Improve Your Understanding of College Level Statistics Problems

Let ${\displaystyle x_1,\cdots ,x_n}$ be a sample from a normal population having unknown mean and variance. Let ${\displaystyle \bar{x}}$ be the average of the first n of them.
What is the distribution of ${\displaystyle x_{n+1}-\bar{x}}$?
If ${\displaystyle \bar{x}=4}$, give an interval that, with ${\displaystyle 90\mathrm{\%}}$ confidence, will contain the value of ${\displaystyle x_{n+1}}$.
The distribution of ${\displaystyle \bar{x}}$ would be normal with mean equal to the population mean and standard deviation equal to the population standard deviation divided by ${\displaystyle \sqrt{n}}$. But how do we find the distribution of ${\displaystyle x_{n+1}}$?

Let ${\displaystyle p_1}$= population proportion for population 1, ${\displaystyle p_2}$= population proportion for population 2, ... and ${\displaystyle p_k}$= population proportion for population k. Consider the following null hypothesis: ${\displaystyle H_0}$: ${\displaystyle p_1}$=${\displaystyle p_2}$=...=${\displaystyle p_k}$. Which of the following statements is correct?
a. The alternative hypothesis to the null hypothesis stated above must be: ${\displaystyle H_{\alpha }}$: Not all population proportions are equal.
b. If the sample data and the chi-square test computations indicate ${\displaystyle H_0}$ cannot be rejected, we cannot detect a difference among the k population proportions.
c. If the sample data and the chi-square test computations indicate ${\displaystyle H_0}$ can be rejected, we have the statistical evidence to conclude that one or more population proportions differ from the other population proportions.
d. All of the above.

Let ${\displaystyle f:\mathrm{\Omega }\to \mathbb{R}}$ be a Borel-measurable function, X a random variable with values in ${\displaystyle \mathrm{\Omega }}$ and ${\displaystyle X_i\in \mathbb{R},i\in \mathbb{N}}$ realizations of X.
Literature
In their book on Monte Carlo Methods (Simulation and the Monte Carlo method, Second Edition) Rubinstein and Kroese give in Section 4.2.1 an approximate confidence interval for the mean E[f(X)]:
${\displaystyle (\hat{\mu }\pm z_{1-\frac{\alpha }{2}}\frac{\widehat{{\sigma }_f}}{\sqrt{N}}),}$
where we write N for the number of Monte Carlo samples, ${\displaystyle z_{1-\frac{\alpha }{2}}}$ for the ${\displaystyle 1-\frac{\alpha }{2}}$ quantile of the standard normal distribution, ${\displaystyle \hat{\mu }=\frac{1}{N}\sum _{i=1}^{N}f\left(X_i\right)}$ for the unbiased estimator of the mean and ${\displaystyle {\hat{\sigma }}_f=\sqrt{\frac{1}{N-1}\sum _{i=1}^N{(f\left(X_i\right)-\hat{\mu })}^2}}$ for the empirical standard deviation derived from the unbiased estimator of the variance.
Extension to variance
In my opinion we can derive in a similar way a confidence interval for the variance var[f(X)] if we define ${\displaystyle g\left(X\right)={(f\left(X\right)-\hat{\mu })}^2}$ and repeat the construction. This leads to the interval
${\displaystyle ({\hat{\sigma }}_f^2\pm z_{1-\frac{\alpha }{2}}\frac{{\hat{\sigma }}_g}{\sqrt{N}}),}$
where ${\displaystyle {\hat{\sigma }}_g=\frac{1}{N-1}\sum _{i=1}^N{(g\left(X\right)-\frac{1}{N}\sum _{i=1}^{N}g\left(X_i\right))}^2}$
How can I transfer this to give a confidence interval for the standard deviation? Surely I can't just take the square root of the boundaries. Any literature, help and thoughts are welcome. Thanks.

Given a set {13, x, -1, 11}, for what x would the mean of the set be 3?

Given a set {-3,2, -1,x}, for what x would the mean of the set be -5?

Let X be a Poisson (${\displaystyle \theta }$) random variable. Unbiased estimator
Show that ${\displaystyle {(-1)}^X}$ is an unbiased estimator for ${\displaystyle e^{-2\theta }}$ This is a fairly bad estimator for a number of reasons - so this exercise helps show why unbiasedness is not the most important criterion for an estimator.

Let ${\displaystyle X_1,\cdots ,X_n}$ be a random sample from a normal distribution with known mean ${\displaystyle \mu }$ and unknown variance ${\displaystyle {\sigma }^2}$. Three possible confidence intervals for ${\displaystyle {\sigma }^2}$ are
a) ${\displaystyle (\sum _{i=1}^n\frac{{(X_i-\bar{X})}^2}{a_1},\sum _{i=1}^n\frac{{(X_i-\bar{X})}^2}{a_2})}$
b) ${\displaystyle (\sum _{i=1}^n\frac{{(X_i-\mu )}^2}{b_1},\sum _{i=1}^n\frac{{(X_i-\mu )}^2}{b_2})}$
c) ${\displaystyle (\frac{n{(\bar{X}-\mu )}^2}{c_1},\frac{n{(\bar{X}-\mu )}^2}{c_2})}$
where ${\displaystyle a_1,a_2,b_1,b_2,c_1,c_2}$ are constants.
Find values of these six constants which give confidence level 0.90 for each of the three intervals when ${\displaystyle n=10}$ and compare the expected widths of the tree intervels in this case
With ${\displaystyle {\sigma }^2=1}$, what value of n is required to achieve a ${\displaystyle 90\mathrm{\%}}$ confidence interval of expected width less than 1 in cases (b) and (c) above?

Given ${\displaystyle X_1,\dots ,X_{100}}$ and ${\displaystyle Y_1,\dots ,Y_{50}}$ which are independent random samples from identical distribution ${\displaystyle N(\mu ,1)}$. Each of two statisticians is trying to build confidence interval for ${\displaystyle \mu }$ on a confidence level of 0.8 ${\displaystyle 1-\alpha =0.8}$. You have to find probability that those intervals are disjoint. I know i have to find ${\displaystyle P(|\bar{X}-\bar{Y}|>\frac{1\cdot 1.28}{\sqrt{50}}+\frac{1\cdot 1.28}{\sqrt{100}})}$ which is ${\displaystyle 1-P(-0.309\le \bar{X}-\bar{Y}\le 0.309)}$
Question is what should i do next how to calculate this probability?

Given a set {12,x, 17}, for what x would the mean be 14?

Given a set {-15, 1, 200, x, 1}, for what x would the mean of the set be -15?

Given a set {-3, x, 3,4,6}, for what x would the mean of the set be -11?

Given a set {-3, x,3,4,6,5, -3, 2}, for what x would the mean of the set be -1?

Given a set {-1, x, -4, -21}, for what x would the mean of the set be 0?

Let ${\displaystyle X_1,\cdots ,X_n}$ random sample of ${\displaystyle U[\theta -\frac{1}{2};\theta +\frac{1}{2}]}$.Consider ${\displaystyle [X_{\left(1\right)};X_{\left(n\right)}]}$ a confidence interval for ${\displaystyle \theta }$. Find their confidence level and show that result is valid for any distribution symmetric around ${\displaystyle \theta }$

Given a set {11, x, -44, 1}, for what x would the mean be 4?

Given a set {-1, 2, -12,52,x}, for what x would the mean of the set be -3?