Improve Your Understanding of College Level Statistics Problems

For a continuous random variable x, the population mean and standard deviation are 120 and 15 respectively. How do you find the mean and standard deviation of the sampling distribution of the sample mean for a sample of 25 elements?

A coach has made a statement that his players have bigger lung capacity than the average of the population of the same age which is 3.4. (Normal distribution)
The measurements yield the following data: 3.4, 3.6, 3.8, 3.3, 3.4, 3.5, 3.7, 3.6, 3.7, 3.4 and 3.6.
$n=11$
$\stackrel{¯<!--\; ¯\; -->}{X}=3.545$
$S=0.157$
Find the required sample size, which lung capacity should be measured, so coach can state his statement with 99% confidence. (assume ${\mathrm{\sigma <!--\; \sigma \; -->}}^2=0.09$)
I don't even know how should I start. My initial thought was to use the U statistics $U=\frac{\stackrel{¯<!--\; ¯\; -->}{X}-<!--\; -\; -->\mathrm{\mu <!--\; \mu \; -->}}{\mathrm{\sigma <!--\; \sigma \; -->}}\sqrt{n}$~$N(0,1)$. But I don't know the U.

A man tests for HIV. What is the predictive probability that his second test is negative?
In a population, it is estimated HIV prevalence to be $\mathrm{\lambda <!--\; \lambda \; -->}$. For a new test for HIV:
- $\mathrm{\theta <!--\; \theta \; -->}$ is the probability of an HIV positive person to test positive
- $\mathrm{\eta <!--\; \eta \; -->}$ is the probability an HIV negative person tests positive in this test.
A person takes the test to check whether they have HIV, he tests positive.
What is the predictive probability he tests negative on the second test?
Assumption: Repeat tests on the same person are conditionally independent.
From my notes predictive probability is given as:
$P(\stackrel{~<!--\; ~\; -->}{Y}=\stackrel{~<!--\; ~\; -->}{y}|Y=y)=\int <!--\; \int \; -->p(\stackrel{~<!--\; ~\; -->}{y}|\mathrm{\tau <!--\; \tau \; -->})p(\mathrm{\theta <!--\; \theta \; -->}|\mathrm{\tau <!--\; \tau \; -->})$ here $\stackrel{~<!--\; ~\; -->}{Y}$ is the unknown observable, y is the observed data and η the unknown.
I am interested in the probability of the second test is negative, given that the first test is positive,without knowing if the man really has HIV or not.
To facilitate this I define:
$y_1$ as the event of the first test being positive and
$\stackrel{~<!--\; ~\; -->}{y_2}$ as the second test being negative
Would this adaption to the formula given above be the correct/best approach to this problem ?
$p(\stackrel{~<!--\; ~\; -->}{y_2},y_1|\mathrm{\tau <!--\; \tau \; -->})=p(\stackrel{~<!--\; ~\; -->}{y_2}|\mathrm{\tau <!--\; \tau \; -->})p(y_1|\mathrm{\tau <!--\; \tau \; -->})p(\mathrm{\tau <!--\; \tau \; -->})$ and this is really $\propto <!--\; \propto \; -->p(\stackrel{~<!--\; ~\; -->}{y_2}|\mathrm{\tau <!--\; \tau \; -->})p(\mathrm{\tau <!--\; \tau \; -->}|y_1)$
I've gotten for the $p(\mathrm{\tau <!--\; \tau \; -->}|y_1)$ from Bayes' theorem:
$$\begin{gathered} p(\mathrm{\tau <!--\; \tau \; -->}|y_1)=\frac{p(\mathrm{\tau <!--\; \tau \; -->})p(y_1|\mathrm{\tau <!--\; \tau \; -->})}{p(y_1)} \\ =\frac{\mathrm{\lambda <!--\; \lambda \; -->}\mathrm{\theta <!--\; \theta \; -->}}{\mathrm{\lambda <!--\; \lambda \; -->}\mathrm{\theta <!--\; \theta \; -->}+\mathrm{\eta <!--\; \eta \; -->}(1-<!--\; -\; -->\mathrm{\lambda <!--\; \lambda \; -->})} \end{gathered}$$
How could I then find $p(\stackrel{~<!--\; ~\; -->}{y_2}|\mathrm{\tau <!--\; \tau \; -->})$? Is this the correct approach?

There are $115,226,802$ households in the US. $58$% of households have at least $1$ musician. $43$% have $2$ or more musicians. How many musicians are there?

I am fond of astronomy and environment. I want to try to make a "light pollution map" but I haven't my satellites... so I use as approximation of light pollution the cities' population. Let say we have for each city C citizens, each one spreads an average of X Watt of electricity for lightning ( I have these data ). Skip the units ( I need just a rough dimensionless "light power" ): city $\text{city light power}=C\times <!--\; \times \; -->X$
I have a map, with many cities. I know light power is inversely proportional to the square of distance. I don't know about sky, air diffraction, cloud reflections.
Start from the simplest model. A flat terrain map. N light sources, every one at position X(n), Y(n) has a specific $\text{"total light power"}=C(n)\times <!--\; \times \; -->X(n)$
At a specific point of coordinates (x,y) which is the light power, sum of all the cities light ?
I tried to calculate and plot, but it seems weird ( too far from some real satellite night shot ) and too slow to calculate.

Can you please give the more appropriate answer of this question with example?
A population is divided into several clusters and it has been found that all the items within a cluster are alike. Which of the sampling procedure would you adopt? Give appropriate reasons for selecting such method.

COVID19 data statistical adjustment for SIR model and estimation
All of us are coping with the current COVID19 crisis. I hope that all of you stay safe and that this situation will end as soon as possible.
For this sad situation and for my unstoppable curiosity, I've started to read something about the SIR model. The variables of such model are s (the fraction of people susceptible to infection), y (the fraction of infected people) and r (the fraction of recovered people + the sad statistics of deaths). The model reads as:
$\{\begin{array}{l}\stackrel{\dot{}<!--\; \dot{}\; -->}{s}=-<!--\; -\; -->\mathrm{\beta <!--\; \beta \; -->}sy\\ \stackrel{\dot{}<!--\; \dot{}\; -->}{y}=\mathrm{\beta <!--\; \beta \; -->}sy-<!--\; -\; -->\mathrm{\gamma <!--\; \gamma \; -->}y\\ \stackrel{\dot{}<!--\; \dot{}\; -->}{r}=\mathrm{\gamma <!--\; \gamma \; -->}y\end{array},$
where $\mathrm{\beta <!--\; \beta \; -->}$ and $\mathrm{\gamma <!--\; \gamma \; -->}$ are positive parameters. One strong hypothesis of this model is that the population size is constant over time (deaths are assumed to be recovered, births are neglected since, hopefully, they will be the part of the population which for sure will be protected from the disease). The initial conditions are set such that $s(0)+y(0)+r(0)=1$ and $s(0)\ge <!--\; \ge \; -->0$, $y(0)\ge <!--\; \ge \; -->0$ and $r(0)\ge <!--\; \ge \; -->0$. Under this assumption, it can be proven that $s(t)+y(t)+r(t)=1\mathrm{\forall <!--\; \forall \; -->}t>0$.
The news often talk about the coefficient:
$R_0=\frac{\mathrm{\beta <!--\; \beta \; -->}}{\mathrm{\gamma <!--\; \gamma \; -->}},$
which rules the behavior of the system (for $R_0<1$ the disease will be wiped out, for $R_0>1$ it will spread out).
The same news also talk about the estimation of such parameter. Well, given the time series of s, y and r, it is rather easy to estimate the parameters $\mathrm{\beta <!--\; \beta \; -->}$ and $\mathrm{\gamma <!--\; \gamma \; -->}$, and hence $R_0$. My main concern is about the time series. For each country we know the daily count of infected people (let's say Y(t)), of recovered (or dead) people (let's say R(t)).
Anyway, there are several infected people which are not recorded (let's say Y′(t)), and many of them get recovered without knowing that they have been infected (let's say R′(t))! Moreover, day after day, the number of tests on people is increasing.
If we indicate with N the (constant) size of population, we get that:
$y(t)=\frac{Y(t)+Y^{\prime }(t)}{N},r(t)=\frac{R(t)+R^{\prime }(t)}{N}\text{and}s(t)=1-<!--\; -\; -->y(t)-<!--\; -\; -->r(t).$
Here is the question(s). How can we perform the estimation of $\mathrm{\beta <!--\; \beta \; -->}$ and $\mathrm{\gamma <!--\; \gamma \; -->}$ if we don't know the unobserved variables Y′(t) and R′(t)? How do the experts of the field estimate $\mathrm{\beta <!--\; \beta \; -->}$ and $\mathrm{\gamma <!--\; \gamma \; -->}$ even though the available data are not complete? Do they use some data adjustment?

In question I do not get any sigma, I know I can do the t student test based on degrees on freedom if I do not have the population st.dev, but in my case I do not even have the sample st.dev, can it be solved or the teacher made a mistake in the text ?
"The maturation time of maize plants follows the normal distribution. We follow 30 plants and it is seen that they reach maturity in 4 months. Construct a 95% confidence interval for the maturation time of the maize plant population."
In order to find the margin of error i still need S (the sample st.dev) no ?
Otherwise how can i proceed ? Chi squared ?

Standard Error is of Population Total
We have the following data and we are required to obtain the standard error of unbiased estimate of the population total:
$N=160,n=64,{\mathrm{\sigma <!--\; \sigma \; -->}}^2=4$
My approach
We know that:
$SE(\stackrel{¯<!--\; ¯\; -->}{X})=\frac{\mathrm{\sigma <!--\; \sigma \; -->}}{\sqrt{n}}$. So, it can be written as:
$SE(\frac{Total}{n})=\frac{\mathrm{\sigma <!--\; \sigma \; -->}}{\sqrt{n}}$
Which in turn will be equal to:
$SE(Total)=(\mathrm{\sigma <!--\; \sigma \; -->})(\sqrt{n})$.
In the above formula, after plugging in values, I am getting $SE=(2)(8)=16$
But this is not correct. The correct answer is 40. Am I doing it incorrectly? I am not sure.
Any help?

For non-negative data the sample mean is not smaller than its standard error.
(1) Let $X_1,X_2,\dots <!--\; \dots \; -->,X_n$ be a random sample from a population with non-negative values. Then show that $\stackrel{¯<!--\; ¯\; -->}{X}\ge <!--\; \ge \; -->S/\sqrt{n},$ where $S^2=[\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}(X_i-<!--\; -\; -->\stackrel{¯<!--\; ¯\; -->}{X}{)}^2]/(n-<!--\; -\; -->1).$
I have not seen this inequality stated before. It is not difficult to prove directly, but may be implied by some more general result I am overlooking.
(2) Also use this inequality to find a counterexample, showing that the sample mean and variance are not independent for exponential (or beta) data; perhaps use $n=4$ for simplicity. [of course, $\stackrel{¯<!--\; ¯\; -->}{X}$ and S are independent for normal data.]

For a sample of n = 16 scores, what is the value of the population standard deviation $(\mathrm{\sigma <!--\; \sigma \; -->})$ necessary to produce each of the following a standard error values?
${\mathrm{\sigma <!--\; \sigma \; -->}}_M$ = 8 points?

A political analyst would like to survey a sample of the registered voters in a county. The county has 47 election districts. How could the analyst use random numbers to obtain a cluster sample?

I am not a mathematician, so go easy on me. I'm a programmer.
I have a database that I got from the Internet (USDA National Nutrient Database for Standard Reference), detailing the amount of each nutrient in each of a few thousand foodstuffs. I wanted to write a program that would be able to create a maximally nutritious meal based on this data.
For each nutrient, I have a target and two penalties - one for going over and one for going under the target (since, for example, it's a lot worse to get too much saturated fat than not enough). The goal is to minimize the sum of the penalties.
The meal can select from all the thousands of foodstuffs, but can only contain five or six.
I wrote the program in Java, implemented a genetic algorithm, specified my requirements, and let it run. It produced recommendations that were pure poison, and didn't seem to improve with time.
Maybe I just don't get genetic algorithms? Let's see what I did...
1) Create a population of randomly generated meals.
2) Normalize each one so it has 2000 calories, by multiplying the amount of each foodstuff proportionally.
3) Select the best 10% of meals to be parents.
4) Create a new generation - a few random to avoid local minima, the rest created by combining the numbers and amounts from the parents.
5) GOTO 2.
What other algorithm can I try? Someone advised me to use simplex algorithm, but I can't seem to explain to it (the implementation in Apache Commons Math) what my fitness function is. But he claimed it would be a natural fit, and I have even heard of someone who used simplex for exactly this.

If you are given a population of students doing hypothesis tests for a certain condition at a certain significance level, is it possible to calculate:
(a) how many students will fail to reject the null hypothesis given that the null hypothesis is false
(b) how many students will reject the null hypothesis given that the null hypothesis is true.
I have tried searching online but so far all sites only show how to calculate the probability that at least one type I error will be made. Any assistance will be greatly appreciated.

The action for this problem takes place in an island of Knights and Knaves, where Knights always make true statements and Knaves always make false statements and everybody is either a Knight or a Knave. Two friends A and B lives in a house. The census taker (an outsider) knocks on the door and it is opened by A. The census taker says ''I need information about you and your friend. Which, if either of you, is a Knight and which, if either of you, is a Knave?" "We are both Knaves" says A angrily and slams the door. What, if any thing can the census taker conclude?
a.A is a Knight and B is a Knave.
b.A is a Knave and B is a Knight.
c.Both are Knaves.
d.Both are Knights.
e.No conclusion can be drawn.
Not able to grasp it please someone help me

The American Academy of Pediatrics wants to conduct a survey of recently graduated family practitioners to assess why they did not choose pediatrics for their specialization. Provide a definition of the population, suggest a sampling frame, and indicate the appropriate sampling unit.
Please do not provide definitions of the words, but instead describing each word that it pertains to conducting the survey.

Calculate the Standard Deviation for Multiple Combined Distributions
You have the mean and standard deviation data for a population of individual unit pumps' reporting error. Each engine has either 4 or 6 unit pumps and the total reporting error for the combined performance of all pumps on the engine is being evaluated (i.e four pumps on an engine all of which are over reporting by 10%, the engine reporting error is 10%). Considering that each pump installed on the engines will be selected from the distribution data provided for an individual pump, what is the overall mean and standard deviation for the entire engine in terms of reporting error for both 4 and 6 pump engines?

Why is the "wrong" interpretation of confidence intervals still seemingly correct?
According to online sources, if you are operating at 95% confidence, it means if you repeated a sampling process many times and then looked at the 95% confidence intervals over all the results, 95% of the time the brackets would contain the true population mean.
But then they say that it is NOT the same as saying "you can be 95% confident that the intervals you computed contain the population mean."
Isn't it, though? For instance I do my first experiment and get a 95% confidence interval. Then a second. Third, fourth, ..., 100th. 95 of those intervals should contain the population mean. Is this correct so far?
If so, then why isn't it the same as me saying, from the moment I did the very first test, "this particular interval has a 95% chance at being one of the intervals that contain the true population mean"?

Why is standard deviation calculated differently for finding Z scores and confidence intervals
Suppose that as a personnel director, you want to test the perception of fairness of two methods of performance evaluation. 63 of 78 employees rated Method 1 as fair. 49 of 82 rated Method 2 as fair. A 99% confidence interval for $p_1$−$p_2$ (where $p_1$ = 63/78 and $p_2$ = 49/82) is as follows:
$p_1-<!--\; -\; -->p_2\pm <!--\; \pm \; -->2.58\sqrt{\frac{p_1(1-<!--\; -\; -->p_1)}{n_1}+\frac{p_2(1-<!--\; -\; -->p_2)}{n_2}}$
$0.029⩽<!--\; ⩽\; -->p_1-<!--\; -\; -->p_2⩽<!--\; ⩽\; -->0.391$
At the 0.01 level of significance the Z score is
$Z=\sqrt{\frac{p(1-<!--\; -\; -->p)}{n_1}+\frac{p(1-<!--\; -\; -->p)}{n_2}}$
(where p = $(x_1+x_2)/(n_1+n_2)=0.70$ but sometimes a different formula $p=(p_1+p_2)/2$ is also used)
Z = 2.90
Both tests indicate that there is evidence of a difference.
But you could also find the Z score using the standard deviation formula in the first method to be 2.993. Why are the Z scores different? Where do the formulas for finding the standard deviation come from?

Kusam is developing a new psychometric test to assess jealousy, and finds that the the split-half correlation is r = 0.10. This means that…
Select one:
a. the assessment has poor internal consistency.
b. the assessment has poor test-retest reliability.
c. the assessment has poor inter-rater reliability.
d. the assessmen does not have criterion validity.