gaokao 2021 Q17

gaokao · China · national-A-arts Chi-squared test of independence
17. Two machine tools, Machine A and Machine B, produce the same type of product. Products are classified by quality into first-grade and second-grade products. To compare the quality of products from the two machines, 200 products were produced by each machine. The quality statistics are shown in the table below:
First-gradeSecond-gradeTotal
Machine A15050200
Machine B12080200
Total270130400

(1) What are the frequencies of first-grade products produced by Machine A and Machine B, respectively?
(2) Can we conclude with 99\% confidence that there is a difference in product quality between Machine A and Machine B? Attachment: $\mathrm { K } ^ { 2 } = \frac { n ( a d - b c ) ^ { 2 } } { ( a + b ) ( c + d ) ( a + c ) ( b + d ) }$,
$\mathrm { P } \left( \mathrm { K } ^ { 2 } \geqslant k \right)$0.0500.0100.001
$k$3.8416.63510.828
17. Two machine tools, Machine A and Machine B, produce the same type of product. Products are classified by quality into first-grade and second-grade products. To compare the quality of products from the two machines, 200 products were produced by each machine. The quality statistics are shown in the table below:

\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
 & First-grade & Second-grade & Total \\
\hline
Machine A & 150 & 50 & 200 \\
\hline
Machine B & 120 & 80 & 200 \\
\hline
Total & 270 & 130 & 400 \\
\hline
\end{tabular}
\end{center}

(1) What are the frequencies of first-grade products produced by Machine A and Machine B, respectively?\\
(2) Can we conclude with 99\% confidence that there is a difference in product quality between Machine A and Machine B?\\
Attachment: $\mathrm { K } ^ { 2 } = \frac { n ( a d - b c ) ^ { 2 } } { ( a + b ) ( c + d ) ( a + c ) ( b + d ) }$,

\begin{center}
\begin{tabular}{ | c | l | l | c | }
\hline
$\mathrm { P } \left( \mathrm { K } ^ { 2 } \geqslant k \right)$ & 0.050 & 0.010 & 0.001 \\
\hline
$k$ & 3.841 & 6.635 & 10.828 \\
\hline
\end{tabular}
\end{center}