Answer
The mathematical model for increasing revenue of internet publishing and broadcasting is:
\[y=2.37x-2.02\]
The mathematical model for decreasing basic cable television subscriber is\[y=-.62x+68.28\].
Yes, the fast growing technology can change the current trend.
Work Step by Step
Consider the year\[2006\], \[x=6\].
From the provided table, the data points are\[\left( 6,11.5 \right)\],\[\left( 7,15.0 \right)\], \[\left( 8,17.8 \right)\], \[\left( 9,19.1 \right)\], and \[\left( 10,21.3 \right)\].
Use the TI83 calculator, enter the \[x\]-coordinates list \[{{\text{L}}_{\text{1}}}\] and corresponding \[y\]-coordinates as a list\[{{\text{L}}_{\text{2}}}\].
Steps to plot the data
Step 1: Press STAT EDIT, enter the \[x\]-coordinates as a list\[{{\text{L}}_{\text{1}}}\] and \[y\]-coordinates as\[{{\text{L}}_{\text{2}}}\].
Step 2: For the regression line equation, go to STAT CALC menu and choose LINREG press \[{{\text{L}}_{\text{1}}},{{\text{L}}_{\text{2}}}\]and\[{{\text{y}}_{1}}\] (\[{{y}_{1}}\]can be chosen by Y-VARS FUNCTION).
The results shown in the calculator are:
\[\begin{align}
& y=ax+b \\
& a=2.37 \\
& b=-2.02
\end{align}\]
So, the mathematical model is\[y=2.37x-2.02\].
From this mathematical model, calculate the revenue for the year\[2008\].
Put \[x=8\]in the obtained mathematical model.
\[\begin{align}
& y=2.37x-2.02 \\
& \ =2.37\cdot \left( 8 \right)-2.02 \\
& =16.98
\end{align}\]
The estimated value \[16.98\] is close to the actualvalue \[17.8\].
So in the future, revenue can be forecasted by this mathematical model.
Yes, there are some circumstances that might affect the accuracy of the prediction.
In the era of fast growing technology, there may be immersion of new technology, which will change the current trend.
For steadily decreasing data with respect to years.
Consider the year\[2006\], \[x=6\].
From the provided table, the data points are\[\left( 2,66.50 \right)\],\[\]\[\left( 4,65.7 \right)\], \[\left( 6,65.3 \right)\], \[\left( 8,64.3 \right)\], and \[\left( 10,61.0 \right)\].
Use the TI83 calculator, enter the \[x\]-coordinates list \[{{\text{L}}_{\text{1}}}\] and corresponding \[y\]-coordinates as a list\[{{\text{L}}_{\text{2}}}\].
Step1: Press STAT EDIT, enter the \[x\]-coordinates as a list\[{{\text{L}}_{\text{1}}}\] and \[y\]-coordinates as\[{{\text{L}}_{\text{2}}}\].
Step 2: For the regression line equation, go to STAT CALC menu and choose LINREG press \[{{\text{L}}_{\text{1}}},{{\text{L}}_{\text{2}}}\]and \[{{\text{y}}_{1}}\] (\[{{y}_{1}}\]can be chosen by Y-VARS FUNCTION).
The resultsshown in the calculator are:
\[\begin{align}
& y=ax+b \\
& a=-.62 \\
& b=68.28
\end{align}\]
So, the mathematical model is\[y=-.62x+68.28\].
From this mathematical model, calculate the number of basic cable subscribers for the year\[2008\].
Put \[x=8\]in the obtained mathematical model.
\[\begin{align}
& y=-.62x+68.28 \\
& \ =-0.62\cdot \left( 8 \right)+68.28 \\
& =63.32
\end{align}\]
The estimated number of basic cable subscribers for the year \[2008\]is \[63.32\] close to the actualvalue \[64.3\].
So, in the future revenue can be forecasted by this mathematical model.
Yes, there are some circumstances that might affect the accuracy of the prediction.
In the era of fast growing technology there may be immersion of new television connection technology which will change the current trend. For example DTH, TV WHITE SPACE etc.
Hence, the mathematical model for increasing revenue of internet publishing and broadcasting is\[y=2.37x-2.02\] and for decreasing basic cable television subscriber is \[y=-.62x+68.28\] and yes, the fast growing technology can change the current trend.
