csat-suneung 2021 Q12

csat-suneung · South-Korea · csat__math-science 3 marks Normal Distribution Algebraic Relationship Between Normal Parameters and Probability
The random variable $X$ follows a normal distribution with mean 8 and standard deviation 3, and the random variable $Y$ follows a normal distribution with mean $m$ and standard deviation $\sigma$. The two random variables $X$ and $Y$ satisfy $$\mathrm { P } ( 4 \leq X \leq 8 ) + \mathrm { P } ( Y \geq 8 ) = \frac { 1 } { 2 }$$ Find the value of $\mathrm { P } \left( Y \leq 8 + \frac { 2 \sigma } { 3 } \right)$ using the standard normal distribution table on the right.
$z$$\mathrm { P } ( 0 \leq Z \leq z )$
1.00.3413
1.50.4332
2.00.4772
2.50.4938

[3 points]
(1) 0.8351
(2) 0.8413
(3) 0.9332
(4) 0.9772
(5) 0.9938 
The random variable $X$ follows a normal distribution with mean 8 and standard deviation 3, and the random variable $Y$ follows a normal distribution with mean $m$ and standard deviation $\sigma$. The two random variables $X$ and $Y$ satisfy
$$\mathrm { P } ( 4 \leq X \leq 8 ) + \mathrm { P } ( Y \geq 8 ) = \frac { 1 } { 2 }$$
Find the value of $\mathrm { P } \left( Y \leq 8 + \frac { 2 \sigma } { 3 } \right)$ using the standard normal distribution table on the right.

\begin{center}
\begin{tabular}{ | c | c | }
\hline
$z$ & $\mathrm { P } ( 0 \leq Z \leq z )$ \\
\hline
1.0 & 0.3413 \\
\hline
1.5 & 0.4332 \\
\hline
2.0 & 0.4772 \\
\hline
2.5 & 0.4938 \\
\hline
\end{tabular}
\end{center}

[3 points]\\
(1) 0.8351\\
(2) 0.8413\\
(3) 0.9332\\
(4) 0.9772\\
(5) 0.9938
Paper Questions
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