gaokao 2015 Q17

gaokao · China · chongqing-arts 13 marks Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence
With the development of China's economy, residents' savings deposits have increased year by year. The table below shows the year-end balance of urban and rural residents' RMB savings deposits in a certain region:
Year20102011201220132014
Time code $t$12345
\begin{tabular}{ l } Savings deposits $y$
(hundred billion yuan)
& 5 & 6 & 7 & 8 & 10 \hline \end{tabular}
(I) Find the regression equation $\hat { y } = \hat { b } t + \hat { a }$ for $y$ with respect to $t$.
(II) Use the regression equation to predict the RMB savings deposits in this region for 2015 ($t = 6$). Note: In the regression equation $\hat { y } = \hat { b } t + \hat { a }$,
$$\hat { b } = \frac { \sum _ { i = 1 } ^ { n } t _ { i } y _ { i } - n \overline { t } \overline { y } } { \sum _ { i = 1 } ^ { n } t _ { i } ^ { 2 } - n \bar { t } ^ { 2 } } , \hat { \mathrm { a } } = \overline { \mathrm { y } } - \hat { \mathrm { b } } \overline { \mathrm { t } }$$
With the development of China's economy, residents' savings deposits have increased year by year. The table below shows the year-end balance of urban and rural residents' RMB savings deposits in a certain region:

\begin{center}
\begin{tabular}{ | l | l | l | l | l | l | }
\hline
Year & 2010 & 2011 & 2012 & 2013 & 2014 \\
\hline
Time code $t$ & 1 & 2 & 3 & 4 & 5 \\
\hline
\begin{tabular}{ l }
Savings deposits $y$ \\
(hundred billion yuan) \\
\end{tabular} & 5 & 6 & 7 & 8 & 10 \\
\hline
\end{tabular}
\end{center}

(I) Find the regression equation $\hat { y } = \hat { b } t + \hat { a }$ for $y$ with respect to $t$.

(II) Use the regression equation to predict the RMB savings deposits in this region for 2015 ($t = 6$). Note: In the regression equation $\hat { y } = \hat { b } t + \hat { a }$,

$$\hat { b } = \frac { \sum _ { i = 1 } ^ { n } t _ { i } y _ { i } - n \overline { t } \overline { y } } { \sum _ { i = 1 } ^ { n } t _ { i } ^ { 2 } - n \bar { t } ^ { 2 } } , \hat { \mathrm { a } } = \overline { \mathrm { y } } - \hat { \mathrm { b } } \overline { \mathrm { t } }$$