Understanding Inferential Statistics through Examples

Two investments X and Y give returns as follows (expectation and variance): $E(X)=0.5$ , $E(Y)=0.4$ and $V(X)=2$ , $V(Y)=1$ . The correlation between X and Y is $\mathrm{\rho <!--\; \rho \; -->}(X,Y)=0.6$ .
Find the expectation and variance of the "combined" investments $U=X+Y$ and $W=2Y$, and finally find the correlation between U and W. Hint: First find $V(U+W)$ .

If I have that X and Y are independent r.v with mean 0 and variance ${\mathrm{\sigma <!--\; \sigma \; -->}}^2$ and I have that
$Z=X\sin <!--\; \; -->\mathrm{\alpha <!--\; \alpha \; -->}+Y\cos <!--\; \; -->\mathrm{\alpha <!--\; \alpha \; -->}$ , what is then the covariance and the correlation between X and Z?

Suppose we have a multivariate normal random variable $X=[X_1,X_2,X_3,X_4{]}^{\top }$ . And here $X_1$ and $X_4$ are independent (not correlated). Also $X_2$ and $X_4$ are independent. But $X_1$ and $X_2$ are not independent. Assume that $Y=[Y_1,Y_2{]}^{\top }$ is defined by
$Y_1=X_1+X_4,Y_2=X_2-X_4.$
If I know the covariance matrix of X, what would be the covariance matrix of Y?

What can we conclude from correlation?I just got my statistics test back and I am totally confused about one of the questions!
A study was done that took a simple random sample of 40 people and measured whether the subjects were right-handed or left-handed, as well as their ages. The study showed that the proportion of left-handed people and the ages had a strong negative correlation. What can we conclude? Explain your answer.
I know that we can't conclude that getting older causes people to become right-handed. Something else might be causing it, not the age. If two things are correlated, we can only conclude association, not causation. So I wrote:
We can conclude that many people become right-handed as they grow older, but we cannot tell why.

A health department estimates that 8 vials of malaria serum will treat 100 people. At this rate, how many vials are required to treat 175 people?

A company estimates that it costs $0.03x^2+4x+1000$ dollars to produce x units of a product. How do you find the expression for the average cost per unit?

Conditional Probability and Independence nonsense in a problem
The statement:
Suppose that a patient tests positive for a disease affecting 1% of the population. For a patient who has the disease, there is a 95% chance of testing positive, and for a patient who doesn't has the disease, there is a 95% chance of testing negative. The patient gets a second, independent, test done, and again tests positive. Find the probability that the patient has the disease.
The problem:
I can solve this problem, but I'm unable to understand what is wrong with the following:
Let $T_i$ be the event that the patient tests positive in the i-th test, and let D be the event that the patient has the disease.
The problem says that $P(T_1,T_2)={0.95}^2\ast <!--\; \ast \; -->0.01+{0.05}^2\ast <!--\; \ast \; -->0.99=0.0115$, because the tests are independent.
By law of total probability we know that:
$P(T_1,T_2)={0.95}^2\ast <!--\; \ast \; -->0.01+{0.05}^2\ast <!--\; \ast \; -->0.99=0.0115$
Replacing, and assuming conditional independence given D, we have:
$P(T_1,T_2)={0.95}^2\ast <!--\; \ast \; -->0.01+{0.05}^2\ast <!--\; \ast \; -->0.99=0.0115$
This is the correct result, but now let's consider that:
$P(T_1,T_2)=P(T_1{)}^2$
We know that $P(T_1,T_2)=P(T_1{)}^2$ for all i because of symmetry, so we have $P(T_1,T_2)=P(T_1{)}^2$. Again, by law of total probability:
$P(T_1)=0.95\ast <!--\; \ast \; -->0.01+0.05\ast <!--\; \ast \; -->0.99\approx <!--\; \approx \; -->0.059$
$P(T_1)=0.95\ast <!--\; \ast \; -->0.01+0.05\ast <!--\; \ast \; -->0.99\approx <!--\; \approx \; -->0.059$
So we have:
$P(T_1,T_2)=P(T_1{)}^2\approx <!--\; \approx \; -->{0.059}^2\approx <!--\; \approx \; -->0.003481$
The second approach is wrong, but it seems legitimate to me, and I'm unable to find what's wrong.
Thank's for your help, you make self studying easier.

For a data set of weights and highway fuel consumption amounts of four types of automobile, the linear correlation coefficient is found and the P-value is 0.001. Please, write a statement that interprets the P-value and includes a conclusion about linear correlation.

Naima's pedometer recorded 43,498 steps in one week. Her goal is 88,942 steps. Naima estimates she has about 50,000 more steps to meet her goal. Is Naima's estimate reasonable?

Why is the correlation between two data series negative when the top half and bottom half are both strongly correlated?

Dimensionality of datasets in multiple regression
As an example, let's say that a linear regression is performed of the form
$Y={\mathrm{\beta <!--\; \beta \; -->}}_0+{\mathrm{\beta <!--\; \beta \; -->}}_1{X}_1+{\mathrm{\beta <!--\; \beta \; -->}}_2{X}_2+\cdots <!--\; \cdots \; -->+{\mathrm{\beta <!--\; \beta \; -->}}_n{X}_n+\mathrm{\epsilon <!--\; \epsilon \; -->}$
where $Y$ is a vector of $10,000$ measurements of peak acceleration of different car models, and the regressors correspond to different technical features of the cars.
From a linear algebra standpoint $Y$ lives in ${\mathbb{R}}^{10000}$, and the coefficients are found by minimizing the sum of the square distances of this vector on a hyperplane.
Now, from the point of view of dimension being the number of linearly independent vectors that span a space, this vector $Y$ is just $1$ dimension.
If it is truly $1$-dimension of a ${\mathbb{R}}^{10000}$ ambient space, the Euclidean projection on the hyperplane that underpins the process of finding the coefficients does not have any dimensionality issues (collinearity between the regressors being a separate topic). Otherwise, $L^2$ norms in high dimensions do pose problems.
"So is $Y$ (the vector of $10,000$ observations) $1$-mimensional or high dimensional?"

(a) Give examples of two variables that have a strong positive linear correlation and two variables that have strong negative linear correlation.

(b) Explain in your own words why the linear correlation coefficient should not be used when its absolute value is too low or close to zero. Give an example.

(c) In the passage below identify the explanatory variable and the response variable. Explain why.
A nutritionist wants to determine if the amounts of water consumed each day by persons of the same weight and on the same diet can be used to predict individual weight loss.

Conflicting Results from t-test and F-based stepwise regression in multiple regression.
I currently am tasked with building a multiple regression model with two predictor variables to consider. That means there are potentially three terms in the model, Predictor A (PA), Predictor B (PB) and PA*PB.
In one instance, I made a LS model containing all three terms, and did simple t-tests. I divided the parameter estimates by their standard errors to calculate t-statistics, and determined that only the intercept and PA*PB coefficients were significantly different from zero.
In another instance, I did stepwise regression by first creating a model with only PA, and then fit a model to PA and PB, and did an F-test based on the Sum of Squares between the two models. The F-test concluded that PB was a significant predictor to include in the model, and when I repeated the procedure, the PA*PB coefficient was found to reduce SSE significantly as well.
So in summary, the t-test approach tells me that only the cross-product term PA*PB has a significant regression coefficient when all terms are included in the model, but the stepwise approach tells me to include all terms in the model.
Based on these conflicting results, what course of action would you recommend?

Jason estimates that his car loses 12% of its value every year. The initial value is 12,000. Which best describes the graph of the function that represents the value of the car after X years ?

Do future events influence the probability of past events?
This question came to mind when I was dealing with this question: A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn at random and are found to be clubs. Find the probability that the lost card is a club.
My query: The probability that a club is lost is 13/52. The next event of taking two cards from this incomplete set of cards happens after one of the cards is lost. How can this affect a past event of losing a card from the pack?

What is J in while calculating SST in multiple regression?
I am little confused what actually is the J in the formula of the SST and SSR for multiple regression
SST= $Y^T[1-<!--\; -\; -->\frac{1}{n}J]Y$
SSR=$Y^T[H-<!--\; -\; -->\frac{1}{n}J]Y$

A set of data with correlation coefficient of -0.55 has a
a) strong negative linear correlation
b) weak negative linear correlation
c) moderate negative linear correlation
d) little or no linear correlation

Determining correlation given 2 other correlation
Hi I'm given a rather ambigious question on correlations. My question is how do we determine the correlation between 2 varibles given their correlation with other variables?
Correlation between chocolate consumption per capita and number of Nobel laureates per 10 million persons for a broader list of 90 countries = 0.45
Perfect positive association bewteen chocolate consumption and chili consumption per capita (correlation = 1)
Correlation between chili consumption per capita and nobel laureates per 10 million persons: ?
1) > 0.45
2) = 0.45
3) < 0.45
4) cannot determine

Based on the estimates $\log <!--\; \; -->(2)=.03$ and $\log <!--\; \; -->(5)=.7$, how do you use properties of logarithms to find approximate values for $\log <!--\; \; -->(80)$?