Find ?_x,if ? = 400, $\mathrm { P } ( \bar { X }

Question

Find ?_x,if ? = 400, $\mathrm { P } ( \bar { X } < 396 ) = .0228$ And n = 100. A)0.20 B)200 C)20 D)2.00

Quizplus · Accepted Answer

We know that
μ = 400 ,
$\mathrm { P } ( \bar { X } < 396 ) = .0228$ ,
n = 100,
σ = ?
Now, we can use the formula for the standardized variable (z-score):
$$
z = \frac{\bar{X}-\mu}{\sigma/\sqrt{n}}
$$
We can rearrange this to solve for sigma:
$$
\sigma = \frac{(\bar{X}-\mu)}{z_{\alpha/2} /\sqrt{n}}
$$
where $z_{\alpha/2}$ is the z-score corresponding to the given confidence level and is calculated as:
$$
z_{\alpha/2} = -z_\alpha = -z_{0.0228/2} \approx -2.19
$$
Now, substituting the values:
$$
\sigma = \frac{(396-400)}{-2.19/\sqrt{100}}\approx 20
$$
Therefore, the answer is C) 20.

Find ?x,if ? = 400 P(Xˉ<396)=.0228\mathrm { P } ( \bar { X } < 396 ) = .0228P(Xˉ<396)=.0228

Find ?_x,if ? = 400 $\mathrm { P } ( \bar { X } < 396 ) = .0228$