 # Get help with high school statistics and probability problems

Recent questions in Statistics and Probability Marcelo Maxwell 2022-09-27

### What is the z-score of sample X, if ? 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 $\frac{m}{gcd\left(m,x\right)}$ distinct elements in the set $\left\{ax\phantom{\rule{0.444em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}m\right):a\in \left\{0,...,m-1\right\}\right\}$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 $\varphi$ 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 systemI came through this problem while studying for an exam in control systems:Consider the following discrete time system$\stackrel{\to }{x}\left(k+1\right)=A\stackrel{\to }{x}\left(k\right)+b\stackrel{\to }{u}\left(k\right),\phantom{\rule{thickmathspace}{0ex}}\stackrel{\to }{y}\left(k\right)=c\stackrel{\to }{x}\left(k\right)$where $b=\left(0,1{\right)}^{T},\phantom{\rule{thickmathspace}{0ex}}c=\left(1,0\right),\phantom{\rule{thickmathspace}{0ex}}A=\left[\begin{array}{ccc}2& & 1\\ 0& & -g\end{array}\right]$ for some $g\in \mathbb{R}$Find a feedback regulation (if there is any) of the form $u\left(k\right)=-K\stackrel{^}{x}\left(k\right)$ where $\stackrel{^}{x}\left(k\right)$ 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 $e\left(k\right)=\stackrel{\to }{x}\left(k\right)-\stackrel{^}{x}\left(k\right)$ go to zero after a few finite time. layout the kingdom observer and the block diagram.My methodit is clean that the eigenvalues of the machine are ${\lambda }_{1}=2,{\lambda }_{2}=-g$ (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$\left[\stackrel{\to }{x}\left(k+1\right)\phantom{\rule{thickmathspace}{0ex}}\stackrel{\to }{e}\left(k+1\right){\right]}^{T}=\left[\begin{array}{ccc}A-bK& & Bk\\ O& & A-LC\end{array}\right]\left[\stackrel{\to }{x}\left(k\right)\phantom{\rule{thickmathspace}{0ex}}\stackrel{\to }{e}\left(k\right){\right]}^{T}$With characteristic equation$\chi \left(z\right)=|zI-A+bK|\phantom{\rule{thickmathspace}{0ex}}|zI-A+LC|={\chi }_{K}\left(z\right){\chi }_{L}\left(z\right)$Also consider$K=\left[\begin{array}{ccc}{k}_{1}& & {k}_{2}\\ {k}_{3}& & {k}_{4}\end{array}\right]$and let $a={k}_{1}+{k}_{3},\phantom{\rule{thickmathspace}{0ex}}\beta ={k}_{2}+{k}_{4}$Then ${\chi }_{K}\left(z\right)=\left(z-2\right)\left(z+g+\beta \right)+a$.So we can select some eigenvalues inside the unit circle and determine $a,\beta$ in terms of g. Choosing e.g. ${\lambda }_{1,2}=±1/2$ we get $a=3g+33/8,\phantom{\rule{thickmathspace}{0ex}}\beta =9/4-g,\phantom{\rule{thickmathspace}{0ex}}g\in \mathbb{R}$QuestionsI 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 $|{\lambda }_{1,2}|<1$?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 InterpretationLet X=(1,2,3,...,20). Suppose that $Y=\left({y}_{1},{y}_{2},...,{y}_{20}\right)$ with ${y}_{i}={x}_{i}^{2}$ and $Z=\left({z}_{1},{z}_{2},...,{z}_{20}\right)$ with ${z}_{i}={e}^{{x}_{i}}$. Pearson correlation coefficient is defined by formula$\rho \left(X,Y\right)=\frac{\sum _{i=1}^{20}\left({x}_{i}-\overline{x}\right)\left({y}_{i}-\overline{y}\right)}{\sqrt{\left(\sum _{i=1}^{20}\left({x}_{i}-\overline{x}{\right)}^{2}\right)\left(\sum _{i=1}^{20}\left({y}_{i}-\overline{y}{\right)}^{2}\right)}}$If $\rho \left(X,Y\right)=1$, we can say that X and Y have a linear correlation. If $0.7\le \rho \left(X,Y\right)<1$ then X and Y has a strong linear correlation, if $0.5\le \rho \left(X,Y\right)<0.7$ then X and Y has a modest linear correlation, and if $0\le \rho \left(X,Y\right)<0.5$ then X and Y has a weak linear correlation. Using this formula, we get $\rho$(X,Y)=0.9 and $\rho$(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 $\rho$(X,Z)=0.5 indicate the modest correlation coefficient, what is another intepretation of this value? What is the difference between $\rho$(X,Y) and $\rho$(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 termThe coefficient of x in the expansion of $\left(2x+4{\right)}^{5}$640512025601280 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:$\sum _{i=1}^{k}P\left({A}_{i}^{c}\right)$ Medenovgj 2022-09-26

### Meta-analysis examine:a. I don't knowb. Standard deviationc. Effect sizesd. 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 $n×n$ 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:$U=L\left(\left(3,2,-6,4\right),\left(0,4,1,-5\right)\right)$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 $g\left(x\right)=\sqrt{x-1}$? Jazmyn Pugh 2022-09-25

### Unable to understand correlation coefficient / auto correlationSuppose 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 $\frac{\sum \left({x}_{i}-\overline{x}\right)\left({Y}_{i}-\overline{Y}\right)}{\sqrt{\sum \left({x}_{i}-\overline{x}{\right)}^{2}\sum \left({Y}_{i}-\overline{Y}{\right)}^{2}}}$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