Answer:
(a) The value of P (X < 230) is 0.9998.
(b) The value of P (180 < <em>X</em> < 245) is 1.
(c) The value of P (X > 190) is 0.9998.
(d) The value of <em>c</em> is 199.7.
(e) The value of <em>c</em> is 213.79.
Step-by-step explanation:
It is provided that random variable <em>X</em> follows a Normal distribution with parameters <em>μ</em> = 210 and <em>σ</em> = √32.
(a)
Compute the probability of the event <em>X</em> < 230 as follows:
![P(X](https://tex.z-dn.net/?f=P%28X%3C230%29%3DP%28%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3C%5Cfrac%7B230-210%7D%7B%5Csqrt%7B32%7D%7D%29%3DP%28Z%3C3.54%29%3D0.9998)
*Use the <em>z</em>-table for the probability.
Thus, the value of P (X < 230) is 0.9998.
(b)
Compute the probability of the event 180 < <em>X</em> < 245 as follows:
![P(180](https://tex.z-dn.net/?f=P%28180%3CX%3C245%29%3DP%28%5Cfrac%7B180-210%7D%7B%5Csqrt%7B32%7D%7D%20%3C%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%20%3C%5Cfrac%7B245-210%7D%7B%5Csqrt%7B32%7D%7D%29%5C%5C%3DP%28-5.30%3CZ%3C6.19%29%5C%5C%5Capprox%201)
*Use the <em>z</em>-table for the probability.
Thus, the value of P (180 < <em>X</em> < 245) is 1.
(c)
Compute the probability of the event <em>X</em> > 190 as follows:
![P(X>190)=P(\frac{X-\mu}{\sigma}>\frac{190-210}{\sqrt{32}})=P(Z>-3.54)=P(Z](https://tex.z-dn.net/?f=P%28X%3E190%29%3DP%28%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3E%5Cfrac%7B190-210%7D%7B%5Csqrt%7B32%7D%7D%29%3DP%28Z%3E-3.54%29%3DP%28Z%3C3.54%29%3D0.9998)
*Use the <em>z</em>-table for the probability.
Thus, the value of P (X > 190) is 0.9998.
(d)
It is provided that P (X < c) = 0.0344.
![P(X](https://tex.z-dn.net/?f=P%28X%3Cc%29%3D0.0344%5C%5CP%28%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3C%5Cfrac%7Bc-210%7D%7B%5Csqrt%7B32%7D%7D%20%20%29%3D0.0344%5C%5CP%28Z%3Cz%29%3D0.0344)
The value of <em>z</em> for which P (Z < z) = 0.0344 is -1.82.
Compute the value of <em>c</em> as follows:
![-1.82=\frac{c-210}{\sqrt{32}} \\c=210-1.82\times/\sqrt{32}\\=210-10.30\\=199.70](https://tex.z-dn.net/?f=-1.82%3D%5Cfrac%7Bc-210%7D%7B%5Csqrt%7B32%7D%7D%20%5C%5Cc%3D210-1.82%5Ctimes%2F%5Csqrt%7B32%7D%5C%5C%3D210-10.30%5C%5C%3D199.70)
Thus, the value of <em>c</em> is 199.7.
(e)
It is provided that P (X > c) = 0.7486.
![P(X>c)=0.7486\\P(\frac{X-\mu}{\sigma}>\frac{c-210}{\sqrt{32}} )=0.7486\\P(Z>z)=0.7486\\P(Z](https://tex.z-dn.net/?f=P%28X%3Ec%29%3D0.7486%5C%5CP%28%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3E%5Cfrac%7Bc-210%7D%7B%5Csqrt%7B32%7D%7D%20%20%29%3D0.7486%5C%5CP%28Z%3Ez%29%3D0.7486%5C%5CP%28Z%3Cz%29%3D1-0.7486%3D0.2514)
The value of <em>z</em> for which P (Z < z) = 0.2514 is 0.67.
Compute the value of <em>c</em> as follows:
![0.67=\frac{c-210}{\sqrt{32}} \\c=210+0.67\times/\sqrt{32}\\=210+3.79\\=213.79](https://tex.z-dn.net/?f=0.67%3D%5Cfrac%7Bc-210%7D%7B%5Csqrt%7B32%7D%7D%20%5C%5Cc%3D210%2B0.67%5Ctimes%2F%5Csqrt%7B32%7D%5C%5C%3D210%2B3.79%5C%5C%3D213.79)
Thus, the value of <em>c</em> is 213.79.