Answer:
(a) The value of P (X < 21 | <em>μ </em> = 23 and <em>σ</em> = 3) is 0.2514.
(b) The value of P (X ≥ 66 | <em>μ </em> = 50 and <em>σ</em> = 9) is 0.0427.
(c) The value of P (X > 47 | <em>μ </em> = 50 and <em>σ</em> = 5) is 0.7258.
(d) The value of P (17 < X < 24 | <em>μ </em> = 21 and <em>σ</em> = 3) is 0.7495.
(e) The value of P (X ≥ 95 | <em>μ </em> = 80 and <em>σ</em> = 1.82) is 0.
Step-by-step explanation:
The random variable <em>X</em> is Normally distributed.
(a)
The mean and standard deviation are:
![\mu=23\\\sigma=3](https://tex.z-dn.net/?f=%5Cmu%3D23%5C%5C%5Csigma%3D3)
Compute the value of P (X < 21) as follows:
![P(X](https://tex.z-dn.net/?f=P%28X%3C21%29%3DP%28%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3C%5Cfrac%7B21-23%7D%7B3%7D%29)
![=P(Z](https://tex.z-dn.net/?f=%3DP%28Z%3C-0.67%29%5C%5C%3D1-P%28Z%3C0.67%29%5C%5C%3D1-0.74857%5C%5C%3D0.25143%5C%5C%5Capprox0.2514)
Thus, the value of P (X < 21 | <em>μ </em> = 23 and <em>σ</em> = 3) is 0.2514.
(b)
The mean and standard deviation are:
![\mu=50\\\sigma=9](https://tex.z-dn.net/?f=%5Cmu%3D50%5C%5C%5Csigma%3D9)
Compute the value of P (X ≥ 66) as follows:
Use continuity correction.
P (X ≥ 66) = P (X > 66 - 0.5)
= P (X > 65.5)
![=P(\frac{X-\mu}{\sigma}>\frac{65.5-50}{9})](https://tex.z-dn.net/?f=%3DP%28%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3E%5Cfrac%7B65.5-50%7D%7B9%7D%29)
![=P(Z>1.72)\\=1-P(Z](https://tex.z-dn.net/?f=%3DP%28Z%3E1.72%29%5C%5C%3D1-P%28Z%3C1.72%29%5C%5C%3D1-0.9573%5C%5C%3D0.0427)
Thus, the value of P (X ≥ 66 | <em>μ </em> = 50 and <em>σ</em> = 9) is 0.0427.
(c)
The mean and standard deviation are:
![\mu=50\\\sigma=5](https://tex.z-dn.net/?f=%5Cmu%3D50%5C%5C%5Csigma%3D5)
Compute the value of P (X > 47) as follows:
![P(X>47)=P(\frac{X-\mu}{\sigma}>\frac{47-50}{5})](https://tex.z-dn.net/?f=P%28X%3E47%29%3DP%28%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3E%5Cfrac%7B47-50%7D%7B5%7D%29)
![=P(Z>-0.60)\\=P(Z](https://tex.z-dn.net/?f=%3DP%28Z%3E-0.60%29%5C%5C%3DP%28Z%3C0.60%29%5C%5C%3D0.7258)
Thus, the value of P (X > 47 | <em>μ </em> = 50 and <em>σ</em> = 5) is 0.7258.
(d)
The mean and standard deviation are:
![\mu=21\\\sigma=3](https://tex.z-dn.net/?f=%5Cmu%3D21%5C%5C%5Csigma%3D3)
Compute the value of P (17 < X < 24) as follows:
![P(17](https://tex.z-dn.net/?f=P%2817%3CX%3C24%29%3DP%28%5Cfrac%7B17-21%7D%7B3%7D%3C%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3C%5Cfrac%7B24-21%7D%7B3%7D%29)
![=P(-1.33](https://tex.z-dn.net/?f=%3DP%28-1.33%3CZ%3C1%29%5C%5C%3DP%28Z%3C1%29-P%28Z%3C-1.33%29%5C%5C%3D0.8413-0.0918%5C%5C%3D0.7495)
Thus, the value of P (17 < X < 24 | <em>μ </em> = 21 and <em>σ</em> = 3) is 0.7495.
(e)
The mean and standard deviation are:
![\mu=80\\\sigma=1.82](https://tex.z-dn.net/?f=%5Cmu%3D80%5C%5C%5Csigma%3D1.82)
Compute the value of P (X ≥ 95) as follows:
Use continuity correction:
P (X ≥ 95) = P (X > 95 - 0.5)
= P (X > 94.5)
![=P(\frac{X-\mu}{\sigma}>\frac{94.5-80}{1.82})](https://tex.z-dn.net/?f=%3DP%28%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3E%5Cfrac%7B94.5-80%7D%7B1.82%7D%29)
![=P(Z>7.97)\\=1-P(Z](https://tex.z-dn.net/?f=%3DP%28Z%3E7.97%29%5C%5C%3D1-P%28Z%3C7.97%29%5C%5C%3D1-%28%5Capprox1%29%5C%5C%3D0)
Thus, the value of P (X ≥ 95 | <em>μ </em> = 80 and <em>σ</em> = 1.82) is 0.