Understanding Inferential Statistics through Examples

The gradient of the regression line $x$ on $y$ is $-<!--\; -\; -->0.2$ and the line passes through $(0,3)$. If the equation of the line is $x=c+dy$, find the value of $c$ and $d$.

Constructing a sample by correlation
Suppose we have two samples with known correlation (should be relatively high). Say both samples have n data points. What if now we still know the correlation factor but one sample only consistent of the first 5 data point.
Could one still construct the remaining data points solely using the correlation with the other sample?
My idea would be to look at the relative differences in the known sample and compensate by the correlation. Could this work? Thanks for any assistance.

Can we determine if two random variables are independent if I know their expected values and the variances?

$Y=AX+B{X}^3$ . Compute $Var(Y|X=x)X$ has distribution $N(1,1)$ and $A\mathrm{\&<!--\; \&\; -->}B$ both have distribution $U(-<!--\; -\; -->1,1)$ .

Let $H$ be an Hilbert space and let $M,N$ closed subspaces of $H$. Let$P_M$ and $P_N$ orthogonal projections with range($$M$$) and range($$N$$) respectively. Prove that $P_M+P_N=P_{M+N}$ if and only if $M\perp <!--\; \perp \; -->N$.

In statistics using a regression analysis in SPSS - - variables are hunger and amount of dancing. Which would be the dependent and independent variables?

How can adjusted r squared be negative?

An simple formula, an example, and an explanation for what all the symbols and variables are for basic linear regression?

If two variables have a negative correlation, can they have a positive covariance?

How to reduce equation?
$$p=\frac{e^{{\mathrm{\beta <!--\; \beta \; -->}}_0+{\mathrm{\beta <!--\; \beta \; -->}}_1\ast <!--\; \ast \; -->age}}{e^{{\mathrm{\beta <!--\; \beta \; -->}}_0+{\mathrm{\beta <!--\; \beta \; -->}}_1\ast <!--\; \ast \; -->age}+1}$$
to $${\log }_e<!--\; \; -->\frac{p}{1-<!--\; -\; -->p}={\mathrm{\beta <!--\; \beta \; -->}}_0+{\mathrm{\beta <!--\; \beta \; -->}}_1\ast <!--\; \ast \; -->age$$

What is the total sum of squares?

What the properties of $$t$$-values are?

How can a t-statistic be used to determine statistical significance?

How should one interpret the standard errors in the output of regression analysis?

Can you use bivariate analysis in a multiple regression problem
I have one response variable and around 10 potential explanatory variables. I am looking for simple ways to visualize and explore my data before modelling. As well as carrying out PCA and looking for correlations between my explanatory variables I wanted to look at bi-variate scatter plots of the response with each explanatory variable - but I have been told this wont work.
This is they way I am picturing it -
Imagine you have n independent, uncorrelated explanatory variables.
You propose to use these variables in a multiple regression. You want to simplify the analysis by not including more variables than is necessary.
So , you want to know if the explanatory variables have any correlation with the response variable to decide whether to include the variable in the regression.
My feeling is 1) that you can look at bi-variate plots for the response variable with each explanatory variable to see if there is a correlation.
2) If a single explanatory variable has no correlation with a the response in bi-variate correlation, it cannot have any affect on the response if included in the multiple regression.
(The only way this variable could affect the response variable would through an interaction with another explanatory variable - but it has already been determined that all the explanatory variables are uncorrelated.)
I have been told that the above is incorrect and a variable showing no correlation in a bi-variate plot can affect the response variable in a multiple regression.
Could someone help me see where I am going wrong? Many Thanks.

How do you calculate r squared by hand?

Correlation and heteroscedasticityI'm studying a dataset and observed a positive correlation between two variables, but when I plot them, it seems that they are heteroscedastics, what conclusions can I get from it ? (Can I really assume that the positive correlation is real ?)

How to get this one:$$b_1=\frac{\sum <!--\; \sum \; -->(X_i-<!--\; -\; -->\stackrel{¯<!--\; ¯\; -->}{X})(Y_i-<!--\; -\; -->\stackrel{¯<!--\; ¯\; -->}{Y})}{\sum <!--\; \sum \; -->{(X_i-<!--\; -\; -->\stackrel{¯<!--\; ¯\; -->}{X})}^2}$$

How does r squared related to standard deviation?

Would it be any concern if we find correlation between intercept and other regression coefficients?