Answer
See below:
Work Step by Step
(a)
Consider the linear model,
\[f\left( x \right)=782x+6564\]
To get the average cost of a family health insurance plan in\[2008\]. Then, in \[2008\]the value of \[x\]will be \[8\]years. So, substitute the value of \[x\] in the above linear model.
Then,
\[\begin{align}
& f\left( x \right)=782x+6564 \\
& =782\cdot 8+6564 \\
& =6256+6564 \\
& =12820
\end{align}\]
Hence, the average cost of a family health insurance plan in\[2008\]is\[\$12820\].
(b)
Consider the exponential model,
\[g\left( x \right)=6875{{e}^{0.077x}}\]
To get the average cost of a family health insurance plan in\[2008\]. Then, in \[2008\]the value of \[x\]will be \[8\]years. So, substitute the value of \[x\] in the above exponential model.
Then, to solve the above expression, use TI83 calculator.
Steps are used in the TI83 calculator is as follows:
Step (1): Press enter on the ON button.
Step (2): Enter the expression that is\[6875{{e}^{0.077\left( 8 \right)}}\].
Step (3): Press enter to get the value of the expression.
In TI83, the expression is as follows:
\[6875{{e}^{\wedge \left( 0.077 \right)\left( 8 \right)}}=12729\]
Hence,the average cost of a family health insurance plan in\[2008\]is\[\$12729\].
(c)
Consider the linear model,
\[f\left( x \right)=782x+6564\]
To get the average cost of a family health insurance plan in\[2008\]. Then, in \[2008\]the value of \[x\]will be \[8\]years. So, substitute the value of \[x\] in the above linear model.
Then,
\[\begin{align}
& f\left( x \right)=782x+6564 \\
& =782\cdot 8+6564 \\
& =6256+6564 \\
& =12820
\end{align}\]
Further, from exponential model
Consider the exponential model,
\[g\left( x \right)=6875{{e}^{0.077x}}\]
To get the average cost of a family health insurance plan in\[2008\]. Then in \[2008\], the value of \[x\]will be \[8\]years. So, substitute the value of \[x\] in the above exponential model.
Then to solve the above expression use TI83 calculator.
Steps are used in the TI83 calculator is as follows:
Step (1): Press enter on the ON button.
Step (2): Enter the expression that is\[6875{{e}^{0.077\left( 8 \right)}}\].
Step (3): Press enter to get the value of the expression.
In TI83, the expression is as follows:
\[6875{{e}^{\wedge \left( 0.077 \right)\left( 8 \right)}}=12729\]
From both the models, exponential model value is less than linear model value as it represents the average cost of a family health insurance plan in\[2008\]. It should be better, when it is minimum.
Hence, the exponential model is a better model for data in\[2008\].
