Understanding Inferential Statistics through Examples

Why do we care if a set forms a basis?

What is explained variance?

There is a Theorem on "Orthogonal Partitioned Regression" which says:
"In the multiple linear least squares regression of $y$ on two sets of variables $X_1$ and $X_2$, if the two sets of variables are orthogonal, then the separate coefficient vectors can be obtained by separate regressions of $y$ on $X_1$ alone and $y$ on $X_2$ alone. ..."

It is important to know that "Causation is not the same as Correlation." Which of the following gives a reason that causation is not the same as correlation?
A) Correlation is only a number between -1 and 1.
B) Sometimes the two variables with a correlation are both the results of a third factor or variable.
C) Even with a strong statistical correlation, there are usually some people who don't fit the pattern.
D) All of the above.

Questions about multiple linear regression
I have a couple true/false questions basically, one of them is this
In the multiple linear regression model the coefficient of multiple determination gives the proportion of total variability due to the effect of a single predictor
I know the coefficient of multiple determination indicates the amount of total variability explained by the model, but I'm not sure about the single predictor part, I don't think this is true because it uses x1, x2... as predictors no?
The other question is this;
In the multiple linear regression model
$y_i={\beta }_0+{\beta }_1{x}_{i,1}+{\beta }_2{x}_{i,2}+{\beta }_3{x}_{i,3}+{\epsilon }_i$
the parameter ${\beta }_1$ represents the variation in the response corresponding to a unit increase in the variable $x_1$
I don't think this question is true but can't really explain why
All help would be greatly appreciated

How do you calculate the t-score of the slope of a time series regression?

How can we manipulate the expression
$$\underset{i=1}{\overset{m}{\sum <!--\; \sum \; -->}}(\underset{j=0}{\overset{n}{\sum <!--\; \sum \; -->}}{a}_j{x}_i^j{)}^2$$
into
$$\underset{j=0}{\overset{n}{\sum <!--\; \sum \; -->}}\underset{k=0}{\overset{n}{\sum <!--\; \sum \; -->}}{a}_j{a}_k(\underset{i=1}{\overset{m}{\sum <!--\; \sum \; -->}}{x}_i^{j+k})$$

There are 20 guests at a party. The host has 8 gallons of punch. He estimates that each guest will drink 2 cups of punch. If his estimate is correct, how much punch will be left over at the end of the party?

Suppose we have the regression model: $y_i$ = ${\mathrm{\beta <!--\; \beta \; -->}}_0$ + ${\mathrm{\beta <!--\; \beta \; -->}}_1{x}_i$ + ${\mathrm{ϵ<!--\; ϵ\; -->}}_i$
where $y_i$ = ($Y_i$ - $\stackrel{¯<!--\; ¯\; -->}{Y}$) and $x_i$ = ($X_i$ - $\stackrel{¯<!--\; ¯\; -->}{X}$).
This will be true iff ${\mathrm{\beta <!--\; \beta \; -->}}_0$ = 0. We immediately see that ${\mathrm{\beta <!--\; \beta \; -->}}_0$ = ($Y_i$ - $\stackrel{¯<!--\; ¯\; -->}{Y})$) - ${\mathrm{\beta <!--\; \beta \; -->}}_1$($X_i$ - $\stackrel{¯<!--\; ¯\; -->}{X})$). Where ${\mathrm{\beta <!--\; \beta \; -->}}_1$ is given by $\frac{COV(X,Y)}{VAR(X)}$. I don't believe this quantity is guaranteed to be 0, so would the answer be that we are unable to determine if the regression line passes through the origin?

Regression analysis is a statistical process for estimating the relationships among variables. Regression analysis is widely used for prediction and forecasting. So why is regression analysis also used as statistical test?

Given this regression model: $y_i={\mathrm{\beta <!--\; \beta \; -->}}_0+{\mathrm{\beta <!--\; \beta \; -->}}_1{x}_i+E_i$.
All the assumptions are valid except that now: $E_i\sim <!--\; \sim \; -->N(0,x_i{\mathrm{\sigma <!--\; \sigma \; -->}}^2)$
Find Maximum likelihood parameters for ${\mathrm{\beta <!--\; \beta \; -->}}_0$, ${\mathrm{\beta <!--\; \beta \; -->}}_1$

Explain in simple words the difference between the Causal and Correlated metric types.

Variable and its dynamics in one multiple regression model
I am trying to find the dependence between default rates in bank and macroeconomic variables with linear regression. To do so I created a code which estimates every possible model - every combination of variables is tested. As an output I obtain R-squared, statistics for Chow, Breusch-Pagan, Breusch-Godfrey, RESET and Shapiro-Wilk tests as well as VIF. The only model which pass all tests, has satisfying R-squared and low VIF is as follows: $y_t={\mathrm{\beta <!--\; \beta \; -->}}_0+{\mathrm{\beta <!--\; \beta \; -->}}_1{x}_t+{\mathrm{\beta <!--\; \beta \; -->}}_2\mathrm{\Delta <!--\; \Delta \; -->}{x}_t$ where $\mathrm{\Delta <!--\; \Delta \; -->}{x}_t=x_t-<!--\; -\; -->x_{t-<!--\; -\; -->1}$. Altough using variable and its dynamics in one model seems a bit strange, I did not find any reason to reject the model. I would be grateful if someone could help me motivate accepting or rejecting such a model.

Correlation Coefficient - statementsThis is in relation to a scatterplot of 400 data points, with a very strong negative correlation.
For the statements, state whether they are correct or incorrect and why.
If each value of the y variable is divided by four, then the value of the correlation coefficient will also be divided by four. - Assume this one is true? As the correlation coefficient then wouldn't change?
The scatterplot shows that the variable x causes the variable y to take the values it does. - not too sure about this one?

Kelly wants to create a rectangular playpen for his cats in his backyard, and estimates that the cats will need 500 square feet. The length of the playpen should be 20 ft longer than the width. What dimensions should the playpen be?

What is meant by a 95% confidence interval?

Multiple linear regression linear relationship or not
How should a multiple linear regression be interpreted that has statistically significant predictors, but an R Square value of $0.004$? Does that mean that there is a significant linear relationship (because statistically significant predictors), even though there is close to no linear relationship ($R^2$ of $0.004$ indicates close to no linear relationship).

2-dimensional representation of a multiple regression function
Supposing I have a multiple regression population function of the form:
$Y_i={\mathrm{\beta <!--\; \beta \; -->}}_1+{\mathrm{\beta <!--\; \beta \; -->}}_2{X}_{2i}+{\mathrm{\beta <!--\; \beta \; -->}}_3{X}_{3i}+u_i$
with $X_{3i}$ a dummy variable (only takes values $0$ and $1$).
I am given a sample of points. Although the latter takes place in 3 dimensional space, the question states "its results can be represented in $Y$ vs $X_2$ space". I don't understand how graphing $Y$ vs $X_2$ will give us a 2 dimensional representation of our population regression function. Isn't $X_{3i}$ being completely omitted?