Understanding Inferential Statistics through Examples

Kyle has 314 stamps in his collection. He is placing his stamps in an album with pages that each hold 42 stamps. He estimates he will need about 8 pages. Is he right? Why or why not?

There are 20 teams in a league.
Each team has a squad of 30 players but each game they only play 11 players.
I am after a measure/ ranking of which teams play a more consistent lineup of 11 together (with the hypothesis that when players play together more, they are more cohesive).
Eg an older team that has played multiple seasons together with the same lineup.
I could sum the games played individually but not sure if there is a more “correct” way of doing this.

Multiple Linear Regression in Matrix Form
I am currently studying for my exams and came across the following question:
State the multiple linear regression equation in matrix form. Write down the order of each matrix and explain what the elements of each matrix and vector stand for. Write down the standard assumptions for the multiple linear regression.
I understand how order in a matrix normally depending on the amount of rows and columns of the matrix, but don't understand when I am talking in terms of the multiple linear regression equation. If you could include what the elements of each matrix and vector standard for as well that would be much appreciated.
I am perfectly fine with the assumptions.
Thank you

Determine the point estimate of the population proportion, the margin of error for the following
Lower bound $=0.531$, upper bound $=0.789,n=1000$
The point estimate of the population proportion is (_)
The margin of error is (_)
The number of individuals in the sample with the specified characteristic is (_)

Specify the regression equation to estimate the DiD as:
$Y={\beta }_0+{\beta }_1\ast <!--\; \ast \; -->[Time]+{\beta }_3\ast <!--\; \ast \; -->[Time\ast <!--\; \ast \; -->Intervention]+{\beta }_4\ast <!--\; \ast \; -->[Covariates]+\epsilon$
instead of:
$Y={\beta }_0+{\beta }_1\ast <!--\; \ast \; -->[Time]+{\beta }_2\ast <!--\; \ast \; -->[Intervention]+{\beta }_3\ast <!--\; \ast \; -->[Time\ast <!--\; \ast \; -->Intervention]+{\beta }_4\ast <!--\; \ast \; -->[Covariates]+\epsilon$
Our ${\beta }_3$ coefficient would still be yielding the DiD estimator right?

What a weighted least squares regression and when is it used?

How many units of each type should be sold in order to maximize total profit?
A store sells two types of sodas, A and B. The store owner pays $8 and $10 for each one unit of soda A and B respectively. One unit of sodas A yields a profit of $2 while a unit of sodas B yields a profit of $3. The store owner estimates that no more than 2000 sodas will be sold every month and he does not plan to invest more than $20,000 in inventory of these sodas.
How many units of each type of toys should be stocked in order to maximize his monthly total profit?

Statistics for Behavioral Sciences - Correlation
What does a negative sign tell you about the relationship between two variables, X and Y?
Select all that apply
1) a negative correlation means X and Y change in the same direction
2) a negative correlation means as X increases, Y decreases
3) a negative correlation means X and Y change in opposite directions
4) a negative correlation means that X and Y are weakly related to each other

How can we derivate with a constant in trying to prove that correlation coefficient is between -1 and 1?

Finding Multiple regression coefficients
If I have a multiple regression like this $Y=a+b_1.X_1+b_2.X_2,$ how can I calculate the values of $b_1$ and $b_2$? I have searched on the web but couldn't find an answer.

Is there a good reason to use covariance and not correlation?

Assume that we have a sequence of n realizations $x_1,x_2,...,x_n$ of an i.i.d random variable X with cdf $F_X(x)$ and pdf $f_X(x)$ . Now define Y as the sum of k consecutive realizations of X
$Y_i=\underset{j=i}{\overset{k}{\sum <!--\; \sum \; -->}}{x}_j$
which generates the sequence $Y_1,...,Y_{n-<!--\; -\; -->k+1}$ where $Y_i$ contains $k-<!--\; -\; -->1$ elements of $Y_{i-<!--\; -\; -->1}$ (and $Y_{i+1}$ ).
What is the disribution $F_Y(x)$ of Y?

Statistics：How to prove that one variable's change is influenced by another variable,that is to say ,they are related?
I major in Bioinformatics. Now,I am in a problem: we all know that temperature changes during a year , I find that a disease incidence is really high when temperature is relatively high , while it becomes really low when the temperature is relatively low , that is to say ,they are related. So ,I want to find out a way to prove that they are related ,not just intuitively feel that they are related. So, any suggestions?

My book problem states that U and V are independent variables, each with expected value $\mathrm{\mu <!--\; \mu \; -->}$ and variance ${\mathrm{\sigma <!--\; \sigma \; -->}}^2$ . We then have $\mathrm{\alpha <!--\; \alpha \; -->}U+V\sqrt{1-<!--\; -\; -->{\mathrm{\alpha <!--\; \alpha \; -->}}^2}$ , and I am asked to calculate the correlation coefficient for U and Z, ${\mathrm{\rho <!--\; \rho \; -->}}_{uz}$ . I know correlation is derived with $\frac{cov(U,Z)}{\sqrt{var(U)var(Z)}}$ , and I'm able to calculate $var(U)$ and $var(Z)$ but not $cov(U,Z)$ . I'm aware $Cov(U,Z)=E(UZ)-<!--\; -\; -->E(U)E(Z)$ , and again I have $E(U)$ and $E(Z)$ , but I have no way of getting $E(UZ)$ that I'm aware of...what am I missing?

Continuous random variable $R(0,2)$ . Describe the dependency of Y on X for $Y=X^2$ . Write a covariance matrix and correlation matrix of a random vector $(X,Y)$ .

Among 240 cell phone owners aged 18-24 surveyeah, 108 said their phone was an Android phone.
Find a point estimate for the proportion of cell phone owners aged 18-24 who have an Android phone. Round the answer to at three decimal places.

Is there such a thing as a weighted multiple regression?
I'm new to linear algebra, but I know how multiple linear regressions work. What I want to do is something slightly different.
As an example, let's say that I have a list of nutrients I want to get every day. Say I have a list of foods, and I want to know how much of each food to eat to get the best fit for my nutrition plan. Assume I'm fine with using a linear model.
However, some nutrients are more important than others. The errors on protein and calcium might be equal in a typical linear regression, but that's no use. Protein has higher priority than calcium (in this model), so I'd want a model that is better fitting to the higher priority points than to the lower ones.
I tried putting weights on the error function, and I end up with a matrix of matrices. At that point, I'm not sure if I'm minimising for the weights or for the coefficients on the nutrients. I think both, but I wasn't sure how to minimise for both at the same time.
Is it possible to solve this with linear algebra, or does this require some numerical approximation solution?

Multiple linear regression with interaction
I'm doing a multiple linear regression with interacting variables. I'll give you an example:
$y$=value, $x_1$=material, $x_2$=weight, $x_3$=color
$x_1$ and $x_2$ are interacting variables but $x_3$ is not. Right now I'm using something like:
$y=a_0+a_1{x}_1+a_2{x}_2+a_3{x}_3+a_{12}{x}_1{x}_2+u$
I'm pretty new to regression analysis so I wonder if there is any way to convert this formula to something like
$y=a_0+a_1{x}_1+a_2{x}_2+a_3{x}_3+u$
so I can see how much effect $x_1$ and $x_2$ have simply by looking at $a_1$ and $a_2$? What I want to do is to just be able to look at the equation and understand how much 1 kg of extra weight adds in value without needing to calculate y. Splitting up the interaction term $a_{12}$ and distributing the effect over $a_1$ and $a_2$ if you guys understand what I mean. Maybe it's not possible or maybe there is a better regression method that is more suited for this, I don't know. I'd love to get some pointers from you guys.
Thanks.

Why is the standard deviation important for non-normal distributions?
In the case of the normal distribution, I know that the standard deviation tells me that 78% of my sample is in the interval $[\mathrm{\mu <!--\; \mu \; -->}-<!--\; -\; -->\mathrm{\sigma <!--\; \sigma \; -->},\mathrm{\mu <!--\; \mu \; -->}+\mathrm{\sigma <!--\; \sigma \; -->}]$
Suppose I have another sample which is not normally distributed. Is there any valuable information I get from knowing the standard deviation of that sample?

What is the standard error?