Stepwise Descriptive Statistics Learning

Existence of solution for matrix equation $(I-<!--\; -\; -->\mathrm{\alpha <!--\; \alpha \; -->}A)\stackrel{¯<!--\; ¯\; -->}{x}=\stackrel{¯<!--\; ¯\; -->}{b}$
This is my first question in here and I would be really thankful if someone could help me with understanding the matter.
I am solving a matrix equation $(I-<!--\; -\; -->\mathrm{\alpha <!--\; \alpha \; -->}A)\stackrel{¯<!--\; ¯\; -->}{x}=\stackrel{¯<!--\; ¯\; -->}{b}$for a positive vector $\stackrel{¯<!--\; ¯\; -->}{x}$. I don't know anything about the signs of the scalar $\mathrm{\alpha <!--\; \alpha \; -->}$ and vector $\stackrel{¯<!--\; ¯\; -->}{b}$ (but I can always split the solution into several cases). Matrix A is a symmetric matrix with tr(A)=0 (basically it is an adjacent matrix of an undirected graph so it is built only of 0 and 1 with 0 on its diagonal).
My approach would be just to write it down as $\stackrel{¯<!--\; ¯\; -->}{x}=(I-<!--\; -\; -->\mathrm{\alpha <!--\; \alpha \; -->}A{)}^{-<!--\; -\; -->1}\stackrel{¯<!--\; ¯\; -->}{b}$ whenever $(I-<!--\; -\; -->\mathrm{\alpha <!--\; \alpha \; -->}A)$ is nonsingular (as far as I understand that happens for a finite number of α and thus Lebesgue measure of this set of "inappropriate" $\mathrm{\alpha <!--\; \alpha \; -->}$ is 0).
In most of the literature, however, it is required that $1>\mathrm{\alpha <!--\; \alpha \; -->}\cdot <!--\; \cdot \; -->{\mathrm{\lambda <!--\; \lambda \; -->}}_{max}(A)$ for the convergence of $I+\mathrm{\alpha <!--\; \alpha \; -->}A+{\mathrm{\alpha <!--\; \alpha \; -->}}^2{A}^2+\dots <!--\; \dots \; -->$ and for the solution $\stackrel{¯<!--\; ¯\; -->}{x}$ to exist (where ${\mathrm{\lambda <!--\; \lambda \; -->}}_{max}(A)$ is a maximum eigenvalue of A) . Moreover, then $det(I-<!--\; -\; -->\mathrm{\alpha <!--\; \alpha \; -->}A)=1-<!--\; -\; -->\mathrm{\alpha <!--\; \alpha \; -->}\cdot <!--\; \cdot \; -->det(A)=1-<!--\; -\; -->\mathrm{\alpha <!--\; \alpha \; -->}\cdot <!--\; \cdot \; -->\underset{i}{\prod <!--\; \prod \; -->}{\mathrm{\lambda <!--\; \lambda \; -->}}_i>0$ and hence it is also invertible.
So I have two questions here:
Q1: Why do I need convergence of $I+\mathrm{\alpha <!--\; \alpha \; -->}A+{\mathrm{\alpha <!--\; \alpha \; -->}}^2{A}^2+\dots <!--\; \dots \; -->$ in here? I understand that if it converges then $\underset{k=0}{\overset{+\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}{\mathrm{\alpha <!--\; \alpha \; -->}}^k{A}^k=(I-<!--\; -\; -->\mathrm{\alpha <!--\; \alpha \; -->}A{)}^{-<!--\; -\; -->1}$, but why do I need that for the existence of a solution x¯? For instance, if we take a simple example:
$A=\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right|\mathrm{\alpha <!--\; \alpha \; -->}=30$
${\mathrm{\lambda <!--\; \lambda \; -->}}_{max}(A)=1$ and hence, convergence condition is violated: 30>1. However, I can still calculate $\stackrel{¯<!--\; ¯\; -->}{x}$. Inverse of $(I-<!--\; -\; -->\mathrm{\alpha <!--\; \alpha \; -->}A)$ exists and is equal to $-<!--\; -\; -->\frac{1}{899}\left|\begin{array}{cc}1& 30\\ 30& 1\end{array}\right|$. If, for instance, $\stackrel{¯<!--\; ¯\; -->}{b}$ was negative, then it would give me a positive $\stackrel{¯<!--\; ¯\; -->}{x}$. Or is the condition $1>\mathrm{\alpha <!--\; \alpha \; -->}\cdot <!--\; \cdot \; -->{\mathrm{\lambda <!--\; \lambda \; -->}}_{max}(A)$ needed to guarantee that inverse will give a positive solution with both positive α and $\stackrel{¯<!--\; ¯\; -->}{b}$?
Q2: What will happen if $\mathrm{\alpha <!--\; \alpha \; -->}<0$ (and potentially $\stackrel{¯<!--\; ¯\; -->}{b}<0$)? What are the conditions for existence of solution $\stackrel{¯<!--\; ¯\; -->}{x}$? What are the conditions for it to be positive?

Sara Gordon is heading a fund raising drive for a college. She wishes to concentrate on the current 10th reunion class and hopes to get contributions from 36% of the 250 members of that class. Past data indicate that those who contribute to the 10th year reunion will donate 4% of their annual salaries. Sara believes that the reunion class members' average salaries have an avg of 32000 dollars and a std. deviation of 9600 dollars. If her expectations are met (36% of class contributing 4% of their salaries), what is the probability that the reunion gift will be between 110000 dollars and 120000 dollars?
Should I take the std error of sampling distribution and calculate the z score? If I do that, the z score comes to $\sim <!--\; \sim \; -->6.1$

Text:Decide if each of the following statements is True or False.
Transforming an entire distribution of scores into z-scores will not change the shape of the distribution.

What is the z-score of sample X, if

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A. Metasynthesis
B. Meta-analysis
C. Literature review
D. Mixed methods synthesis

Name two different ways to measure effect size.
Effect size measures the size of the effect of what on what?
How do the two measures of effect size differ?
Why do we measure effect size?
When do we measure effect size?

in some exam past papers I have been doing I have come across the statistics equation
z=(sample mean - mean)/standard deviation
as well as the equation
z = (sample mean - mean)/(standard deviation/sqrt n).
could anyone please explain when to use which equation and what the difference is? thanks so much!

In a planned​ study, there is a known population with a normal​distribution, $\mu =20$​,and $\sigma =4$.
What is the predicted mean if the researcher predicts​
(a) a small positive effect​ size,
(b) a medium negative effect​ size,
(c) a large positive effect​ size,
(d) an effect size of $d=0.55$​,
(e) an effect size of $\sigma =4$?

What is the z-score of sample X, if $n=256,\mathrm{\mu <!--\; \mu \; -->}=67,\text{St.Dev}=80,{\mathrm{\mu <!--\; \mu \; -->}}_X=56$?

If an entire population with $\mathrm{\mu <!--\; \mu \; -->}=60,\mathrm{\sigma <!--\; \sigma \; -->}=8$ is transformed into z-scores, then the distribution of z-scores will have a mean of ___ and a standard deviation of ___
a) 60; 8
b) 1; 0
c) 0;1
d) 0;8

I know how to calculate question which are phrased like so
A study of data collected at a company manufacturing flashlight batteries shows that a batch of 8000 batteries have a mean life of 250 minutes with a standard deviation of 20 minutes. Assuming a Normal Distribution, estimate:
(i) How many batteries will fail before 220 minutes?
Answer = 534.4
But I can not figure questions phrased like this:
Support call times at a technical support center are Normally distributed with a mean time of 8 minutes and 45 seconds and a standard deviation of 1 minute and 5 seconds. On a particular day, a total of 500 calls are taken at the centre. How many of these calls are likely to last more than 10 minutes
I dont understand how to find the z-score in this question as its to do with time?

How do you find the domain and range of y-4=1(x-4)?

A die is rolled 60 times.
Find the normal approximation to the chance that the face with six spots appears between 9, 10, or 11 times.
The exact chance that the face with six spots appears 9, 10, or 11 times?
my solution; in one roll of the die $P(6\text{ spots})=\frac{1}{6}$ In 60 rolls of the die the expected number of times the face with 6 spots appears is given by $E(X)=\text{Mean}=\frac{60}{6}=10$ times.
$SE(6spots)=SD=SQR(\frac{60}{6}\cdot <!--\; \cdot \; -->\frac{5}{6})=2.8868$
Applying the normal approximation to the chance that the face with six spots appears 10 times, I obtain the following: mean = 10
$\text{Standard deviation}=2.8868$
Using the continuity correction of 0.5 the following z-scores are found: $z_1=(9.5-10)/2.8868=-0.1732$ and $z_2=(10.5-10)/2.8868=0.1732$ Subtracting the cumulative probabilities for the two z-scores gives $0.5688-<!--\; -\; -->0.4312=0.1376$ which is the approximate chance that the face with six spots appears 10 times. find the six spots appears between 9,10, or 11?
The exact chance that the face with six spots appears 10 times is found from
$60C10{\left(\frac{1}{6}\right)}^{10\times }{\left(\frac{5}{6}\right)}^{50}=0.137$ find six spots appears 9, 10 or 11 times?

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What is the unit of a z-score?

How do you identify the terms, like terms, coefficients and constants in each expression: -5x+4-x-1?

A commonly used form of statistical analysis in meta-analysis is:
- the two standard deviation method
- the proportion/frequency method
- the effect size
- all of the above

What is the domain and range of $y=3\sqrt{x-<!--\; -\; -->4}+2$?

Find the​ z-scores that separate the middle 48​% of the distribution from the area in the tails of the standard normal distribution.
The​ z-scores are?

I have two cases:
1. If I add a constant to them, does the mean, standard deviation and z score change?
What I think: Mean changes, std deviation doesn't change and z score doesn't change.
2. If I blackuce all the numbers by 5% does the mean, standard deviation and z score change?
What I think: Mean changes by 5%, standard deviation changes but the percentage difference remains the same, and the z score changes by 2.5%
Am I thinking correctly?