Find an answer to any high school probability and statistics problem

Recent questions in Statistics and Probability

Marcelo Maxwell 2022-09-27

What is the z-score of sample X, if $n = 256, μ = 42, St.Dev = 80, μ_{X} = 89$ ?

sombereki51 2022-09-27

The 10th grader Jack and the 11th grader Michael have identical math scores (i.e., they both got 95 on Exam 1). Thus, they also have identical z-scores.
a) True. That's what z scores are.
b) Not necessarily. Depends on the standard deviation in each of their classes.

tamnicufl 2022-09-27

There are $\frac{m}{gcd (m, x)}$ distinct elements in the set ${a x (\mod m) : a \in {0, . . ., m - 1}}$
I have only known these by using a computer to generate the number of distinct elements. But I am not sure how to prove this conjecture.
And is there any way that we can connect this problem to Euler's phi function so that we can simply use properties of $ϕ$ function to prove it?
And can we also use some counting principle here to give an exact answer?

zakownikbj 2022-09-26

Question on designing a state observer for discrete time system
I came through this problem while studying for an exam in control systems:
Consider the following discrete time system
$\vec{x} (k + 1) = A \vec{x} (k) + b \vec{u} (k), \vec{y} (k) = c \vec{x} (k)$
where $b = (0, 1)^{T}, c = (1, 0), A = [\begin{matrix} 2 & 1 \\ 0 & - g \end{matrix}]$ for some $g \in R$
Find a feedback regulation (if there is any) of the form $u (k) = - K \hat{x} (k)$ where $\hat{x} (k)$ is the country estimation vector that is produced via a linear complete-order state observer such that the nation of the system and the estimation blunders $e (k) = \vec{x} (k) - \hat{x} (k)$ go to zero after a few finite time. layout the kingdom observer and the block diagram.
My method
it is clean that the eigenvalues of the machine are $λ_{1} = 2, λ_{2} = - g$ (consequently it is not BIBO solid) and that the pair (A,b) is controllable for every fee of g, as nicely a the pair (A,c) is observable for all values of g. consequently we will shift the eigenvalues with the aid of deciding on a benefit matrix okay such that our device is strong, i.e. it has its eigenvalues inside the unit circle $| z | = 1$ .
The state observer equation is
$[\vec{x} (k + 1) \vec{e} (k + 1)]^{T} = [\begin{matrix} A - b K & B k \\ O & A - L C \end{matrix}] [\vec{x} (k) \vec{e} (k)]^{T}$
With characteristic equation
$χ (z) = | z I - A + b K | | z I - A + L C | = χ_{K} (z) χ_{L} (z)$
Also consider
$K = [\begin{matrix} k_{1} & k_{2} \\ k_{3} & k_{4} \end{matrix}]$
and let $a = k_{1} + k_{3}, β = k_{2} + k_{4}$
Then $χ_{K} (z) = (z - 2) (z + g + β) + a$ .
So we can select some eigenvalues inside the unit circle and determine $a, β$ in terms of g. Choosing e.g. $λ_{1, 2} = \pm 1 / 2$ we get $a = 3 g + 33 / 8, β = 9 / 4 - g, g \in R$
Questions
I want to ask the following:
Is my approach correct? Should I select the eigenvalues myself since I am asked to design the observer or should I just solve the characteristic equation and impose $| λ_{1, 2} | < 1$ ?
Should I determine L matrix as well since the error must also vanish? (because it is not asked)

Melina Barber 2022-09-26

Pearson Correlation Coefficient Interpretation
Let X=(1,2,3,...,20). Suppose that $Y = (y_{1}, y_{2}, . . ., y_{20})$ with $y_{i} = x_{i}^{2}$ and $Z = (z_{1}, z_{2}, . . ., z_{20})$ with $z_{i} = e^{x_{i}}$ . Pearson correlation coefficient is defined by formula
$ρ (X, Y) = \frac{\sum_{i = 1}^{20} (x_{i} - \bar{x}) (y_{i} - \bar{y})}{\sqrt{(\sum_{i = 1}^{20} (x_{i} - \bar{x})^{2}) (\sum_{i = 1}^{20} (y_{i} - \bar{y})^{2})}}$
If $ρ (X, Y) = 1$ , we can say that X and Y have a linear correlation. If $0.7 \leq ρ (X, Y) < 1$ then X and Y has a strong linear correlation, if $0.5 \leq ρ (X, Y) < 0.7$ then X and Y has a modest linear correlation, and if $0 \leq ρ (X, Y) < 0.5$ then X and Y has a weak linear correlation. Using this formula, we get $ρ$ (X,Y)=0.9 and $ρ$ (X,Z)=0.5. However, the relationship between X and Y is actually quadratic but they have the high correlation coefficient that indicate linear correlation.
So, my question is what is "linear correlation" actually between X and Y ? Since $ρ$ (X,Z)=0.5 indicate the modest correlation coefficient, what is another intepretation of this value? What is the difference between $ρ$ (X,Y) and $ρ$ (X,Z), noting that Y and Z is not a linear function of X.

Alan Sherman 2022-09-26

What is the lower bound of a random variable's variance?

Elias Heath 2022-09-26

While calculating percentile rank there are two formula
1st )
percentile ={ ( number of values below x + 0.5)/total number of observation } 100
2nd )
percentile ={ ( number of values below x )/total number of observation } 100
I have to know when to used 1st formula and when to used 2nd formula
propert explanation needed

gemauert79 2022-09-26

Use the Binomial Theorem to find the indicated coefficient or term
The coefficient of x in the expansion of $(2 x + 4)^{5}$
640
5120
2560
1280

hommequidort0h 2022-09-26

Suppose that, for two populations, the distributions of the variable under consideration have the same shape. Further suppose that you want to perform a hypothesis test based on independent random samples to compare the two population means. In each case, decide whether you would use the pooled t-test or the Mann-Whitney test and give a reason for your answer. You know that the distributions of the variable are a. normal. b. not normal.

malaana5k 2022-09-26

Meaning of 'obligatory disclaimer'
Can you explain to me what 'obligatory disclaimer' means in probability. Is it like a lack of information or what?
The context is the following:
Second, we found that the marginal distribution of Y is Bern(0.08), whereas the conditional distribution of Y given X = 1 is Bern(0.2) and the conditional distribution of Y given X = 0 is Bern(0.04). Since conditioning on the value of X alters the distribution of Y , X and Y are not independent: learning whether or not the sampled individual is a current smoker gives us information about the probability that he will develop lung cancer. This example comes with an obligatory disclaimer. Although we have found that X and Y are dependent, we cannot make conclusions about whether smoking causes lung cancer based on this association alone. (Joseph K. Blitzstein, Jessica Hwang--Introduction to Probability)

Conrad Beltran 2022-09-26

List contains events $A_{1}$ , $A_{2}$ , …, $A_{5}$ which are mutually exclusive and collectively exhaustive.
Compute the following:
$\sum_{i = 1}^{k} P (A_{i}^{c})$

Medenovgj 2022-09-26

Meta-analysis examine:
a. I don't know
b. Standard deviation
c. Effect sizes
d. Statistical significance

Jean Farrell 2022-09-25

Number of undirected graphs with n vertices and k edges (inclusive of simple, non-simple, isomorphic, and disconnected graphs)
Given the constraints (or non-constraints rather), is there a closed solution on a set of labeled vertices?
One way I looked at this problem is by trying to implement a constrained stars-and-bars technique on the diagonal and diagonal-exclusive half of the $n \times n$ adjacency matrix, but I'm not getting any good leads.
It seems there are multiple definitions of a self-loop in undirected graphs. In the context of this question, I define a self-loop to represent a degree of 2 in an undirected graph.

easternerjx 2022-09-25

NCAE shows that the average score of the students is 80 and sd is 15. What is the percentile rank of a score of 84?

Krish Crosby 2022-09-25

Based on the effect size conventions, $d = 0.60$ is a
(a)small effect size
(b)medium effect size
(c)large effect size

basaltico00 2022-09-25

This might be a really simple question, but I just didn't find an answer from anywhere.
I'm teaching linear algebra to myself and in my study material I came upon notation that I just don't understand. I can't find an explanation for it from my material and it is hard to find on the internet as well it seems.
Example:
$U = L ((3, 2, - 6, 4), (0, 4, 1, - 5))$
What does the L() notation mean? U itself should be a subspace for $R^{4}$ . I would assume that those are vectors within the L().

Zachariah Norris 2022-09-25

How do you find the domain and range for $g (x) = \sqrt{x - 1}$ ?

Jazmyn Pugh 2022-09-25

Unable to understand correlation coefficient / auto correlation
Suppose I have a vector say:
[5 5 5 5 4 5]
then common sense says that there is a very high auto-correlation for the vector because it is more or less the same values. But when I try to calculate the auto-correlation coefficient, I'm getting a very low value(<0.3) for all lags. What does this mean? shouldn't it be higher because the series is very similar?Am I missing something?does correlatiom mean not similarity but similarity in rate(Rate of change)?

deiluefniwf 2022-09-24

The formula that I know for correlation coefficient
$\frac{\sum (x_{i} - \bar{x}) (Y_{i} - \bar{Y})}{\sqrt{\sum (x_{i} - \bar{x})^{2} \sum (Y_{i} - \bar{Y})^{2}}}$
If the only given values I have are $\sum x_{i}, \sum x_{i}^{2}, \sum y_{i}, \sum x_{i} y_{i}$ is it even possible to compute the correlation coefficient?

Kelton Molina 2022-09-24

Let $L$ be the tangent line to $y = \tan (2 x)$ at $(\frac{π}{2}, 0)$ . What is the $y$ -intercept of $L$ ?
(a) $0$
(b) $\frac{π}{2}$
(c) $- π$
(d) $1$
(e) $2$

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