gaokao 2020 Q18

gaokao · China · national-III-science 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 on each of 100 days in a certain city. The organized data is shown in the table below (unit: days):
\backslashbox{Air Quality Level}{Number of Exercisers}[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 value of the 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 based on the contingency table, 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 level of the city on that day?
A student interest group randomly surveyed the air quality level and the number of people exercising in a certain park on each of 100 days in a certain city. The organized data is shown in the table below (unit: days):

\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
\backslashbox{Air Quality Level}{Number of Exercisers} & [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 value of the 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 based on the contingency table, 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 level of the city on that day?