gaokao 2020 Q18

gaokao · China · national-III-arts 12 marks Chi-squared test of independence
A student interest group randomly surveyed the air quality level and the number of people exercising in a certain park each day over 100 days in a city. The data is organized in the following table (unit: days):
Air Quality Level$[ 0,200 ]$$( 200,400 ]$$( 400,600 ]$
1 (Excellent)21625
2 (Good)51012
3 (Slight Pollution)678
4 (Moderate Pollution)720

(1) Estimate the probability that the air quality level on a given day in the city is 1, 2, 3, or 4 respectively;
(2) Find the estimated average number of people exercising in the park on a given day (use the midpoint of each interval as the representative value for data in that interval);
(3) If the air quality level on a given day is 1 or 2, the day is called ``good air quality''; if the air quality level is 3 or 4, the day is called ``poor air quality''. Based on the given data, complete the following $2 \times 2$ contingency table and determine whether there is 95\% confidence to conclude that the number of people exercising in the park on a given day is related to the air quality of the city on that day.
Number of people $\leqslant 400$Number of people $> 400$
Good air quality
Poor air quality

Attachment: $K ^ { 2 } = \frac { n ( a d - b c ) ^ { 2 } } { ( a + b ) ( c + d ) ( a + c ) ( b + d ) }$,
$P \left( K ^ { 2 } \geqslant k \right)$0.0500.0100.001
$k$3.8416.63510.828
.
A student interest group randomly surveyed the air quality level and the number of people exercising in a certain park each day over 100 days in a city. The data is organized in the following table (unit: days):

\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
Air Quality Level & $[ 0,200 ]$ & $( 200,400 ]$ & $( 400,600 ]$ \\
\hline
1 (Excellent) & 2 & 16 & 25 \\
\hline
2 (Good) & 5 & 10 & 12 \\
\hline
3 (Slight Pollution) & 6 & 7 & 8 \\
\hline
4 (Moderate Pollution) & 7 & 2 & 0 \\
\hline
\end{tabular}
\end{center}

(1) Estimate the probability that the air quality level on a given day in the city is 1, 2, 3, or 4 respectively;\\
(2) Find the estimated average number of people exercising in the park on a given day (use the midpoint of each interval as the representative value for data in that interval);\\
(3) If the air quality level on a given day is 1 or 2, the day is called ``good air quality''; if the air quality level is 3 or 4, the day is called ``poor air quality''. Based on the given data, complete the following $2 \times 2$ contingency table and determine whether there is 95\% confidence to conclude that the number of people exercising in the park on a given day is related to the air quality of the city on that day.

\begin{center}
\begin{tabular}{ | c | c | c | }
\hline
 & Number of people $\leqslant 400$ & Number of people $> 400$ \\
\hline
Good air quality & & \\
\hline
Poor air quality & & \\
\hline
\end{tabular}
\end{center}

Attachment: $K ^ { 2 } = \frac { n ( a d - b c ) ^ { 2 } } { ( a + b ) ( c + d ) ( a + c ) ( b + d ) }$, \begin{tabular}{ c | r r r }
$P \left( K ^ { 2 } \geqslant k \right)$ & 0.050 & 0.010 & 0.001 \\
\hline
$k$ & 3.841 & 6.635 & 10.828 \\
\hline
\end{tabular}.