Understanding Z Scores with Practical Examples

A local newspaper in a large city wants to assess support for the construction of a highway bypass around the central business district to reduce downtown traffic. They survey a random sample of 1152 residents and find that 543 of them support the bypass. Construct and interpret a 95% confidence interval to estimate the proportion of residents who support construction of the bypass.

What is the z-score of sample X, if

How do you find the area under the normal distribution curve to the right of z = –3.24?

A distribution has a standard deviation of 5. What is the z score for a score that is above the mean by 10 points

I am looking at Central Limit Theorem. It says that for a series of random sample $W_1,\cdots <!--\; \cdots \; -->,W_2$ under the same mean $\mathrm{\mu <!--\; \mu \; -->}$ and standard deviation $\mathrm{\sigma <!--\; \sigma \; -->}$, then the random variable
${\displaystyle \frac{\stackrel{¯<!--\; ¯\; -->}{W}-<!--\; -\; -->\mathrm{\mu <!--\; \mu \; -->}}{\mathrm{\sigma <!--\; \sigma \; -->}/\sqrt{n}}}$
follows normal distribution. However, it seems like for Binomial samples,
${\displaystyle \frac{\stackrel{¯<!--\; ¯\; -->}{W}-<!--\; -\; -->\mathrm{\mu <!--\; \mu \; -->}}{\mathrm{\sigma <!--\; \sigma \; -->}}}={\displaystyle \frac{\stackrel{¯<!--\; ¯\; -->}{W}-<!--\; -\; -->np}{\sqrt{np(1-<!--\; -\; -->p)}}}$
is normal. I want to know where the n−−√ term in the denominator went?

Describe exactly what information is provided by a z-score. A positively skewed distribution has $\mathrm{\mu <!--\; \mu \; -->}=80,\mathrm{\sigma <!--\; \sigma \; -->}=12$. If this entire distribution is transformed into z-scores, describe the shape, mean, and standard deviation for the resulting distribution of z-scores.

I have a problem solving this exercise. I have this:
1. $P(0\le <!--\; \le \; -->Z\le <!--\; \le \; -->z_2)=0.3$
2. $P(Z\le <!--\; \le \; -->z_1)=0.3$
3. $P(z_1\le <!--\; \le \; -->Z\le <!--\; \le \; -->z_2)=0.8$
I need to find the z values for each given probability. I already solved the first and the second like this:
1. I calculated the inverse standard normal distribution (with LibreOffice Calc) and I found that $z_2$ is 0.841
2. I calculated the inverse standard normal distribution (with LibreOffice Calc) and I found that $z_1$ is −0.524
How can I find $z_1$ and $z_2$ of the third point of the exercise?

What is the z-score of sample X, if $n=54,\mathrm{\mu <!--\; \mu \; -->}=12,\text{St.Dev}=30,{\mathrm{\mu <!--\; \mu \; -->}}_X=11$?

What is the z-score of sample X, if $n=36,\mathrm{\mu <!--\; \mu \; -->}=8,\text{St.Dev}=9,{\mathrm{\mu <!--\; \mu \; -->}}_X=11.2$?

Can a z-score be negative?

Z-scores are useful with regard to inferential statistics primarily because they offer a tool by which to determine whether _____.
a. an individual is different from a sample.
b. a sample is noticeably different from a population.
c. a distribution of scores is standardized.
d. a distribution of scores is symmetrically shaped.

Susan is 4' 11 tall (59 inches). Given that the average height for women is 63.5" and the standard deviation is 2.5", find Susan's z-score.
Hint: Z-score can be positive or negative. Make sure you determine which appropriately.

I want to apply to a university where the 25th /75th percentiles for the SAT Math are 490 / 620 respectively, but I am curious how would I find the mean and the standard deviation assuming that the data is normally distributed? I know that we are talking about the middle 50th-percentile and $z_{0.25}=620$ and $z_{0.75}=490$

Assume the average of laptop computer is $875 with a standard deviation of $65. The following data represent the prices of a sample of laptops at an electronics store. Calculate the z-score for each of the following prices.

Use the standard normal table to find the z-score that corresponds to the given percentile. If the area is not in the table, use the entry closest to the area. If the area is halfway between two entries, use the z-score halfway between the corresponding z-scores. If convenient, use technology to find the z-score.
$P_{80}$
The z-score that corresponds to $P_{80}$ is ...
(Round to two decimal places as needed.)

A random variable X is normally distributed with $\mathrm{\mu <!--\; \mu \; -->}=60$ and $\mathrm{\sigma <!--\; \sigma \; -->}$ = 3. What is the value of 2 numbers a,b so that $P(X=a)=P(X=b)$.
The solution is $a=60$ and $b=65$.
However, I do not know how to come up with that answer. As far as I understand $P(X=a)$ and $P(X=b)$ have to be both 0 since you always have to give a range e.g. $P(a<X)$. Moreover if I insert the values 60 and 65 in the formula $Z=(X-<!--\; -\; -->\mathrm{\mu <!--\; \mu \; -->})/\mathrm{\sigma <!--\; \sigma \; -->}$ than I would end up with 0,1.667 and z-scores 0.5, 0.952 respectively.

What is the z-score of sample X, if $n=36,\mathrm{\mu <!--\; \mu \; -->}=12.3,\text{St.Dev}=21,{\mathrm{\mu <!--\; \mu \; -->}}_X=18.5$?

What is the z-score of sample X, if

A distribution has a standard deviation of ${\displaystyle \sigma =8}$. How do you find the z-score below the mean by 8 points?

Trying to understand statistics/hypothesis testing. The example in the book discusses using a Z-Test. I am familiar with and on an intuitive level, I understand a Z-score, a Z-score basically measures how many standard deviations from the mean a point in a sample space is. This makes sense to me.
What I do not understand is why for a Z-test, we seemingly take the Z-score and divide it by $\sqrt{n}$ ?
Can anyone explain the difference, it seems like we are moving away from the intuitive explanation of "number of standard deviations from the mean" which is how we measure the probability via the area under the curve