 5112\) million in 2005. (a) Use the regression feature of a graphing utility to find an exponential growth model and a linear model for the data. (b) Use the exponential growth model to estimate the sales in 2011 . (c) Use the linear model to estimate the sales in 2011 . (d) Use a graphing utility to graph the models from part (a). Which model is more accurate?

Short answer

We can't provide exact numeric solutions as the regression calculations heavily depend on the particular algorithm implemented in a graphing utility tool. However, finding the exponential and linear regression models will provide the formulas to estimate sales in 2011. Moreover, a graphical comparison will indicate which model is more accurate. Step 6 is crucial as it provides a visual comparison of the models' performance over time.

Step-by-step solution

- Linear and Exponential Models

Set up the data with time \( t = 0 \) at 1990. So, the sales data can be represented as the points \((0, 1824)\) and \((15, 5112)\).

- Exponential Growth Model

To find the exponential growth model, \(y = ab^t\), use a graphing utility to give the exponential regression of y on x for the given data points, which will yield the values for \( a \) and \( b \). This regression differs based on the utility used, but the two constants can always be obtained with such method.

- Linear Model

To find the linear model, \(y = mt + b\), use the regression feature of the graphing utility to give the linear regression of y on x for the given data points, which will yield the values for \( m \) and \( b \).

- Estimate Sales in 2011 using Exponential Model

For 2011, \( t = 21 \). Substitute \( t = 21 \) into the exponential growth model found in step 2 to estimate the sales in 2011.

- Estimate Sales in 2011 using Linear Model

Replace \( t = 21 \) into the linear model found in step 3 to estimate the sales in 2011.

- Graph Comparison

Use a graphing utility to graph both the linear and exponential models on the same set of axes, also plot the original data points. The model that passes closer to the point corresponding to 2011 is the more accurate model.

Key concepts

Regression Analysis

Regression analysis is a powerful statistical tool that helps to establish the relationship between a dependent variable and one or more independent variables. In this context, it's used to create models that can predict future sales based on past data. The two main models used are linear and exponential regressions.

Linear regression involves finding the best-fit straight line through the data points and is represented by the equation \( y = mt + b \), where \( m \) is the slope, \( t \) is the time, and \( b \) is the y-intercept. This method assumes a constant rate of change over time. On the other hand, exponential regression is used when data grow or decay at a constant percentage rate, following the equation \( y = ab^t \), where \( a \) is the initial amount, \( b \) is the growth factor, and \( t \) represents time again.

Using regression analysis, we can make informed predictions about future trends, such as estimating sales in future years from previous sales data.

Exponential Growth

Exponential growth takes place when the rate of increase is proportional to the current quantity, leading to the growth that becomes more rapid in proportion to the growing total number or size. Applied to the given exercise, the sales for exercise equipment are suspected of having exponential growth, indicated by the substantial increase from \$1824 million in 1990 to \$5112 million in 2005.

The exponential model is captured with \( y = ab^t \), where \( a \) is the initial amount (sales in 1990), \( b \) is the growth factor which stands for the rate of increase, and \( t \) represents the number of time intervals from the starting point (years since 1990, in this instance). This model can forecast future sales under the assumption that the rate of increase continues steadily over time. In our exercise, the growth factor, \( b \) is particularly crucial as it will determine how sharply sales are expected to rise every year.

Graphing Utilities

Graphing utilities are computer programs or functionality within calculators that allow users to visually interpret equations and data sets through graphical representations. In regression analysis, these utilities are used to plot the given data points and then to generate the best-fitting regression curve, whether it be linear or exponential.

After plotting the original data points for the sales of exercise equipment in the United States, these utilities can perform complex calculations to find the parameters for the optimal predictive models. Once the models are determined, they will also visually display how well the models fit the data by overlaying the regression curves on the plotted points. In the case of our exercise, the graphing utility will not only help to create the models but also to estimate which model more accurately predicts sales for the year 2011.

Estimation Methods

Estimation methods in regression analysis involve determining the parameters that produce the best fit for the observed data. These methods are crucial for making predictions based on the models developed. In the exercise provided, estimation is used in two ways: first to calculate the coefficients of the exponential and linear regression models, and second to predict future values given a specific time.

Once the model parameters are identified using a graphing utility, in this case, the constants \( a \) and \( b \) for the exponential model and \( m \) and \( b \) for the linear model, we would apply these constants to estimate the sales for the year 2011. For the exponential model, the calculations exhibit the compound nature of growth, while the linear model projects a straight-line increase over time.

These estimated values can then be compared to assess which model more accurately fits the future sales, which, in real-world applications, is key for making data-driven decisions.