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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Minimal sufficiency and UMVUE in a pseudo-Normal distribution

I already asked a(n stupid)question about this problem here thinking I wouldn't have problems to continue it but I was pretty wrong. I'm finding several more problems trying to solve it. I'll try to ...
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2answers
31 views

A woman has n keys

A woman has n keys, of which one will open her door. After trying one she discards it and tries again if it does not work. What is the expected number of attempts needed? Its straight forward to see ...
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1answer
10 views

Nonparametric Skew of Data

Recently in my studies of statistics, I have come across the second skewness coefficient to determine the skewness of the set of data. The formula is given by: $$ \frac{3(\mu - \nu)}{\sigma}$$ ...
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12 views

function of independent random variables

I have following question: If $X$ and $Y$ are independent, then are $g(X)$ and $g(Y)$ independent as well, for any real function $g$?
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20 views

Finding probability of Type I & II error

Past studies have shown that 35% of drivers received parking citations while on vacation. Paula believes the actual percentage is higher. She decides to carry out a test with sample size n. Let Y be ...
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18 views

Hypothesis testing: does it have the highest frequency?

Let's suppose, for example, that a fisherman catches $N$ fishes, with a total of $r$ different species: $E_1$, $E_2$, ..., $E_r$. The fisherman caught $O_1$ fishes of species $E_1$, $O_2$ fishes of ...
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6 views

What measurement should be used to determine per unit capacity of a given volume ish question

I am reselling data packages and would like to add a sense of scale to the packages. Without disputing actual numbers, an email message is between (3-20)Kbytes, a photo 270kb, a web page ...
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31 views

Asymptotically normal but biased estimator

This is the problem 2.11 from Lehman book "Theory of point estimation" 2-nd edition. Construct a sequence $\{\delta_{n}\}$ of estimators of $g(\theta)$, satisfying $$ \sqrt{n}[\delta_{n} - ...
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18 views

Statistical Multiple Linear Regression Log Transformation

If for example we have a multiple linear regression as follows: $$hydrcarb=x_1+x_2tanktemp+x_3disptemp+x_4tankpres+x_5disppres+x_6tankpres^2+x_7dispres^2$$ And I am trying to do a backward ...
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21 views

Minimizing Type I error population proportion estimate

Past studies have shown that 35% of drivers received parking citations while on vacation. Paula believes the actual percentage is higher. She decides to carry out a test with sample size $ n $. Let $ ...
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1answer
26 views

Hypothesis testing for difference of two means

The weight of 11 schoolchildren was measured before and after six months on the proposed lunch plan. The weights before were: 132, 146, 135, 141, 139, 162, 128, 137, 145, 151, 131 The weights ...
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25 views

Maximum likelihood estimator in terms of $\sum \frac{(x_i-\mu)^2}{x_i}$

I'm trying to solve this problem Let X be a random absolutely continuous variable with probability density function $$f_{\lambda\mu}(x) = \sqrt{\frac{\lambda}{2\pi ...
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1answer
40 views

Chi-square test of independence: show that sum of squared standard normals has chi-square distribution

I'm studying the chi-square test of independence. According to my understanding, we first hypothesize independence between variables and consider them as being normally distributed. Then we go on to ...
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1answer
20 views

Correction factor for the variance

In an exercise they asked me: "Why could we use the following correction factor? $\text{varianceX} = \frac{n-1}{n}*\text{varianceY}$ What I said was basically, because the unbiased sample variance ...
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1answer
20 views

Statistics - Unbiased estimators

Let $X_1, X_2, \ldots, X_N$ be an i.i.d. random sample of Bernoulli random variables, with $\mathbb{P}(X_i =1) = p$ and $\mathbb{P}(X_i = 0) = 1 − p$. I'm confused as to why $1 − X^{-}$ is an ...
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27 views

Minimizing Unintegrable Exponential Function

I am trying to develop an algorithm which minimizes an unintegrable function. I don't have a strong mathematics background and am unaware of such strategies. My integral is of the following form: ...
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1answer
10 views

In hypothesis testing..

... we use two tail test and divide the area by $\frac{\alpha}{2}$ each. Why don't we divide it into some other propotion say $\frac{3}{4} \alpha$ and $\frac{\alpha}{4}$? Here $\alpha$ means type 1 ...
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18 views

Sampling distribution question with unknown n.

Suppose that 53% of the population of voters were in favor of fighting the global warming. If we wanted to conduct a random sample of size $n$ of voters, how many should I survey if I want the ...
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31 views

Expectations of squared sum question

I can't seem to figure out why these expectations turn out the way they do, I am currently studying about the Fisher Information. If $X_1,X_2,...,X_n $ are all iid Poission($\lambda$) , then going ...
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25 views

Is it necessary to normalize likelihood within an event space before further multiplication among events?

Say I have observed data, and parameters $A,B$: Parameter $A$ contains possible values: $a_1,a_2,a_3$ Parameter $B$ contains possible values: $b_1,b_2,b_3$ Now, assume I know the likelihood of ...
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1answer
98 views

Does color affect purchase decisions across items?

My question relates to Determine if being on honor roll gives an advantage in being assigned to the math class (I've seen similar questions involving hits on a webpage) which I believe should be ...
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19 views

Inferring Probabilities from relative frequencies

I have an question concerning the converse strong law of large numbers By the Converse Strong Law of large numbers, i mean the general principle (2) which is the converse of the standard strong law ...
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21 views

Random Variables and Statistic

I'm studying Statistical Inference by Casella and I'm confused with the definitions of random variable & statistic. So let we have the probability space $(\Omega, F, P)$ where $\Omega$ is the ...
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24 views

Deriving a formula for a confidence interval

Derive a formula for a $(1-\alpha)100\%$ C.I. for $\mu_x -\mu_y $ for data that has the following properties: A random sample $X_1,X_2...X_n \ are \ i.i.d ~N(\mu_x, \sigma^2 ) $ Another random ] ...
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32 views

How sample size affects confidence interval.

Suppose the weight of n primary one students has sample mean of 20KG. If n = 40, a certain percentage of confidence interval for the population mean is (15.5,24.5). Find the confidence interval if we ...
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16 views

F-test and T-test produce the same results

I am modelling a stochastic process by two different methodologies and I expect the results of each to be normally distributed with identical means and stdevs. To test that the distributions after ...
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35 views

Interval estimate to infer the population mean with a 95% confidence level

An industrial designer wants to determine the average time it takes for an adult to assemble a toy. 24 people were randomly chosen to assemble the toy and the time taken (in minutes) were as follows: ...
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12 views

Forecasting disputed transaction frequencies

Problem I would like to forecast credit card chargeback/dispute frequencies using historical dispute data I have recorded over time. The data I currently store includes: Disputed transaction date ...
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15 views

Intuitive explanation of requirement for achieving the Cramer Rao Lower Bound

this question relates to the requirement for achieving CRLB. I know that for a random sample $Y_1, \ldots, Y_n$, an estimator $U$ of $g(\theta)$ is MVUE (i.e. it is unbiased and also ...
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1answer
24 views

Reading P value from ANOVA table generated by R [duplicate]

I generated an ANOVA table in R and every boxes are in number except the P value shows "3.387e-05". What does that really mean?
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23 views

Definition of true density

I am reading a paper and it talks about true densities, I mean they talk about obtaining densities from data and later compare them with the true density I want to know how can I obtain a true density ...
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16 views

Required sub-chapters or more materials needed to learn Statistical Inference other than my textbooks

My school is using new curriculum now and chapter "Statistical Inference" appears in my textbooks. Now I'm at second level of senior high school. I have two books, each of them has own sub-chapters ...
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54 views

Kruskal Wallis - Effect size

I analyse 4 algorithms and 3 sets of metrics for each algorithm in which I apply the non-parametric Kruskal-Wallis test for each metric to detect any differences in performance between these ...
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Testing $H_0 : \mu_x \neq \mu_y $, in a company that markets two brands of latex paint.

A company markets two brands of latex paint regular and a more expensive brand that claims to dry an hour faster. A consumer magazine decides to test this claim by painting ten panels with each ...
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1answer
25 views

Subpopulations of an island (Bayes theorem?)

Help appreciated here. An island with 2 regions, I and II, has 4 types of individuals: AX, AY, BX and BY, for which we know their exact total nos. Here A-B-X-Y are simply traits, e.g., A=Male, ...
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1answer
15 views

Paired and unpaired data-Statistics/Hypothesis testing

I'm getting a bit confused about paired and unpaired data, for example in this question I don't understand how this data is paired. If it was the same steel pipes that were left uncoated first in ...
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19 views

how to find $X^2_{0.95,14}$, $P(X^2_{9} \geq y) = 0.99$ in a TI-84 calculator?

Can someone please show the steps in how to find $X^2_{0.95,14}$ in a TI-84 calculator? My professor does not want us to use a statistics table, but to know how to use the calculator in quizzes and ...
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10 views

Hypothesis Question

Consider the following hypothesis: $H_{0}:\mu\leq3000$ vs $H_{a}:\mu>3000$ A sample size $n$ must be decided so the risk of a type 1 error is at most 1%, and also so that if the value of ...
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43 views

What's in a name? (Sum of Squares)

I have always believed that in order to fully understand and appreciate the mathematical subtleties and ideas behind a certain concept is to understand the name given to it. For now, can someone ...
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117 views

Baysien Inference and error probablility

Let $\Theta$ be a Bernoulli random variable that indicates which one of two hypotheses is true, and let $P(\Theta=1)=p$. Under the hypothesis $\Theta=0$, the random variable $X$ is uniformly ...
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27 views

Interpreting what this means in a paper - significantly different at the .05 level?

I am having a hard time interpreting what something means in a paper I'm trying to get through. If you care, this is the paper: Gender Differences in the Effect of Education on the Slope of ...
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1answer
34 views

Bias of $\sigma^2$ estimator

I need to find the bias of $\frac{\sum(x_{i}-\bar{x})^2}{n+1}$ for $\sigma^2$. To do so, one must take its expectation but add and minus $\mu$ from the summation part so we can bring $\sigma^2$ into ...
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44 views

t-distribution and Degrees of freedom

Why t- distribution have n-1 degrees of freedom? I know that it is used when population variance is not known but what determines n-1
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How can we have $T_n \xrightarrow{\mathbb P_\vartheta} \vartheta$ if $T_n$ are defined on different spaces?

Here is how I understand the standard parametric model in statistical inference: We have a r.v. $X:\Omega \to \Psi$ which has some known to us distribution yet the exact parameter is unknown to us. ...
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1answer
40 views

When does the variance of a consistent estimator go to zero?

I came across the following statement (marked as true) in multiple-choice section of an old exam: The variance of a consistent estimator goes to zero with the growing sample size. As far as I ...
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15 views

Big O p Question about Eigenvalue of Random Matrix

Suppose $S_1, S_2, \dots$ are a sequence of random symmetric matrices in $\mathbb{R}^{d\times d}$. Suppose we know that $|\lambda_\max(S_n)| = O_p(b_n)$ and also that $|\lambda_\min(S_n)| = O_p(b_n)$ ...
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19 views

How can he assume SD of population equals to SD of sample means?

I'm referring to this KhanAcademy video: https://youtu.be/bekNKJoxYbQ?t=445. My question: How can he approximate the SD of the population to be equal to SD of the sample means? Isn't that SD of the ...
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39 views

showing that $E[(\hat\theta -\theta)^2] \lt Var(\bar X)=\dfrac{1}{n}$. [closed]

Suppose $X_1, X_2, \dots, X_n$ are i.i.d $N(\theta, 1),\theta_0 \lt\theta$ , Find the MLE of $\theta$ and show that it is better than the sample mean $\bar X$ in the sense of having smaller mean ...
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22 views

Conditional Probability/Expectation in the EM algorithm

I'm doing a study in which I measure data under a random censoring process. The observed data which may be interpreted as the lifetime of a subject, is denoted by $t$, with the censoring variable $c$ ...
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15 views

calculating the risk function $max(\bar{X},2)$ under Squared error loss function

suppose $X_1,X_2,\ldots,X_n$ be a random sample of $N(\theta,1), \theta>2$. how can I calculate the risk function $max(\bar{X},2)$ under Squared error loss function