gaokao 2023 Q17

gaokao · China · national-B-science 12 marks Measures of Location and Spread
A factory compares the treatment effects of two processes (Process A and Process B) on the elasticity of rubber products through 10 paired experiments. In each paired experiment, two rubber products of the same material are selected, one is randomly chosen to be treated with Process A and the other with Process B. The elasticity rates of the rubber products treated by Process A and Process B are recorded as $x _ { i } , y _ { i } ( i = 1,2 , \cdots 10 )$ respectively. The experimental results are as follows:
Experiment Number $i$12345678910
Elasticity Rate $x _ { i }$545533551522575544541568596548
Elasticity Rate $y _ { i }$536527543530560533522550576536

Let $z _ { i } = x _ { i } - y _ { i } ( i = 1,2 , \cdots , 10 )$. Let $\bar { z }$ denote the sample mean of $z _ { 1 } , z _ { 2 } , \cdots , z _ { 10 }$ and $s ^ { 2 }$ denote the sample variance.
(1) Find $\bar { z }$ and $s ^ { 2 }$.
(2) Determine whether the elasticity rate of rubber products treated by Process A is significantly higher than that treated by Process B. (If $\bar { z } \geqslant 2 \sqrt { \frac { s ^ { 2 } } { 10 } }$, then it is considered that the elasticity rate of rubber products treated by Process A is significantly higher than that treated by Process B; otherwise, it is not considered to be significantly higher.)
A factory compares the treatment effects of two processes (Process A and Process B) on the elasticity of rubber products through 10 paired experiments. In each paired experiment, two rubber products of the same material are selected, one is randomly chosen to be treated with Process A and the other with Process B. The elasticity rates of the rubber products treated by Process A and Process B are recorded as $x _ { i } , y _ { i } ( i = 1,2 , \cdots 10 )$ respectively. The experimental results are as follows:

\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | c | }
\hline
Experiment Number $i$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
Elasticity Rate $x _ { i }$ & 545 & 533 & 551 & 522 & 575 & 544 & 541 & 568 & 596 & 548 \\
\hline
Elasticity Rate $y _ { i }$ & 536 & 527 & 543 & 530 & 560 & 533 & 522 & 550 & 576 & 536 \\
\hline
\end{tabular}
\end{center}

Let $z _ { i } = x _ { i } - y _ { i } ( i = 1,2 , \cdots , 10 )$. Let $\bar { z }$ denote the sample mean of $z _ { 1 } , z _ { 2 } , \cdots , z _ { 10 }$ and $s ^ { 2 }$ denote the sample variance.\\
(1) Find $\bar { z }$ and $s ^ { 2 }$.\\
(2) Determine whether the elasticity rate of rubber products treated by Process A is significantly higher than that treated by Process B. (If $\bar { z } \geqslant 2 \sqrt { \frac { s ^ { 2 } } { 10 } }$, then it is considered that the elasticity rate of rubber products treated by Process A is significantly higher than that treated by Process B; otherwise, it is not considered to be significantly higher.)