Answer
a) The expression is $\underline{\frac{1}{7}\sum\limits_{i=1}^{7}{{{a}_{i}}}=3.9}$ which represents the mean hour spent on the digital media.
b) The value of $\frac{1}{7}\sum\limits_{i=1}^{7}{{{a}_{i}}}$ using ${{a}_{n}}=0.5n+2$ is $4$ and it is overestimated by 0.1 hour with respect to part (a).
Work Step by Step
(a)
From the given bar graph,
${{a}_{1}}=2.7$, ${{a}_{2}}=3.0$, ${{a}_{3}}=3.2$, ${{a}_{4}}=3.7$, ${{a}_{5}}=4.4$ ${{a}_{6}}=4.9$ and ${{a}_{7}}=5.3$
Therefore,
$\begin{align}
& \frac{1}{7}\sum\limits_{i=1}^{7}{{{a}_{i}}}=\frac{1}{7}\left( 2.7+3.0+3.2+3.7+4.4+4.9+5.3 \right) \\
& =\frac{27.2}{7} \\
& =3.885 \\
& \approx 3.9\simeq
\end{align}$
Hence, $\frac{1}{7}\sum\limits_{i=1}^{7}{{{a}_{i}}}=3.9$ represents the mean hours that U.S. adult users spent on the digital media.
(b)
We have the equation ${{a}_{n}}=0.5n+2$, For $ n=1$,
$\begin{align}
& {{a}_{1}}=0.5\times 1+2 \\
& =2.5
\end{align}$
For $ n=2$
$\begin{align}
& {{a}_{2}}=0.5\times 2+2 \\
& =3.0
\end{align}$
For $ n=3$
$\begin{align}
& {{a}_{3}}=0.5\times 3+2 \\
& =3.5
\end{align}$
For $ n=4$
$\begin{align}
& {{a}_{4}}=0.5\times 4+2 \\
& =4.0
\end{align}$
For $ n=5$
$\begin{align}
& {{a}_{5}}=0.5\times 5+2 \\
& =4.5
\end{align}$
For $ n=6$
$\begin{align}
& {{a}_{6}}=0.5\times 6+2 \\
& =5.0
\end{align}$
For $ n=7$
$\begin{align}
& {{a}_{7}}=0.5\times 7+2 \\
& =5.5
\end{align}$
Therefore,
$\begin{align}
& \frac{1}{7}\sum\limits_{i=1}^{7}{{{a}_{i}}}=\frac{1}{7}\left( 2.5+3.0+3.5+4.0+4.5+5.0+5.5 \right) \\
& =\frac{28}{7} \\
& =4
\end{align}$
Hence, the relation will overestimate the value of $\frac{1}{7}\sum\limits_{i=1}^{7}{{{a}_{i}}}$ by 0.1 hour.
