The area of statistics that focuses on taking information from samples of a population, in order to derive information on the entire population.

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Inferential Statistic -F-ration

If there is no treatment effect, we can expect the F-ratio to be (a) some value between +1 and -1 (b) 1.0 (c) some random valuable (d) 0 (e) cannot be determined
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18 views

hypothesis testing on exponential family

In class, the professor did present the next problem: Suppose you have a random sample $x_1,...,x_n$, but is unknown if the original distribution of the sample is gamma or exponential. You also have, ...
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30 views

Clarifying the assumptions about a paired t-test

I've wrote my question in red ink (see links). There are two questions that I have. Primarily I want to know why they concluded that "there is some evidence that there is some difference in mean ...
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1answer
26 views

Hypothesis test to justify a claim

So I have a question regarding hypothesis tests where i have to justify a claim with statistical evidence. It is as follows: The average number of accidents in previous years in a city has been 15 ...
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45 views

Determine who is the best seller

The numbers below show the number of lollipops Betty and Sharon each month for a total of 12 months or a year. Using the data and plot below, can you determine who is the bestseller? Would it be ...
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1answer
24 views

How to estimate the max of a population using the normal distribution equation on a small sample

I recently watched a documentary on Mathematics. In the show they managed to estimate the weight of the largest fish that the fisherman was likely to of ever caught in his career just by analysing one ...
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23 views

How to calculate survey bias due to preference for first answer?

I was recently given the results to a survey in which participants chose answers to questions they would be likely to randomly answer, and in which the survey population is known to have a preference ...
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8 views

Convergence of Sample Moments and Weak Law of Large Numbers

Given $\{X_i\}_{i=1}^n$, sequence of indepedent random variables such that $E[X_i]=\mu$, prove: $$a_r=\frac{\sum X_i^r}{n}\xrightarrow{p}E[X^r]$$ and prove: $$m_r=\frac{\sum ...
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36 views

How to formulate and test this statistical hypothesis

first of all I'd like to clarify that my biggest problem with this topic is probably my inability to formulate it correctly, maybe the answer is found trivially on the internet but I'm not able to ...
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16 views

Hypothesis testing - comparing data with a function

I am a beginner in statistics. My problem is as follows: I have a set of $N$ observations $\{(x_i,y_i,e_i)\}$, where $e_i$ is the error in the $i^\mathrm{th}$ observation. The individual observations ...
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2answers
40 views

An Estimator Based on Exponential RVs

Let $X_1$, $X_2$, $\cdots$, $X_n$ be $n$ random variables independently sampled from the exponential distribution $\text{exp}(1)$. Suppose $k \leq n$, and $X_{(k)}$ is the $k$-th order statistic, ...
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29 views

Interpter P-value. Is the following statement true or false, and where is the mistake?

I have the following question, Statement: A given exercise has the p-value of 0.08 and my alpha is 5%. The exercise was using a linear regression model to predict some future value ...
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39 views

Can the correlation of a random variable $X$ and $g(X)$ be $0$?

Question: Can the correlation of a random variable $X$ and $g(X)$ be $0$? My attempt: I don't believe it can because they are dependent by definition therefore $Cov(X,g(X)) \ne 0$ which means the ...
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63 views

Deriving Joint probability density functions

Question: Let X and Y be two continuous random variables with joint probability density function $$f(x,y)=\begin{cases}\frac{1}{2} & \text{if} \ \lvert x \rvert + \lvert y \rvert \le 1 & \\ 0 ...
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1answer
23 views

$U$ as a random variable? And what's with this integration becoming $1$?

I'm confused about two things. For the set up, I am told that Likelihood function is $L(\theta)=\Pi f(y_i;\theta)$ for a distribution with pdf $f(y;\theta)$. Log likelihood function is ...
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12 views

Horseshoe estimator posterior

Suppose given the Horseshoe estimator: $Y|\beta,\sigma^2 \sim N(X\beta,\sigma^2 I)$ $\beta|\sigma^2,\tau_{1}^2,...,\tau_{p}^2 \sim N(0,\sigma^2 D)$ $\tau_{j} \sim C+(0,1)$ $\sigma^2 \sim \pi ...
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1answer
17 views

how to find AIC values for both models using R software?

I'm studying survival analysis. I estimated both Cox regression model and Buckley&James regression model. In order to determine which model is better for my dataset, I used Akaike Information ...
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24 views

Using head-to-head results and Bayes' Theorem to modify predictions of sport/game contests that are initially derived from Elo-type ratings

I am working on an extension of the Glicko2 rating system to use in predicting the outcome of sport/game contests that uses the actual head-to-head results of previous meetings of competitors to ...
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1answer
29 views

Determining the degree of freedom for a $\chi$-squared test

I have read that the degree of freedom is calculated by subtracting $1$ from the number of states a random variable can be in. I am performing a goodness of fit test on a $64\times 32$ matrix where ...
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2answers
55 views

Help: SPSS and Data Interpretation of Voters. Republican vs. Democrats (1993 election)(Almost finished)

Hello everyone, I am Julieta this time I get stuck in the following exercise. It is a statistical analysis of pools, the statement is quite long I will try to keep it short and put some links. Note: ...
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1answer
9 views

Comparing expected counts to observed counts

The question is as follows: A restaurant offers 7 different dishes and predicts the dishes will be ordered in the following proportions: 1 (25%), 2 (20%), 3 (10%), 4 (15%), 5 (5%), 6 (6%), 7 (15%). ...
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1answer
15 views

$MLE$ of a multinomial. [CORRECTED]

It might be a very silly question, but I just can't figure it out. Let $X_1,...,X_n$ be random variables with pmfs: $$f(k,p)= \begin{cases} p_1, \hspace{0.5cm} \text{if } k=a\\ p_2, \hspace{0.5cm} ...
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22 views

Existence of asymptotic variance for an estimator when it doesn't converge to normal distribution.

The definition of an asymptotic variance says: For sequence of estimators $\mathbf{U}=(U_1, U_2,\ldots)$, where: $U_i=U_i(X_1,\ldots,X_i)$, if for a sequence of constants $\{k_n\}$: ...
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How to find n=# of samples

My question is what is the equation to find n if you know standard deviation, and the CI of 95%. Like what is the equation for n?
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Sample Estimate of the Mean of a Population Lying in [a,b]

Suppose I have a population of $N\ge 1$ real numbers, all known to lie in the real interval $[a,b]$, where $a,b\in\mathbb{R}$, $a\le b$, and $a$ and $b$ are known values. I know nothing else about the ...
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22 views

Two sample testing, when all the parameters are unknown

Suppose $(X_1,...,X_p)$ is a sample of independent $\mathcal{N}(μ_1, σ_{1}^2)$ random variables, and $(Y_1, . . . , Y_q)$ is a sample of independent $\mathcal{N}(μ_2, σ_2^2)$ random variables. If all ...
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12 views

Existence of asymptotic variance for an estimator when it doesn't converge to normal.

The definition of an asymptotic variance says: For sequence of estimators $\mathbf{U}=(U_1, U_2,...)$, where: $U_i=U_i(X_1,...,X_i)$, if for a sequence of constants $\{k_n\}$: ...
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1answer
42 views

Asymptotic variance of MLE of normal distribution.

I am trying to explicitly calculate (without using the theorem that the asymptotic variance of the MLE is equal to CRLB) the asymptotic variance of the MLE of variance of normal distribution, i.e.: ...
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Comparing definitions of limiting and asymptotic variances - what is the intuition behind?

In Casella's inference, it says: Definition 10.1.7: For an estimator $T_n$, if $\lim_{n\to \infty} k_n Var T_n = \tau^2 < \infty$, where $\{k_n\}$ is a sequence of constants, then ...
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Relationship between 0-1 Loss and Type I and II error in Neyman Pearson

In the context of hyphotesis test $$H_0:\theta\in \Theta_0$$ $$H_1:\theta\notin \Theta_0$$. Find the relationship between the 0-1 loss defined by $$L(\theta,\delta)=1-\delta \theta\in\Theta_0$$ ...
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38 views

Clarification on tests for independence and homogeneity?

If the null hypothesis for a test of independence is true, what distribution does the test statistic have? Would it still follow a chi-square distribution, or a normal distribution? In a ...
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20 views

Can't figure out the correct degrees of freedom for my goodness of fit

I am trying to solve the following question: The number of goals scored by a certain football team was recorded for each of 100 matches and the results are summarised in the following table. ...
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1answer
37 views

why equal variance assumption is necessary in T-test

So I generally understand the basis for the t-test: i.e. you take advantage of the fact that you can make $\bar{X}-\bar{Y}$ standard normal: $$Z = \frac{((\bar{X}-\bar{Y})-(\mu_X-\mu_Y))}{\sigma ...
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Desirable properties of statistical estimators?

What are some of the properties that people will consider when designing a statistical estimators? For example, unbiasedness and sufficiency are some of the factors considered. Please give some ...
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Drawing conclusions from given data as percentiles

For a problem I want to determine the number of errors. I have data from two observers, let's say observer A and B. This data is given as $5\%$, $50\%$ and $95\%$ percentiles. Observer A: ...
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1answer
71 views

Difference beteween ANOVA and ANCOVA

In the context of using only experiment data for ANOVA analysis, ANCOVA offers post hoc statistical control. Is this a valid conclusion and why?
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Understanding the shape of T distributions

I'm trying to understand why a T distribution with a small sample size has fatter tails and what this means. My textbook says "...t distributions have more probability in the tails and less in the ...
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Calculate calculate $E(\frac{1}{\bar X})$, Var$(\frac{1}{\bar X})$

Consider n i.i.d. observations from a Poisson (λ) distribution. Suppose $\bar X =\frac1n\sum_{i=1}^n X_i.$ How do I calculate Var$(\frac{1}{\bar X})$ or even $E(\frac{1}{\bar X})$ for that matter. ...
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Multilinear loss in Uniform-Exponential model

Let a prior $\pi(\theta)=\frac{1}{3}(\mathbb{I}_{[0,1]}(\theta)+\mathbb{I}_{[2,3]}(\theta)+\mathbb{I}_{[4,5]}(\theta))$ and $f(x\mid\theta)=\theta e^{-\theta x}$. Taking the multilinear loss ...
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1answer
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using the Cramer-Rao bound to find the approximate variance of a bernoulli trial

Given you have an independent random sample $X_1, X_2,..., X_n$ of a Bernoulli random variable with parameter $p$, estimate the variance of the maximum likelihood estimator of $p$ using the ...
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22 views

What is the standard deviation of the sample distribution for the sample standard deviation?

I know that for the sample distribution for the sample mean given a large sample or a normal underlying distribution, the mean of the sample distribution is the population mean of the underlying ...
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1answer
40 views

rigorous statistics book recommendations

I am learning statistical inference by myself, I have skim through a few books like Casella Hoggs and I find it omitted lots of details, for example, they didn't introduce the conditional expectation, ...
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What statistical tests should I perform to determine significance in differences between control, treatment, and sham groups?

Say I have 3 groups of scores: one control, one treatment, and one sham. I want to determine whether there is a significant increase (or decrease) in the mean scores with the treatment compared to the ...
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Hypothesis testing using proportions

Suppose that you interview 100 exiting voters about who they voted for governor. Of the 100 voters, 55 reported that they voted for the Democratic candidate. Is there sufficient evidence to suggest ...
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Hypothesis testing and the size of critical region

I'm studying stats in order to obtain a certificate and I'm struggling with this question. I've searched a lot and I really didn't understand how to procede in this question. I even don't know how to ...
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84 views

How to solve an integratation involved an unknown function?

Can anyone have any suggestions how to solve this equation for $w_i$, that is, what is the solution of $w_i$? $$ \int_0^\infty e^{\Phi^{-1}(w_i)ε_i}P(r_i│ε_i )f(ε_i )dε_i=δ $$ Where, $f(ε_i)$ is the ...
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2answers
28 views

How can I determine the best relationship for 3 variables, given several data points?

What is the best way to determine the relationship for three apparently related variables? The relationship does not appear to be linear, and may follow a combination of non-linear functions. I have ...
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Bayesian Inferences: Finding Posterior HPD Interval

I am currently working with Beta-Bernoulli and Beta-Binomial models. I have been searching around for the specific steps in obtaining the Posterior HPD intervals for both. Does anyone know how to find ...
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Finding the Cramer-Rao Lower Bound

Given the probability density function $$f(x; \theta) = \frac{ \left(\ln(\theta)\right)^{x}}{\theta x!}, \quad x = 0,1,\ldots ; \theta > 1$$ and $0$ otherwise, find the Cramer-Rao Lower Bound ...
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Transformation of continuous random variables

Let's say that $Y = \log T = \alpha + \sigma W$. I know that If $W$ has logsitic distribution, the $T$ will have the log-logistic distribution. Also, if $W$ has the standard normal distribution then ...