For an F(2, 237) distribution, how should I determine a critical F value from a statistical table? Would I look up (2, 150) or (2, 300)? Should it simply be the closest degrees of freedom?
71540
Often people use programs to obtain p-values, but sometimes - for whatever reason - it may be necessary to obtain a critical value from a set of tables.   Given a statistical table with a limited number of significance levels, and a limited number of degrees of freedom, how do I obtain approximate critical values at other significance levels or degrees of freedom (such as with $t$, chi-square, or $F$ tables)?  That is, how do I find the values "in between" the values in a table?
111746
Suppose I have the minimum, mean, and maximum of some data set, say, 10, 20, and 25. Is there a way to:   create a distribution from these data, and know what percentage of the population likely lies above or below the mean   Edit:  As per Glen's suggestion, suppose we have a sample size of 200.
116793
I'm a mathematician self-studying statistics and struggling especially with the language.   In the book I'm using, there is the following problem:  A random variable $X$ is given as $\text{Pareto}(\alpha,60)$-distributed with $\alpha&gt;0$. (Of course, you could take any distribution depending on one parameter for the sake of this question.) Then a sample of five values $14$, $21$, $6$, $32$, $2$ is given.   First part: "Using the method of maximum likelihood, find an estimate $\hat{\alpha}$ of $\alpha$ based on [the sample]." This was no problem. The answer is $\hat{\alpha}\approx 4.6931$.  But then: "Give an estimate for the standard error of $\hat{\alpha}$."  What is meant by this? Since $\hat{\alpha}$ is just a fixed real number, I don't see in what way it could have a standard error. Am I to determine the standard deviation of $\text{Pareto}(\hat{\alpha},60)$?  If you think the question is not clear, this information would help me as well.
89719
I'm confused as to the correct formula for approximate degrees of freedom to use for Welch's t-test. Satterthwaite's (1946) formula is the most commonly cited formula, but Welch gave an alternative in 1947. I'm not sure which is preferable (or used by most statistical software).  Satterthwaite's formula: $$\frac{\left(s_x^2/n_x +s_y^2/n_y\right)^2}{(s_x^2/n_x )^2/(n_x-1)+(s_y^2/n_y )^2/(n_y-1)}$$  Welch's formula: $$-2+ \frac{\left(s_x^2/n_x +s_y^2/n_y\right)^2}{(s_x^2/n_x )^2/(n_x+1)+(s_y^2/n_y )^2/(n_y+1)}$$  References:   Satterthwaite, F.E. (1946). "An Approximate Distribution of Estimates of Variance Components". Biometrics Bulletin, 2, 6, pp. 110–114.  Welch, B.L. (1947). "The generalization of "Student's" problem when several different population variances are involved". Biometrika, 34, 1/2, pp. 28–35.
22448
Prove that t-distribution is close to normal distribution when degree of freedom is large.   $$f(x)=\frac{\Gamma[(n+1)/2]}{\sqrt{n\pi}\Gamma(n/2)}\left(1+\frac{x^2}{n}\right)^{-(n+1)/2}$$  using limit of density and Law of large number.  Anyone can help me with this question please? Thank you and appreciate your help!
35174
I am studying for my upcoming stats final exam and I come across this question:  Let X be a single observation from the uniform distribution:   $$f(x) = \begin{cases}       0 &amp; \text{otherwise} \\       \frac{1}{\theta} &amp;  0 ≤x ≤ \theta \end{cases}$$  Suppose we use $(X, 1.5X)$ as a confidence interval for $\theta$. What is the confidence level?  I have no idea how to start.  Since I am given a confidence interval, I believe I need to integrate the function from $X$ to $1.5X$, but I'm not sure what to do next, should I even integrate.  How can I solve this problem?
8183
Suppose we have i.i.d. $n$ observations $(X_1,X_2,...X_n)$ from a   population with density $$f_\theta(x)=\begin{cases}\theta x^{\theta-1}&amp;\text{ if }0\leq x\leq 1\\0&amp;\text{otherwise.}\end{cases}$$ Note that $\theta&gt;0$. If $T_n$ denotes the   maximum likelihood estimator of $\theta$ given that $n$ is the sample   size, then show that $$\text{var}(T_n)\stackrel{n\to\infty}{\longrightarrow} 0$$   This is the problem I am trying to solve: I computed $$T_n=\dfrac{-n}{\sum_{i=1}^{n}\ln(X_i)}$$ But I am stuck in finding $\text{var}(T_n)$ and showing that $\text{var}(T_n)\to0$ as $n\to\infty$.
110178
So if Pearson's Chi Squared Statistic is given for a $1 \times N$ table, then its form is:  $$\sum_{i=1}^n\frac{(O_i - E_i)^2}{E_i}$$  Then this approximates $\chi_{n-1}^2$, the Chi-Squared Distribution with $n-1$ degrees of freedom, as the sample size $N$ gets larger.    What I don't understand is how this asymptotic approximation works. I feel like the $E_i$'s in the denominators should be replaced with $\frac{s_i^2}{n_i}$.  Since that would give you $\chi_n^2 = \sum_{i=1}^nZ_i^2$, for $Z_i\sim n(0,1)$.  But of course this has $n$ degrees of freedom, not $n-1$, so clearly something else is going on.
54299
Regarding s, I am wondering why $1$% and $5$% seem to be the gold standard for "statistical significance". Why not other values, like $6$% or $10$%?  Is there a fundamental mathematical reason for this, or is this just a widely held convention?
84761
Assume ${{X}_{1}}$, ${{X}_{2}}$, ${{X}_{3}}$,...,${{X}_{n}}$  are i.i.d. samples from distribution with density f, and cdf F.  Let V=min( ${{X}_{1}}$, ${{X}_{2}}$, ${{X}_{3}}$,...,${{X}_{n}}$)  To find the minimum $${B}_{v}={\{V\ge{v}\}}=\{{X}_{1}\ge{v}\}\cap\{{X}_{2}\ge{v}\}\cap...\cap\{{X}_{n}\ge{v}\}$$  So, $$P(V\ge{v})=[1-F(v)]^{n}$$  The cdf is  $$F_{v}(V)=1-[1-F(v)]^{n}$$   The density is $${f}_{v}(V)=\frac{d}{dv}[1-F(v)]^{n}=nf(v)[1-F(v)]^{n-1}$$   To find the density, why did we use $[1-F(v)]^{n}$ instead of $1-[1-F(v)]^{n}$, shouldn't we used the cdf to derive the pdf?
113413
Let $X$ be a random variable that follows $Uniform(-1,2)$ distribution.   Then what will be the cdf of $X^2$?   What i did was $$P(X^2\le y)=P(X\le y^{0.5})= \int_{-1}^{y^{0.5}}(1/3)dx=(1/3)(y^{0.5}+1) $$ $0\lt y \lt 4$ But this cdf doesnt give the probability $0$ at $y=0$ also the pdf which i get doesn't integrate to 1 What is wrong can somebody explain?? Am i doing something wrong??
122046
What are the maximum-likelihood estimators for the parameters of Student's t-distribution? Do they exist in closed form? A quick Google search didn't give me any results.  Today I am interested in the univariate case, but probably I will have to extend the model to multiple dimensions.  EDIT: I am actually mostly interested in the location and scale parameters. For now I can assume that the degrees of freedom parameter is fixed, and possibly use some numeric scheme to find the optimal value later.
54864
Let $X_1,...,X_n$ be iid as $U \sim [0,1]$ with pdf $f(x) = 1(0 &lt;x&lt; 1)$. Determine the range distribution.   I have managed to determine the distribution of the minimum and the maximum, however I cannot/do not know how to determine the distribution of range.   Minimum:  $f_N = n(1-x)^{n-1}$ and maximum $f_M= nx^{n-1}$   I am aware that the calculation of range distribution is based on convolution: $f_R=\int f_{N,M}(x,x-r)dx$, However I cannot get the correct result, which according to my lecture slides is : $f_R= n(n-1)r^{n-2}(1-r)$  Could someone break this down for me ? My math background is not great
766
X has a Uniform distribution $U(0,1)$. I have to find the distribution of the variable $U =|X-1/3|$.  This is what i can get so far :  $F(U) \;{= P\{U &lt; u\} \\ = P\{| X-1/3 | &lt; u\} \\ \ddots}$  I know that i shall get something like $P\{-(U-1/3) &lt; X &lt; U -1/3\}$. Could anyone explain how to get the PDF and the CDF for this problem and explain how can i easily change this absolute values for other problems ( other examples $Z=1-|X|,~~G=|X|-1$ etc..).
37140
Does anyone have an intuitive explanation (no formulas, just words! :D) about the "$n-1$" instead of "$n$" in the unbiased variance estimator $$S_n^2 = \dfrac{\sum\limits_{i = 1}^n \left(X_i-\bar{X}\right)^2}{n-1}?$$
109917
I am stuck with a stats question..... help please     A television manufacturer is studying television remote control usage. One of the criteria they are measuring is the distance at which people attempt to activate the television set with the remote control. They have discovered that the activation distances are normally distributed with a mean distance of 6 feet and a standard deviation of 2.75 feet. If a remote’s maximum range is 10 feet, what percentage of the time will users attempts to operate the remote outside of its operating limit?
29256
Inspired by my other , I would like to ask how does one find the mode of a probability density function (PDF) of a function $f(x)$?  Is there any "cook-book" procedure for this? Apparently, this task is much more difficult than it seems at first.