Master Confidence Intervals: Examples, Equations, and Questions

Let ${\displaystyle X_1,\dots X_n}$ be iid ${\displaystyle N(\theta ,1)}$. A ${\displaystyle 95\mathrm{\%}}$ confidence interval for ${\displaystyle \theta }$ is ${\displaystyle \bar{X}\pm \frac{1.96}{\sqrt{n}}}$. Let p denote the probability that an additional independent observation, ${\displaystyle X_{n+1}}$, will fall in this interval. Is p greater than, less than, equal to ${\displaystyle 0.95}$? Prove your answer.

Let ${\displaystyle \bar{X}}$ be the mean of a random sample of size n from ${\displaystyle N(\mu ,9)}$. Find n so that ${\displaystyle P(\bar{X}-1<\mu <\bar{X}+1)=.9}$
This problem is given at the end of a chapter in my statistics textbook concerning confidence intervals for means.

Can a 2 paired sample sets of size 15 (30 readings in total) be assumed to have normal distribution when finding confidence intervals?
I've seen that for large sample sizes (about 30 or more) the distribution tends to a normal distribution, but what if I have 2 sets of 15 which are paired? Would that count as a normal distribution, or would I need to use a t-distribution to approximate confidence intervals for the difference between their means?

Are they both used to calculate the confidence interval at ${\displaystyle 95\mathrm{\%}}$ for a population proportion?
Here are two questions from my book.
In a survey in a large city, 170 households out of 250 owned a pet. Find the ${\displaystyle 95\mathrm{\%}}$ confidence interval for the proportion of households in the city who own a pet.
From a random sample, 136 out of 400 people experience discomfort after receiving a vaccine. Construct a ${\displaystyle 95\mathrm{\%}}$ confidence interval for the population proportion who might experience discomfort.
For each question do I use: ${\displaystyle \hat{p}\pm \frac{1}{\sqrt{n}}}$ or ${\displaystyle \hat{p}\pm \sqrt{\frac{p(1-p)}{n}}}$?

A sample of 25 circuit boards yields 2 defective boards after testing. What would be confidence interval for the true proportion of circuit boards in the lot from which the sample was drawn? Use the Clopper-Pearson Interval and also a normal approximation interval. In yur own words which do feel is better to use and why?

Assuming you are summing 10 poisson variables, and the sum of their results is 60, how would you approximate a ${\displaystyle 95\mathrm{\%}}$ CI for theta?

Calculate confidence interval when only given alpha?
Is there a formula to properly calculate the confidence interval when you're only given alpha and sample data?

Calculating the Confidence Interval
It is found that in a random sample of 100 Science students, there are 48 studying statistics . To test whether the true proportion of students in statistics is 50% or not, suitable null and alternative hypotheses are:
(a) ${\displaystyle H0:p=0.48;H1:p̸=0.48}$
(b) ${\displaystyle H0:p=0.5;H1:p>0.5}$
(c) ${\displaystyle H0:p=0.5;H1:p<0.5}$
(d) ${\displaystyle H0:p=0.5;H1:p=0.48}$
(e) ${\displaystyle H0:p=0.5;H1:p̸=0.5}$
Find a 95% Confidence Interval for p in Q1.
I have determined the answer to Q
1), which I believe is e).
For question 2), I have been told the CI= estimator +/- (table value)(Standard error). The estimator is 0.48, and I have found the SE to be 0.05. However, I am unsure about how to calculate the "table value".

Can a prediction interval be interpreted as a probability?
Suppose I find a ${\displaystyle 90\mathrm{\%}}$ prediction interval for some data distribution. This implies that if I sample large enough data from this distribution, then 90% of such data will lie inside the prediction interval. Is it same as saying that any randomly sampled data point from the distribution will lie inside the prediction interval with 0.9 probability?

Calculate ${\displaystyle P[A,B,C]}$ from ${\displaystyle P[A,B]}$ and ${\displaystyle P[B,C]}$

Calculating confident intervals for the expected daily returns
${\displaystyle SV=\frac{1}{n-1}\cdot \sum _i{({\left\{DR\right\}}_i-SM)}^2}$

An approximate ${\displaystyle 95\mathrm{\%}}$ confidence interval for the true unemployment rate p is
${\displaystyle (\hat{p}-z_{2.5}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}},\hat{p}-z_{97.5}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}})}$
${\displaystyle =(0.079-1.96\sqrt{\frac{0.079(1-0.079)}{4148}},0.079+1.96\sqrt{\frac{0.079(1-0.079)}{4148}})}$
${\displaystyle =(0.071,0.087)}$
What I'm mainly confused about is: how did they jump immediately to their first line?

A t - distribution's convergence to a Normal distribution
I'm currently messing around with confidence intervals and I can't really understand how a t distribution converges to a normal distribution for large n.
For example, suppose we want to construct a 95% confidence interval when we have a sample mean ${\displaystyle \bar{X}=74.8}$ and sample variance ${\displaystyle S=1.23\text{with}n=143}$.
I would construct the confidence interval using,
${\displaystyle (\bar{X}-z_{1-\frac{\alpha }{2}}\frac{S}{\sqrt{n}},\bar{X}+z_{1-\frac{\alpha }{2}}\frac{S}{\sqrt{n}})}$
since ${\displaystyle n>30}$. If ${\displaystyle n<30}$ I would have used a t-distribution.
My question is why does the t - distribution approach a normal distribution for relatively large n?

What is the rule of 1.96 for estimating confidence intervals?

What sample size is necessary for 95% CI?
A state legislator wishes to survey residents of her district to see what proportion of the electorate is aware of her position on using state funds to pay for abortions.
a. What sample size is necessary if the 95% CI for p is to have a width of at most 0.10 irrespective of p?
b. If the legislator has strong reason to believe that at least ${\displaystyle \frac{2}{3}}$ of the electorate know of her position, how large a sample size would you recommend?

What is the radius of a "Gaussian" sphere such that approx. all the population lie within?
Let ${\displaystyle \mathbf{X}\in B{\mathbf{R}}^n}$ be a random vector distributed normally, i.e. ${\displaystyle \mathbf{X}\sim \mathcal{N}(\mathbf{0},{\sigma }^2{I}_n)}$, where ${\displaystyle \sigma >0}$ is the standard deviation and ${\displaystyle I_n}$ is the identity matrix of order n.SNKIn the case of the univariate Gaussian distribution, i.e.,
${\displaystyle X\sim N(0,{\sigma }^2)}$, we say that approximately 99.7% of the population lie within three standard deviations.
What would be an appropriate constant in the case of the multivariate "isotropic" Gaussian distribution, i.e., ${\displaystyle \mathbf{X}\sim \mathcal{N}(\mathbf{0},{\sigma }^2{I}_n)}$? That is, what would be the radius of the n-sphere in terms of the standard deviation ${\displaystyle \sigma }$, such that approximately all the population lie within?
P.s.: Please feel free to edit the title (and/or the body of this question). I was not sure how to express what I wanted in the title.

Wilson interval with zero trials
I'm calculating Wilson score interval for some number of trials, so I have a question:
is it correct to assume this interval as [0 , 1] when you have zero trials?

A random sample of professional wrestlers was obtained, and the annual salary(in dollars) for each was recorded. The summary statistics were ${\displaystyle x=47,500}$ and ${\displaystyle n=18}$. Assume the distribution of annual salary is normal with ${\displaystyle \sigma =8,500}$
a. Find a ${\displaystyle 90\mathrm{\%}}$ confidence interval for the true mean annual salary for all professional wrestlers.
b. How large a sample is necessary in order for the bound on the error of estimation to be 3000?

95% confidence interval for ${\displaystyle {\theta }_2-{\theta }_1}$ from uniform ${\displaystyle ({\theta }_1,{\theta }_2)}$
where ${\displaystyle X_1,\dots ,X_n}$ are the random sample from ${\displaystyle U({\theta }_1,{\theta }_2)}$, the uniform distribution with two parameters ${\displaystyle {\theta }_1,{\theta }_2}$.
I have the maximum likelihood estimators ${\displaystyle {\hat{\theta }}_1=X_{\left(1\right)}}$, the minimum order statistic, and ${\displaystyle {\hat{\theta }}_2=X_{\left(n\right)}}$, the maximum order statistic. But I don't know how to compute the confidence interval of ${\displaystyle {\theta }_2-{\theta }_1}$.