Improve Your Understanding of College Level Statistics Problems

Let f and g be functions. Then the domain of f composed with g is the intersection of the domain of f and the domain of g. True or False?

Given there's a number generator which creates irregular numbers to 9. Given after 50 number eras as it were 5 of the created numbers are underneath 3 (i.e. 0,1 or 2) and the importance level is α = 1%. How can I test in the event that for this importance level the number generator is producing all numbers with equal probability? It is obvious to me that in the event that the number generator would be without a doubt producing the numbers with uniform likelihood, one would anticipate 15 created numbers between and 2 - but how do I test the nullhypothesis in this case accurately?

Categorize the type of sampling used in each of the following situations (simple random, stratified, systematic, cluster, convenience). Explain in a sentence or two.
To judge the appeal of a proposed television sitcom, a random sample of 10 people from each of three different age categories was selected and those chosen were asked to rate a pilot show.

I have a question about t-statistics in first place I need to use t-statistics to estimate the population mean from a sample set, as you see I have a data set the mean and standard deviation are known for sample but not for the population.
How can I use t-statistics to determine the population mean I have X bar and S and n values for sample $n=45$how can I compute the degree of freedom and how can I determine the confidence level? thanks for your time huge regards

Discrete version of continuous SIR model
I'm working with a SIR infection model, which is
$$\begin{array}{rcl}\frac{dS}{dt}& =& -<!--\; -\; -->\mathrm{\beta <!--\; \beta \; -->}IS\\ \frac{dI}{dt}& =& \mathrm{\beta <!--\; \beta \; -->}IS-<!--\; -\; -->\mathrm{\gamma <!--\; \gamma \; -->}I\\ \frac{dR}{dt}& =& \mathrm{\gamma <!--\; \gamma \; -->}I\end{array}$$
in continuous time, where S, I, and R are the proportion of Susceptible, Infected, and Recovered, respectively.
However, since I am working with fixed-width discrete-time data, I think it would be more appropriate to modify the equations accordingly. I know this is incorrect (based on getting a negative values for $\stackrel{^<!--\; ^\; -->}{\mathrm{\gamma <!--\; \gamma \; -->}}$ and $\stackrel{^<!--\; ^\; -->}{\mathrm{\beta <!--\; \beta \; -->}}$, neither of which should be negative):
$$\begin{array}{rcl}{S}_{t+1}& =& -<!--\; -\; -->\mathrm{\beta <!--\; \beta \; -->}{I}_t{S}_t\\ I_{t+1}& =& \mathrm{\beta <!--\; \beta \; -->}{I}_t{S}_t-<!--\; -\; -->\mathrm{\gamma <!--\; \gamma \; -->}{I}_t\\ R_{t+1}& =& \mathrm{\gamma <!--\; \gamma \; -->}{I}_t\end{array}$$
Ultimately, I would like to estimate $\mathrm{\beta <!--\; \beta \; -->}$ and $\mathrm{\gamma <!--\; \gamma \; -->}$ by doing regression on
$$I_{t+1}=\mathrm{\beta <!--\; \beta \; -->}(I_t{S}_t)+\mathrm{\gamma <!--\; \gamma \; -->}(-<!--\; -\; -->I_t)+U_t\text{, where }{U}_t\text{~}N(0,{\mathrm{\sigma <!--\; \sigma \; -->}}^2)$$
or whatever the discrete version is.

How do you find the exact value of ${\tan }^{-<!--\; -\; -->1}<!--\; \; -->(\tan <!--\; \; -->(-<!--\; -\; -->\frac{3\mathrm{\pi <!--\; \pi \; -->}}{4}))$?

Let $X$ be a locally comapct and Hausdorff space. We say a positive Radon Measure on $X$ is faithful if
$$0\le <!--\; \le \; -->f,\int <!--\; \int \; -->fd\mathrm{\mu <!--\; \mu \; -->}=0\to <!--\; \to \; -->f(x)=0\mathrm{\forall <!--\; \forall \; -->}x\in <!--\; \in \; -->X$$
True or false: If there is a faithful positive Radon measure on $X$ then $X$ has a countable dense subset ?

How do you find the exact value of $\frac{\sin <!--\; \; -->(5\mathrm{\pi <!--\; \pi \; -->})}{3}$?

We have the Random Variable X, which is $\mathrm{\Gamma <!--\; \Gamma \; -->}(p,\mathrm{\lambda <!--\; \lambda \; -->})$ distributed with the density:
$$f_{p,\mathrm{\lambda <!--\; \lambda \; -->}}(x)=\frac{{\mathrm{\lambda <!--\; \lambda \; -->}}^p}{\mathrm{\Gamma <!--\; \Gamma \; -->}(p)}\cdot <!--\; \cdot \; -->x^{p-<!--\; -\; -->1}\cdot <!--\; \cdot \; -->e^{-<!--\; -\; -->\mathrm{\lambda <!--\; \lambda \; -->}x}$$
with p=10 and $H_0:\mathrm{\lambda <!--\; \lambda \; -->}=2$ or $H_1:\mathrm{\lambda <!--\; \lambda \; -->}=4$ and $\mathrm{\alpha <!--\; \alpha \; -->}=0.001$
I want to apply the Lemma of Neyman Pearson which states:
Be c>0 fixed and chosen in the way that $A(c)=\{x\in <!--\; \in \; -->B:\frac{f_0(x)}{f_1(x)}\ge <!--\; \ge \; -->c\}$ such that ${\mathbb{P}}_{H_0}(X\in <!--\; \in \; -->A(c))=\mathrm{\alpha <!--\; \alpha \; -->}$
Then the test with the region A(c) among all tests with significance level $\mathrm{\alpha <!--\; \alpha \; -->}$ is the most powerful.
I am now trying to calculate A(c), but got stuck. I have:
$${\int <!--\; \int \; -->}_{A(c)}{f}_0(x)dx={\int <!--\; \int \; -->}_{A(c)}\frac{{\mathrm{\lambda <!--\; \lambda \; -->}}^p}{\mathrm{\Gamma <!--\; \Gamma \; -->}(p)}\cdot <!--\; \cdot \; -->x^{p-<!--\; -\; -->1}\cdot <!--\; \cdot \; -->e^{-<!--\; -\; -->\mathrm{\lambda <!--\; \lambda \; -->}x}dx=\mathrm{\alpha <!--\; \alpha \; -->}.$$
But I don't know how to get A(c) from this integral...
$\frac{f_0(x)}{f_1(x)}=\frac{1}{1024}\cdot <!--\; \cdot \; -->e^{2x}$

The concept of a minimal sufficient statistic, which captures nothing more than the essential.
Is it true that any random variable $X$ as above has a sufficient statistic?

Do all pure samples of a given compound have the same percent composition?

A sample is selected from a population with a mean of $\mathrm{\mu <!--\; \mu \; -->}=40$. If the sample mean is $M=45$, which of the following combinations would make the sample mean a good, representative value for the population?​
Question options:
a. ​A small sample and a large population standard deviation
b. ​A large sample and a large population standard deviation
c. ​A small sample and a small population standard deviation
d. ​A large sample and a small population standard deviation

Confidence Intervals Calculation and Interpretation
Could someone verify if my reasoning is correct?
1. Is it true that if a population is normally distributed, then the sample variance for any sample size is also normally distributed, so that we may use an interval of the form [(sample variance) $\pm <!--\; \pm \; -->$ (critical value)(standard deviation of sample variance)] to estimate the population variance?I understand that the form statistic $\pm <!--\; \pm \; -->$ (critical value) ∗ (standard dev. of statistic) is applicable to mean and proportion, but is it applicable to any statistic? I believe there is a more complex formula for variance, as it follows a chi-squared distribution and is not always normally distributed, even when the population is.
2. A 90% confidence interval for the height, in meters, of adults in Switzerland is 1.78 $\pm <!--\; \pm \; -->$ 0.2. Are we 90% confident that the sample mean of the next sample of adults taken in Switzerland will be between 1.58 and 1.98 meters tall?
Would this also be false? I understand that a particular confidence interval of 90% calculated from an experiment does not mean that there is a 90% probability of a sample mean from a repeat of the experiment falling within this interval.

How can I compare two series that are same thing, but at different rates?
Consider the series $1-<!--\; -\; -->\frac{1}{2}+\frac{2}{3}-<!--\; -\; -->\frac{1}{3}+\frac{2}{4}-<!--\; -\; -->\frac{1}{4}+\frac{2}{5}-<!--\; -\; -->\frac{1}{5}$
I can see that if you group the terms in pairs of two, you get $(1-<!--\; -\; -->\frac{1}{2})+(\frac{2}{3}-<!--\; -\; -->\frac{1}{3})+(\frac{2}{4}-<!--\; -\; -->\frac{1}{4})+(\frac{2}{5}-<!--\; -\; -->\frac{1}{5})$...= $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}$...
The partial sums $S_{2n}$= the harmonic series from $$n=2$$ to n.

The concentration of alcoholin a person's bloodstream is measurable.
Suppose that the relative risk R of having an accident while driving a car can be modeled by an equation of the form $R=e^{kx}$ where x is the percent of concentration of alcohol in the bloodstream and k Is a constant Using this value of k, what is the relative risk if the concentration is 0.17 percent?

Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).
a. A pollster interviews all human resource personnel in five different high tech companies.
b. A high school educational researcher interviews 50 high school female teachers and 50 high school male teachers.
c. A medical researcher interviews every third cancer patient from a list of cancer patients at a local hospital.
d. A high school counselor uses a computer to generate 50 random numbers and then picks students whose names correspond to the numbers.

Dr. F. did a study and obtained significant results. The more varied their participants in such charac­teristics as age, ethnic group etc.,
1. the more limited the potential generalizations of his study
2. the better the internal validity of his study
3. the greater the external validity of his study
4. the less representative his sample

As sample size increases, what happens to the standard error of M?

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.