Question

The following table shows a company's annual revenue (in billions of dollars) for 2009 to 2014.

Year	Period (t)	Revenue ($ billions)
2009	1	23.8
2010	2	29.4
2011	3	37.8
2012	4	50.3
2013	5	59.7
2014	6	66.6

(a) Construct a time series plot.

A time series plot contains a series of 6 points connected by line segments. The horizontal axis ranges from 0 to 7 and is labeled: Period. The vertical axis ranges from 0 to 80 and is labeled: Revenue ($ billions). The points are plotted from left to right in a fairly consistent downward, diagonal direction at regular increments of 1 period starting at period 1. The points are between 23 to 67 on the vertical axis.

A time series plot contains a series of 6 points connected by line segments. The horizontal axis ranges from 0 to 7 and is labeled: Period. The vertical axis ranges from 0 to 80 and is labeled: Revenue ($ billions). The points are plotted from left to right at regular increments of 1 period starting at period 1. Initially, the points are plotted in a downward, diagonal direction. However, after the third point, the points are plotted in an upward, diagonal direction that becomes steeper as period increases. The points are between 17 to 67 on the vertical axis.

A time series plot contains a series of 6 points connected by line segments. The horizontal axis ranges from 0 to 7 and is labeled: Period. The vertical axis ranges from 0 to 80 and is labeled: Revenue ($ billions). The points are plotted from left to right at regular increments of 1 period starting at period 1. Initially, the points are plotted in an upward, diagonal direction. However, after the third point, the points are plotted in a downward, diagonal direction that becomes steeper as period increases. The points are between 23 to 71 on the vertical axis.

A time series plot contains a series of 6 points connected by line segments. The horizontal axis ranges from 0 to 7 and is labeled: Period. The vertical axis ranges from 0 to 80 and is labeled: Revenue ($ billions). The points are plotted from left to right in a fairly consistent upward, diagonal direction at regular increments of 1 period starting at period 1. The points are between 23 to 67 on the vertical axis.

What type of pattern exists in the data?

The time series plot shows an upward linear trend.The time series plot shows a downward curvilinear trend.     The time series plot shows an upward curvilinear trend.The time series plot shows a downward linear trend.

(b) Develop a linear trend equation for this time series to forecast revenue (in billions of dollars). (Round your numerical values to three decimal places.)

T_t =

(c) What is the average revenue increase per year (in billions of dollars) that this company has been realizing? (Round your answer to three decimal places.)

$  billion

(d) Compute an estimate of this company's revenue (in billions of dollars) for 2015. (Round your answer to two decimal places.)

$  billion

Answer 1

Answer(a):

The time series plot for given data is as below:

The correct option for description of time series plot is

D. A time series plot contains a series of 6 points connected by line segments. The horizontal axis ranges from 0 to 7 and is labeled: Period. The vertical axis ranges from 0 to 80 and is labeled: Revenue ($ billions). The points are plotted from left to right in a fairly consistent upward, diagonal direction at regular increments of 1 period starting at period 1. The points are between 23 to 67 on the vertical axis.

Pattern in the data: we can observe that there is clear upward linear trend in the data.

The correct option is

A. The time series plot shows an upward linear trend.

Answer(b): To fit a linear trend model, we fit a linear regression using period (t ) as independent variable(x) and revenue as dependent variable(y).

Year	Period (t)=x	Revenue ($ billions)=y
2009	1	23.8	-2.5	-20.8	52	6.25	432.64
2010	2	29.4	-1.5	-15.2	22.8	2.25	231.04
2011	3	37.8	-0.5	-6.8	3.4	0.25	46.24
2012	4	50.3	0.5	5.7	2.85	0.25	32.49
2013	5	59.7	1.5	15.1	22.65	2.25	228.01
2014	6	66.6	2.5	22	55	6.25	484
Total	21	267.6	0	0	158.7	17.5	1454.42
Mean	3.5	44.6

The regression equation between two variables can be given by following equation

The estimate of b0 and b1 can be obtained by least square method.

The least square estimate of b1 can be given by following expression

The estimate of b0 can be given by

The final estimated equation of line can be given as

In terms of given variables, the trend equation will be

Answer(c):

The average revenue increase per year (in billions of dollars) will be equal to the slope of the fitted trend regression model.

Hence the correct answer is

$9.069 Billion

Answer(d):

To compute an estimate of this company's revenue (in billions of dollars) for 2015, we have to predict the T_t for t=7 using the above estimated model

Hence the estimate of this company's revenue (in billions of dollars) for 2015 is $76.34 Billion