gaokao 2021 Q17

gaokao · China · national-A-science 12 marks Chi-squared test of independence
17. (12 points) Two machine tools, A and 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 machine tools, 200 products were produced by each machine tool. The quality statistics are shown in the table below:
First-gradeSecond-gradeTotal
Machine A15050200
Machine B12080200
Total270130400

(1) What are the proportions 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: $k^2 = \frac{n(ac - bd)^2}{(a+b)(c+d)(a+c)(b+d)}$,
$P(K^2 \geq k)$0.0500.0100.001
$k$3.8416.63510.828
17. (12 points) Two machine tools, A and 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 machine tools, 200 products were produced by each machine tool. 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 proportions 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: $k^2 = \frac{n(ac - bd)^2}{(a+b)(c+d)(a+c)(b+d)}$, \begin{tabular}{ c | l l l }
$P(K^2 \geq k)$ & 0.050 & 0.010 & 0.001 \\
\hline
$k$ & 3.841 & 6.635 & 10.828 \\
\hline
\end{tabular}