gaokao 2025 Q15

gaokao · China · national-I 13 marks Chi-squared test of independence
To study the relationship between a certain disease and ultrasound examination results, 1000 people who had undergone ultrasound examination were randomly surveyed, yielding the following contingency table:
\backslashbox{Category}{Ultrasound Result}NormalAbnormalTotal
Has Disease20180200
No Disease78020800
Total8002001000

(1) Let $P$ denote the probability that a person with abnormal ultrasound results has the disease. Find the estimated value of $P$.
(2) Based on the significance level $\alpha = 0.001$ for independence testing, analyze whether the ultrasound examination result is related to having the disease.
Attachment: $\chi^2 = \frac{n(ad - bc)^2}{(a+b)(c+d)(a+c)(b+d)}$,
$P(\chi^2 \geq k)$0.0050.0100.001
$k$3.8416.63510.828
To study the relationship between a certain disease and ultrasound examination results, 1000 people who had undergone ultrasound examination were randomly surveyed, yielding the following contingency table:

\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
\backslashbox{Category}{Ultrasound Result} & Normal & Abnormal & Total \\
\hline
Has Disease & 20 & 180 & 200 \\
\hline
No Disease & 780 & 20 & 800 \\
\hline
Total & 800 & 200 & 1000 \\
\hline
\end{tabular}
\end{center}

(1) Let $P$ denote the probability that a person with abnormal ultrasound results has the disease. Find the estimated value of $P$.\\
(2) Based on the significance level $\alpha = 0.001$ for independence testing, analyze whether the ultrasound examination result is related to having the disease.

Attachment: $\chi^2 = \frac{n(ad - bc)^2}{(a+b)(c+d)(a+c)(b+d)}$,

\begin{center}
\begin{tabular}{ | c | l | l | c | }
\hline
$P(\chi^2 \geq k)$ & 0.005 & 0.010 & 0.001 \\
\hline
$k$ & 3.841 & 6.635 & 10.828 \\
\hline
\end{tabular}
\end{center}