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You collect data from some population with mean $\mathrm{\mu <!--\; \mu \; -->}$ and standard deviation $\mathrm{\sigma <!--\; \sigma \; -->}$ to test the following hypotheses: $H_0:\mathrm{\mu <!--\; \mu \; -->}=14vs.H_A:\mathrm{\mu <!--\; \mu \; -->}\ne <!--\; \ne \; -->14$. You obtain a p-value of 0.053. Which of the following is true?
a. A 95% confidence interval for $\mathrm{\mu <!--\; \mu \; -->}$ will include the value 14.
b. A 95% confidence interval for $\mathrm{\mu <!--\; \mu \; -->}$ will include the value 1.
c. A 90% confidence interval for $\mathrm{\mu <!--\; \mu \; -->}$ will include the value 1.
d. A 90% confidence interval for $\mathrm{\mu <!--\; \mu \; -->}$ will include the value 14.

Which of the following statements about probabilities and components is not true?
Choose the correc answer below:
a) $P(E)-<!--\; -\; -->P(E^{\prime })=1$
b) $P(E)=1-<!--\; -\; -->P(E^{\prime })$
c) $P(E)+P(E^{\prime })=1$
d) $P(E^{\prime })=1-<!--\; -\; -->P(E)$

Expanding $(a+b{)}^{\frac{1}{2}}$
I was wondering if it's possible to expand $(a+b{)}^{\frac{1}{2}}$.For example, $(a+b{)}^2=a^2+2ab+b^2$. But what is the expansion of $(a+b{)}^{\frac{1}{2}}$? I've learned about binomial theorem but I can't figure it out.

What is the standard form of a line that goes through (5,−4) and is perpendicular to ${\displaystyle y=\frac{5}{4}x-5}$?

Determine the number of different ways that the letters of "success" can be arranged

Which option best completes the diagram?
A. More prosperous citizens have money to spend.
B. The country’s government discovers a new natural resource.
C. More educated citizens are more likely to create a democratic government.
D. The country creates alliances with others in the region

Should the independent (or dependent) variables in a linear regression model be normal or just the residual?

If randomly picked from m symbols, each symbol being equally likely, how long on average would it take to pick m/2 different symbols?
Lets take the specific example of the English alphabet. In this case, I have 26 letters (symbols). Lets say these letters are plastic toys in a box. At random, I take one toy letter from the box, and write it down on a piece of paper. I then put that toy letter back into the box, and repeat the process. How many times, on average, would I have to repeat this process for my piece of paper to contain 13 of the 26 letters of the alphabet?
Is there a way to generalize the answer to m symbols instead of 26? What happens when m is odd?

i'm seeking out thoughts for a 15-hour mathematical enrichment course in a chinese language high faculty. What (pretty) simple concern would you advocate as a subject for any such course?
historical past/issues:
My students are generally pretty good at math, but many of them have no longer been uncovered to rigorous or summary mathematical reasoning. an amazing topic would be one that could not be impossibly hard for students who have by no means written or study proofs in English.
i have taught this magnificence three times earlier than. (a part of the purpose that i'm posting that is that i have used up all my thoughts!) the primary semester I taught an introductory range theory elegance (which meandered its way toward a proof of quadratic reciprocity, though I think this become in the end too advanced/abstract for some of the students). the second one semester I taught fundamental graph idea and packages (with a focal point on planarity and coloring). The 1/3 semester I taught a class at the Rubik's dice.
the students' math backgrounds are pretty numerous: a number of them take part in contest math competitions, and so are familiar with IMO-fashion techniques, however many aren't. a number of them may additionally realize some calculus, however I cannot assume it. all of them are superb at what in the united states is on occasion termed "pre-calculus": trigonometry, conic sections, systems of linear equations (though, shockingly, no matrices), and the like. They realize what a binomial coefficient is.
So, any ideas? preferably, i'd like to find some thing a bit "sexy" (like the Rubik's cube) -- tries to encourage wide variety theory through cryptography seemed to fall on deaf ears, however being capable of "see" institution idea on the cube became pretty popular.
(Responses specifically welcome from folks who grew up in the percent -- any mathematical subjects you desire were protected within the excessive college curriculum?)

Relationship among Exponential, Gamma, and Normal distribution
I am studying stochastic processes where I stumbled upon the theorem that says the sum of exponential distributions is gamma distribution. However, from the central limit theorem, we know that sum of a sufficiently large number of IID random variables converges to Normal distribution. So is the following relationship always true?

What is the relationship between the correlation of two variables and their covariance?

I'm a graduate student studying quantum mechanics/quantum information and would like to consolidate my understanding of linear algebra. What are some good math books for that purpose?

Let X be a separable Banach space. The p-Wasserstein space on X is defined as
$W_p(X)=\{\mathrm{\mu <!--\; \mu \; -->}\in <!--\; \in \; -->P(X):{\int <!--\; \int \; -->}_X||x|{|}^{p}d\mathrm{\mu <!--\; \mu \; -->}(x)<\mathrm{\infty <!--\; \infty \; -->}\},p\ge <!--\; \ge \; -->1$

Bayes' Theorem in Conditional Probability
We are testing for a disease D that we think is present, D+, with probability 0.4, and absent, D-, with probability 0.6. We believe that a test has sensitivity $P\{T+|D+\}=0.75$ and specificity $P\{T-<!--\; -\; -->|D-<!--\; -\; -->\}=0.8$.
Q1: What is our probability that the disease is present if we perform the test and it is positive, T+, and our probability the disease is absent if that test is negative, T-?
My ans:
We can apply Bayes' formula to calculate $P\{D+|T+\}$ and $P\{D-<!--\; -\; -->|T-<!--\; -\; -->\}$.
$P\{D+|T+\}=5/7$
$P\{D-<!--\; -\; -->|T-<!--\; -\; -->\}=24/29$
Q2: Suppose we perform three tests, conditionally independent given D. Given each possible number of positive test results, 0, 1, 2, or 3, what is our probability that the disease is present?
My ans:
Let k denote the number of positive test results.
We know that P{ [exactly] k successes in n trials | p } $=(nCk)x(p^k)\times <!--\; \times \; -->((1-<!--\; -\; -->p{)}^{n-k})$, hence we can calculate $P\{k=0|D+\}andP\{k=0|D-<!--\; -\; -->\}$.
We can then apply Bayes' formula to calculate $P\{D+|k=0\}$.
$P\{D+|k=0\}=0.692$
Using the same approach for $k=1:3$, we derive:
$P\{D+|k=1\}=0.5$
$P\{D+|k=2\}=0.308$
$P\{D+|k=3\}=0.165$
Would really appreciate it if someone can verify whether my reasoning and answers are correct.

A survey was carried out to find out if the occurrence of different kinds of natural disasters varies from on part of the town to the other. The town was divided into two as center town and town outskirts, and the natural disasters were divided into tornadoes, lighting, and fires.The survey showed these results:
Tornadoe Lightning Fires
Town Center 10 220 120
Town Outskirts 20 150 200
Do the above findings provide enough proof at 95 confidence level to indicate that the happening of natural disasters depends on the part of town?

What kind of technique is to be adopted if I have to find an equation or model for say, $D$ depends on $C$, $C$ changes for a set of $B$, which changes for different $A$.

Can you determine the correlation coefficient from the coefficient of determination?

What is the z-score of sample X, if

What is the relationship between effect size and sample size ?
- when the effect size is small large sample are needed
- when the effect size is large,large samples are needed
- regardless of effect size,large sample are generally necessary
- effect size and sample size are dependent on level power