1. Developing the equation for the linear trend component of the football attendance time series:
Step 1: Calculate the average attendance for the 7 years
Total attendance = 28,000 + 30,000 + 31,500 + 30,400 + 30,500 + 32,200 + 30,800 = 213,400
Average attendance = 213,400 / 7 = 30,485.71
Step 2: Calculate the deviations from the average for each year
Year 1: 28,000 - 30,485.71 = -2,485.71
Year 2: 30,000 - 30,485.71 = -485.71
Year 3: 31,500 - 30,485.71 = 1,014.29
Year 4: 30,400 - 30,485.71 = -85.71
Year 5: 30,500 - 30,485.71 = 14.29
Year 6: 32,200 - 30,485.71 = 1,714.29
Year 7: 30,800 - 30,485.71 = 314.29
Step 3: Calculate the squared deviations for each year
Year 1: (-2,485.71)^2 = 6,179,578.49
Year 2: (-485.71)^2 = 235,355.10
Year 3: (1,014.29)^2 = 1,028,897.96
Year 4: (-85.71)^2 = 7,346.94
Year 5: (14.29)^2 = 204.08
Year 6: (1,714.29)^2 = 2,940,816.33
Year 7: (314.29)^2 = 98,652.04
Step 4: Calculate the sum of the squared deviations
Sum of squared deviations = 6,179,578.49 + 235,355.10 + 1,028,897.96 + 7,346.94 + 204.08 + 2,940,816.33 + 98,652.04 = 10,490,850.94
Step 5: Calculate the linear trend equation
Linear trend equation = ΣXY / ΣX^2
ΣXY = 1*28,000 + 2*30,000 + 3*31,500 + 4*30,400 + 5*30,500 + 6*32,200 + 7*30,800 = 1,132,700
ΣX^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 = 1 + 4 + 9 + 16 + 25 + 36 + 49 = 140
Linear trend equation = 1,132,700 / 140 = 8,090.71
Therefore, the equation for the linear trend component of the football attendance time series is:
Attendance = 8,090.71 * Year
2. Plot the time series for automobile sales at B.J. Scott Motors, Inc. and comment on the appropriateness of a linear trend.
- From the plot of the time series, it appears that a linear trend may not be appropriate for this data as the sales figures fluctuate significantly over the 10-year period. A more appropriate functional form for the trend pattern of this time series could be a quadratic or exponential trend due to the non-linear nature of the data.
3. Show a graph of the cost per unit time series for the small manufacturing firm and determine if a linear trend is present.
- Based on the graph of the cost per unit time series, it does not appear that a linear trend is present as the data points do not follow a straight line pattern. Instead, there seems to be an overall upward trend with fluctuations.
(b) Develop the equation for the linear trend component of the time series and calculate the average cost increase per year.
- Since a linear trend does not seem to be appropriate for this data, it may be more useful to analyze the average increase in cost per unit over the 8-year period.
- To calculate the average cost increase per year, we can find the difference between the cost per unit in the first year and the last year, and then divide by the total number of years:
Average cost increase per year = (36.00 - 20.00) / 8 = 2.00
Therefore, the average cost increase that the firm has been realizing per year is $2.00 per unit.
Step by step and by showing your work answer the questions below.
The following table gives average attendance figures for home football games at a
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?
1 answer