Answer:
(a) ![X\sim N(\mu = 73, \sigma = 16)](https://tex.z-dn.net/?f=X%5Csim%20N%28%5Cmu%20%3D%2073%2C%20%5Csigma%20%3D%2016%29)
(b) 0.7910
(c) 0.0401
(d) 0.6464
Step-by-step explanation:
Let <em>X</em> = amount of time that people spend at Grover Hot Springs.
The random variable <em>X</em> is normally distributed with a mean of 73 minutes and a standard deviation of 16 minutes.
(a)
The distribution of the random variable <em>X</em> is:
![X\sim N(\mu = 73, \sigma = 16)](https://tex.z-dn.net/?f=X%5Csim%20N%28%5Cmu%20%3D%2073%2C%20%5Csigma%20%3D%2016%29)
(b)
Compute the probability that a randomly selected person at the hot springs stays longer than 60 minutes as follows:
![P(X>60)=P(\frac{X-\mu}{\sigma}>\frac{60-73}{16})\\=P(Z>-0.8125)\\=P(Z](https://tex.z-dn.net/?f=P%28X%3E60%29%3DP%28%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3E%5Cfrac%7B60-73%7D%7B16%7D%29%5C%5C%3DP%28Z%3E-0.8125%29%5C%5C%3DP%28Z%3C0.8125%29%5C%5C%3D0.7910)
*Use a <em>z</em>-table for the probability.
Thus, the probability that a randomly selected person at the hot springs stays longer than an hour is 0.7910.
(c)
Compute the probability that a randomly selected person at the hot springs stays less than 45 minutes as follows:
![P(X](https://tex.z-dn.net/?f=P%28X%3C45%29%3DP%28%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3C%5Cfrac%7B45-73%7D%7B16%7D%29%5C%5C%3DP%28Z%3C-1.75%29%5C%5C%3D1-P%28Z%3C-1.75%29%5C%5C%3D1-0.9599%5C%5C%3D0.0401)
*Use a <em>z</em>-table for the probability.
Thus, the probability that a randomly selected person at the hot springs stays less than 45 minutes is 0.0401.
(d)
Compute the probability that a randomly person spends between 60 and 90 minutes at the hot springs as follows:
![P(60](https://tex.z-dn.net/?f=P%2860%3CX%3C90%29%3DP%28X%3C90%29-P%28X%3C60%29%5C%5C%3DP%28%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3C%5Cfrac%7B90-73%7D%7B16%7D%29-P%28%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3C%5Cfrac%7B60-73%7D%7B16%7D%29%5C%5C%3DP%28Z%3C1.0625%29-P%28Z%3C-0.8125%29%5C%5C%3D0.8554-0.2090%5C%5C%3D0.6464)
*Use a <em>z</em>-table for the probability.
Thus, the probability that a randomly person spends between 60 and 90 minutes at the hot springs is 0.6464