Let $\bar { X }$ be the sample mean of a sample of size 25 randomly extracted from a population that follows a normal distribution with population mean 75 and population standard deviation 5. For a random variable $Z$ following the standard normal distribution, a positive constant $c$ satisfies $$\mathrm { P } ( | Z | > c ) = 0.06$$ Which of the following in are correct? [4 points] ㄱ. For a constant $a$ such that $\mathrm { P } ( Z > a ) = 0.05$, we have $c > a$. ㄴ. $\mathrm { P } ( \bar { X } \leqq c + 75 ) = 0.97$ ㄷ. For a constant $b$ such that $\mathrm { P } ( \bar { X } > b ) = 0.01$, we have $c < b - 75$. (1) ㄱ (2) ㄷ (3) ㄱ, ㄴ (4) ㄴ, ㄷ (5) ㄱ, ㄴ, ㄷ
Let $\bar { X }$ be the sample mean of a sample of size 25 randomly extracted from a population that follows a normal distribution with population mean 75 and population standard deviation 5. For a random variable $Z$ following the standard normal distribution, a positive constant $c$ satisfies
$$\mathrm { P } ( | Z | > c ) = 0.06$$
Which of the following in <Remarks> are correct? [4 points]
<Remarks>\\
ㄱ. For a constant $a$ such that $\mathrm { P } ( Z > a ) = 0.05$, we have $c > a$.\\
ㄴ. $\mathrm { P } ( \bar { X } \leqq c + 75 ) = 0.97$\\
ㄷ. For a constant $b$ such that $\mathrm { P } ( \bar { X } > b ) = 0.01$, we have $c < b - 75$.\\
(1) ㄱ\\
(2) ㄷ\\
(3) ㄱ, ㄴ\\
(4) ㄴ, ㄷ\\
(5) ㄱ, ㄴ, ㄷ