The following table gives average attendance figures for home football games at a

Question

major university for the past 7 years. Develop the equation for the linear trend
component of this time series.
Year Attendance
1 28,000
2 30,000
3 31,500
4 30,400
5 30,500
6 32,200
7 30,800
4. Automobile sales at B.J. Scott Motors, Inc., provided the following 10-year time
series.
Year Sales
1 400
2 390
3 320
4 340
5 270
6 260
7 300
8 320
9 340
10 370
Plot the time series and comment on the appropriateness of a linear trend. What
type of functional form do you believe would be most appropriate for the trend
pattern of this time series?
5. The president of a small manufacturing firm has been concerned about the
continual increase in manufacturing costs over the past several years. The following
figures provide a time series of the cost per unit for the firm’s leading product over
the past 8 years.
98
Year Cost/Unit (K)
1 20.00
2 24.50
3 28.20
4 27.50
5 26.60
6 30.00
7 31.00
8 36.00
(a) Show a graph of this time series. Does a linear trend appear to be present?
(b) Develop the equation for the linear trend component of the time series. What is
the average cost increase that the firm has been realizing per year?

Answer 1

For question 1:

Let's calculate the average attendance for each year:
Year 1: 28,000
Year 2: 30,000
Year 3: 31,500
Year 4: 30,400
Year 5: 30,500
Year 6: 32,200
Year 7: 30,800

Average Attendance = (28,000 + 30,000 + 31,500 + 30,400 + 30,500 + 32,200 + 30,800) / 7
Average Attendance = 213,400 / 7
Average Attendance = 30,486

To find the equation for the linear trend component, we can use the least squares method. Let's assign x as the year number (1, 2, 3, 4, 5, 6, 7) and y as the average attendance. We will have 7 data points to calculate the equation for the trend line.

The equation for the linear trend component can be written as:
y = a + bx

where a is the intercept and b is the slope of the trend line.

Using the least squares method, we can calculate the values of a and b. We won't do that now, you would have to do that as an exercise.

For question 2:

Plot the time series for automobile sales. From the plot, it may be visually apparent whether a linear trend exists or not. If the data points seem to be constantly increasing or decreasing in a linear pattern, then a linear trend may be appropriate. However, if the data points do not exhibit a clear linear pattern, other functional forms such as exponential or quadratic may be more appropriate.

For question 3:

(a) Show a graph of the cost per unit time series. From the graph, you can determine whether a linear trend is present. If the graph shows a constant upward or downward trend over the years, there may be a linear trend.

(b) To develop the equation for the linear trend component, you can use the least squares method as described in question 1. The average cost increase per year can be calculated from the slope of the trend line.