Automobile unit sales at B. J. Scott Motors, Inc., provided the following 10 -year time series.

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Automobile unit sales at B. J. Scott Motors, Inc., provided...

Automobile unit sales at $\mathrm{B}$. J. Scott Motors, Inc., provided the following 10 -year time series. $$\begin{array}{cccc} \text { Year } & \text { Sales } & \text { Year } & \text { Sales } \\ 1 & 400 & 6 & 260 \\ 2 & 390 & 7 & 300 \\ 3 & 320 & 8 & 320 \\ 4 & 340 & 9 & 340 \\ 5 & 270 & 10 & 370\end{array}$$ a. Construct a time series plot. Comment on the appropriateness of a linear trend. b. Using Minitab or Excel, develop a quadratic trend equation that can be used to forecast sales. c. Using the trend equation developed in part (b), forecast sales in year 11 d. Suggest an alternative to using a quadratic trend equation to forecast sales. Explain.

Automobile unit sales at $\mathrm{B}$. J. Scott Motors, Inc., provided the following 10 -year time series.
$$\begin{array}{cccc}
\text { Year } & \text { Sales } & \text { Year } & \text { Sales } \\
1 & 400 & 6 & 260 \\
2 & 390 & 7 & 300 \\
3 & 320 & 8 & 320 \\
4 & 340 & 9 & 340 \\
5 & 270 & 10 & 370\end{array}$$
a. Construct a time series plot. Comment on the appropriateness of a linear trend.
b. Using Minitab or Excel, develop a quadratic trend equation that can be used to forecast sales.
c. Using the trend equation developed in part (b), forecast sales in year 11
d. Suggest an alternative to using a quadratic trend equation to forecast sales. Explain.

Key Concepts

Time Series Plot

A time series plot is a graphical representation of a series of data points ordered in time. It allows analysts to visually inspect the data for trends, cycles, or irregularities. Plotting the data over time is essential for understanding the behavior and potential seasonality of the data, as well as for identifying structural breaks or outliers that may influence the forecasting model.

Linear Trend Model

A linear trend model assumes that the data follows a straight-line pattern over time. This model fits a linear equation to the data, suggesting a constant rate of increase or decrease. The appropriateness of a linear trend is evaluated by checking if the data’s pattern truly conforms to a straight line or if it exhibits curvature, which might indicate that a linear model is too simplistic for capturing the actual dynamics.

Quadratic Trend Model

A quadratic trend model extends the linear trend by incorporating a squared term in the time variable. This allows the model to capture curvature in the data, offering a better fit when the trend is non-linear. The quadratic model can adjust for acceleration or deceleration in the data, making it more flexible for forecasting in cases where the relationship between time and the variable is not strictly linear.

Forecasting

Forecasting in time series analysis involves using historical data to predict future values by applying a model fitted to the time series. Once a trend equation, whether linear or quadratic, is developed, it can be used to forecast future observations. Accurate forecasting relies on the appropriateness of the chosen model and the stability of the underlying data patterns over time.

Alternative Forecasting Methods

When a quadratic trend equation does not adequately capture the data's behavior or when the relationship is complex, alternative forecasting methods may be considered. These alternatives can include moving averages, exponential smoothing, or ARIMA models. Each method offers a different approach to handle trends, seasonality, and fluctuations in the data, providing flexibility in achieving more accurate future forecasts.

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