Given a linear regression model obtained by ordinary least squares, prove that the sample covariance between the fitted values and the residuals is zero.

Tamara Bryan
2022-07-20
Answered

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Jorge Franklin

Answered 2022-07-21
Author has **11** answers

I assume that you meant their dot product is zero. If so, let $\hat{y}=Hy$ where $H=X({X}^{\prime}X{)}^{-1}{X}^{\prime}$ (from OLS). Also, note that $H$ is idempotent, i.e. ${H}^{2}=H.$. Then,

${\hat{y}}^{\prime}.e={y}^{\prime}H(I-H)y={y}^{\prime}(H-{H}^{2})y={y}^{\prime}(H-H)y=0.$

${\hat{y}}^{\prime}.e={y}^{\prime}H(I-H)y={y}^{\prime}(H-{H}^{2})y={y}^{\prime}(H-H)y=0.$

asked 2020-11-07

Which of these shows the strongest correlation?

asked 2022-04-06

$\mu =m=\frac{{q}_{1}+{q}_{3}}{2}$

Let $X$ be a random variable with p.m.f./p.d.f. ${f}_{X}(x)$ that is symmetric about $\mu (\in R),$ i.e., ${f}_{X}(x+\mu )={f}_{X}(\mu -x)$, $\mathrm{\forall}x\in (-\mathrm{\infty},\mathrm{\infty}).$

If ${q}_{1},m$ and ${q}_{3}$ are respectively the lower quartile, the median and the upper quartile of the distribution of $X$ then show that $\mu =m=\frac{{q}_{1}+{q}_{3}}{2}$

How to prove.....

any Hints.

Let $X$ be a random variable with p.m.f./p.d.f. ${f}_{X}(x)$ that is symmetric about $\mu (\in R),$ i.e., ${f}_{X}(x+\mu )={f}_{X}(\mu -x)$, $\mathrm{\forall}x\in (-\mathrm{\infty},\mathrm{\infty}).$

If ${q}_{1},m$ and ${q}_{3}$ are respectively the lower quartile, the median and the upper quartile of the distribution of $X$ then show that $\mu =m=\frac{{q}_{1}+{q}_{3}}{2}$

How to prove.....

any Hints.

asked 2021-11-30

The following is the average daily temperature for Frederick, MaryLand for the month of June:

$$\begin{array}{|cccccc|}\hline 81& 78& 78& 89& 74& 85\\ 85& 84& 80& 90& 73& 86\\ 86& 82& 90& 92& 70& 73\\ 79& 92& 76& 74& 86& 76\\ 71& 81& 73& 77& 72& 79\\ \hline\end{array}$$

a) Complete the frequency distribution for the data.

$$\begin{array}{|ccc|}\hline Temperature& Frequency& Relative\text{}Frequency\\ 70-74\\ 75-79\\ 80-84\\ 85-89\\ 90-94\\ \hline\end{array}$$

a) Complete the frequency distribution for the data.

asked 2022-05-09

Calculating the position of the median

I know this is a simple question, but I cannot find a straight answer anywhere.

When calculating the medium of listed data, the formula is (n+1)/2. My Statistics teacher said for grouped data the position of the medium is n/2. However, this seems contradictory as for discrete grouped data, the data could be written in a list if the original values were known. Therefore two different values for the median are found.

I know this question is probably going to get flagged as it has already been asked, however, it has never been answered. I find it frustrating that such a fundamental concept in Statistics, what is supposed to be precise and never subjective has a wishy-washy answer.

Rant over, I think the position should be (n+1)/2 for discrete data and n/2 for continuous data. Moreover, this then leads into the obvious question of how we should calculate quartiles as well.

It would be nice if everyone could agree on a certain method for calculating the medium as all my textbooks are saying different things.

Someone please dispell the confusion and this statistical mess.

I know this is a simple question, but I cannot find a straight answer anywhere.

When calculating the medium of listed data, the formula is (n+1)/2. My Statistics teacher said for grouped data the position of the medium is n/2. However, this seems contradictory as for discrete grouped data, the data could be written in a list if the original values were known. Therefore two different values for the median are found.

I know this question is probably going to get flagged as it has already been asked, however, it has never been answered. I find it frustrating that such a fundamental concept in Statistics, what is supposed to be precise and never subjective has a wishy-washy answer.

Rant over, I think the position should be (n+1)/2 for discrete data and n/2 for continuous data. Moreover, this then leads into the obvious question of how we should calculate quartiles as well.

It would be nice if everyone could agree on a certain method for calculating the medium as all my textbooks are saying different things.

Someone please dispell the confusion and this statistical mess.

asked 2021-09-16

Find P(A) using the following table.

asked 2021-01-08

You have 11 cubes in 8 different colors (1 color repeats 2 times and another one repeats 3 times) that you want to build in a line. How many different lines can be formed with those cubes?

asked 2021-11-29

As an engineer working for a water bottling company, you collect the following data in order to test the perfomance of the bottling systems. Plot the data and calculate the mean and standard deviation.

$$\begin{array}{|cc|}\hline \text{Milliters of Water in the Bottle}& Frequency\\ 485& 13\\ 490& 17\\ 495& 25\\ 500& 40\\ 505& 23\\ 510& 18\\ 515& 15\\ \hline\end{array}$$