Answer:
(a) 0.7967
(b) 0.6826
(c) 0.3707
(d) 0.9525
(e) 0.1587
Step-by-step explanation:
The random variable <em>X</em> follows a Normal distribution with mean <em>μ</em> = 10 and variance <em>σ</em>² = 36.
(a)
Compute the value of P (X > 5) as follows:
![P(X>5)=P(\frac{x-\mu}{\sigma}>\frac{5-10}{\sqrt{36}})\\=P(Z>-0.833)\\=P(Z](https://tex.z-dn.net/?f=P%28X%3E5%29%3DP%28%5Cfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D%3E%5Cfrac%7B5-10%7D%7B%5Csqrt%7B36%7D%7D%29%5C%5C%3DP%28Z%3E-0.833%29%5C%5C%3DP%28Z%3C0.83%29%5C%5C%3D0.7967)
Thus, the value of P (X > 5) is 0.7967.
(b)
Compute the value of P (4 < X < 16) as follows:
![P(4](https://tex.z-dn.net/?f=P%284%3CX%3C16%29%3DP%28%5Cfrac%7B4-10%7D%7B%5Csqrt%7B36%7D%7D%3C%5Cfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D%3C%5Cfrac%7B16-10%7D%7B%5Csqrt%7B36%7D%7D%29%5C%5C%3DP%28-1%3CZ%3C1%29%5C%5C%3DP%28Z%3C1%29-P%28Z%3C-1%29%5C%5C%3DP%28Z%3C1%29-1%2BP%28Z%3C1%29%5C%5C%3D2P%28Z%3C1%29-1%5C%5C%3D%282%5Ctimes0.8413%29-1%5C%5C%3D0.6826)
Thus, the value of P (4 < X < 16) is 0.6826.
(c)
Compute the value of P (X < 8) as follows:
![P(X](https://tex.z-dn.net/?f=P%28X%3C8%29%3DP%28%5Cfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D%3C%5Cfrac%7B8-10%7D%7B%5Csqrt%7B36%7D%7D%29%5C%5C%3DP%28Z%3C-0.33%29%5C%5C%3D1-P%28Z%3C0.33%29%5C%5C%3D1-0.6293%5C%5C%3D0.3707)
Thus, the value of P (X < 8) is 0.3707.
(d)
Compute the value of P (X < 20) as follows:
![P(X](https://tex.z-dn.net/?f=P%28X%3C20%29%3DP%28%5Cfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D%3C%5Cfrac%7B20-10%7D%7B%5Csqrt%7B36%7D%7D%29%5C%5C%3DP%28Z%3C1.67%29%5C%5C%3D0.9525)
Thus, the value of P (X < 20) is 0.9525.
(e)
Compute the value of P (X > 16) as follows:
![P(X>16)=P(\frac{x-\mu}{\sigma}>\frac{16-10}{\sqrt{36}})\\=P(Z>1)\\=1-P(Z](https://tex.z-dn.net/?f=P%28X%3E16%29%3DP%28%5Cfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D%3E%5Cfrac%7B16-10%7D%7B%5Csqrt%7B36%7D%7D%29%5C%5C%3DP%28Z%3E1%29%5C%5C%3D1-P%28Z%3C1%29%5C%5C%3D1-0.8413%5C%5C%3D0.1587)
Thus, the value of P (X > 16) is 0.1587.
**Use a <em>z</em>-table for the probabilities.