Improve Your Understanding of College Level Statistics Problems

What is a pooled variance?

Degress of freedom are used to correct for the fact that population standard deviation will always be less precise than the sample standard deviation. True or False?

Classify the following random variables as continuous or discrete: the mark out of 50 for a geography test.

A poll from a previous year showed that 10% of smartphone owners relied on their data plan as their primary form of internet access. Researchers were curious if that had changed, so they tested $H_0:p=10\mathrm{\%<!--\; \%\; -->}$ versus $H_a:p\ne <!--\; \ne \; -->10\mathrm{\%<!--\; \%\; -->}$ where p is the proportion of smartphone owners who rely on their data plan as their primary form of internet access. They surveyed a random sample of 500 smartphone owners and found that 13% of them relied on their data plan.
The test statistic for these results was $z\approx <!--\; \approx \; -->2.236$, and the corresponding P-value was approximately 0.025.
Assuming the conditions for inference were met, which of these is an appropriate conclusion?
a) At the $\mathrm{\alpha <!--\; \alpha \; -->}$=0.01 significance level, they should conclude that the proportion has changed from 10%.
b) At the $\mathrm{\alpha <!--\; \alpha \; -->}$=0.01 significance level, they should conclude that the proportion is still 10%.
c) At the $\mathrm{\alpha <!--\; \alpha \; -->}$=0.05 significance level, they should conclude that the proportion has changed from 10%.
d) At the $\mathrm{\alpha <!--\; \alpha \; -->}$=0.05 significance level, they should conclude that the proportion is still 10%.
The correct answer is c but why could it not have been b? Why is it c?

(1) Prove that O(n) is homeomorphic to $\mathrm{S}\mathrm{O}(n)\times <!--\; \times \; -->{\mathbb{Z}}_2$. (2) Are these two isomorphic as topological groups?
(O(n) : Orthogonal group, SO(n) : Special orthogonal group)
My attempt for (1) :Let's pick any element C from $\mathrm{O}(n)-<!--\; -\; -->\mathrm{S}\mathrm{O}(n)$, and let
$$\begin{array}{rl}& f:\mathrm{S}\mathrm{O}(n)\to <!--\; \to \; -->\mathrm{S}\mathrm{O}(n);f(A)=A\\ & g:\mathrm{O}(n)-<!--\; -\; -->\mathrm{S}\mathrm{O}(n)\to <!--\; \to \; -->\mathrm{S}\mathrm{O}(n);g(A)=CA\end{array}$$
Then f and g are homeomorphisms. Since SO(n) is compact, $g^{-<!--\; -\; -->1}(\mathrm{S}\mathrm{O}(n))=\mathrm{O}(n)-<!--\; -\; -->\mathrm{S}\mathrm{O}(n)$ is also compact. So, they are closed in the subspace topology on O(n). Thus, by the glueing lemma, the function
$$\mathrm{\phi <!--\; \phi \; -->}:\mathrm{O}(n)\to <!--\; \to \; -->\mathrm{S}\mathrm{O}(n)\times <!--\; \times \; -->{\mathbb{Z}}_2;\mathrm{\phi <!--\; \phi \; -->}(A)=\{\begin{array}{ll}(f(A),0)& (\text{when }A\in <!--\; \in \; -->\mathrm{S}\mathrm{O}(n))\\ (g(A),1)& (\text{otherwise})\end{array}$$
becomes continuous. We can easily confirm that $\mathrm{\phi <!--\; \phi \; -->}$ is bijective and ${\mathrm{\phi <!--\; \phi \; -->}}^{-<!--\; -\; -->1}$ is continuous.
Questions:
(1) I'm not sure I'm going in the right direction (in proving (1)). I used glueing lemma to justifying continuousness of the function $\mathrm{\phi <!--\; \phi \; -->}$, but I doubt whether I'm using that lemma in the right place.
(2) I tried to prove that two groups are NOT isomorphic, but I can't find any key for that. I tried to compare orders of elements in two groups, but I couldn't find some good elements to compare... Is there any simple way to check whether two groups are isomorphic?

True and False?
Let $A$ and $B$ be nonempty sets and $f:A\to <!--\; \to \; -->B$ be a function. Then if $f(X\cap <!--\; \cap \; -->Y)=f(X)\cap <!--\; \cap \; -->f(Y)$ for all nonempty subsets $X$ and $Y$ of $A$, then $f$ must be $1-<!--\; -\; -->1$.

Which of the following is a probabilistic sampling approach that does not require the use of a sampling frame?
A) Stratified
B) Frame-free
C) Convenience
D) Skipped step
E) Judgment
F) Intercept
G) Cluster
H) Exploratory

What is the exact value of $\cos <!--\; \; -->135$?

Consider the language $L=\{+,\cdot <!--\; \cdot \; -->,0,1\}$ of rings. It is easy to show using compactness that if $\mathrm{\sigma <!--\; \sigma \; -->}$ is a sentence that holds in all fields of characteristic $0$, there is some $N\in <!--\; \in \; -->\mathbb{N}$ such that $\mathrm{\sigma <!--\; \sigma \; -->}$ holds for all fields of characteristic $p\ge <!--\; \ge \; -->N$. A sort of converse would be, if $\mathrm{\sigma <!--\; \sigma \; -->}$ is a sentence that holds in all fields of positive characteristic, $\mathrm{\sigma <!--\; \sigma \; -->}$ is true in all fields of characteristic $0$.

The Ability Test has been proven to predict the writing skills of Senior High School students. What type of test validity is shown in the example?
A. Construct Validity
B. Criterion Validity
C. Content Validity
D. Face Validity

How many different phone numbers are possible in the area code 928? Note: Phone numbers cannot start with 0 or 1.

Are all continuous random variables necessarily infinite? Explain.

Test of confidence intervals?
In one of my assignments I have to "test" if the confidence intervals for a set of parameters in a mixed effect model is accurate. I'm asked to simulate from fittet parameters and there after refit them using the same model many times, and lastly take 2.5% and 97.5% quantiles of them and compare with the original CIs. My question is, how does this procedure in anyway measure how accurate my original confidence intervals are?