Improve Your Understanding of College Level Statistics Problems

A population of values has a normal distribution with ${\displaystyle \mu =192.6}$ and ${\displaystyle \sigma =34.4}$. You intend to draw a random sample of size ${\displaystyle n=173}$.
Find the probability that a single randomly selected value is less than 186.1.
${\displaystyle P\left(X<186.1\right)=}$?
Write your answers as numbers accurate to 4 decimal places.

What is the optimal time for a scuba diver to be on the bottom of the ocean? That depends on the depth of the dive. The U.S. Navy has done a lot of research on this topic. The Navy defines the "optimal time" to be the time at each depth for the best balance between length of work period and decompression time after surfacing. Let ${\displaystyle x=}$ depth of dive in meters, and let ${\displaystyle y=}$ optimal time in hours. A random sample of divers gave the following data.
$\begin{array}{|cccccccc|}\hline x& 13.1& 23.3& 31.2& 38.3& 51.3& 20.5& 22.7\\ y& 2.78& 2.18& 1.48& 1.03& 0.75& 2.38& 2.20\\ \hline\end{array}$
(a)
Find ${\displaystyle \Sigma x,\Sigma y,\Sigma x^2,\Sigma y^2,\Sigma xy,\text{and}r}$. (Round r to three decimal places.)
${\displaystyle \Sigma x=}$
${\displaystyle \Sigma y=}$
${\displaystyle \Sigma x^2=}$
${\displaystyle \Sigma y^2=}$
${\displaystyle \Sigma xy=}$
${\displaystyle r=}$
(b)
Use a $1\mathrm{\%}$ level of significance to test the claim that ${\displaystyle \rho <0}$. (Round your answers to two decimal places.)
${\displaystyle t=}$
critical ${\displaystyle t=}$
Conclusion
Reject the null hypothesis. There is sufficient evidence that ${\displaystyle \rho <0}$.Reject the null hypothesis. There is insufficient evidence that ${\displaystyle \rho <0}$.
Fail to reject the null hypothesis. There is sufficient evidence that ${\displaystyle \rho <0}$.Fail to reject the null hypothesis. There is insufficient evidence that ${\displaystyle \rho <0.}$
(c)
Find ${\displaystyle S_e,a,\text{and}b}$. (Round your answers to four decimal places.)
${\displaystyle S_e=}$
${\displaystyle a=}$
${\displaystyle b=}$

For a population (${\displaystyle \mu =100}$ and ${\displaystyle \sigma =20}$), find the X values. ${\displaystyle z=+1.50}$.