Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [statistical-inference]

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

Statistical inference makes propositions about a population using data sampled from the population. To test a hypothesis about a population, a typical workflow is to select a statistical model of the process that generates the data and then deduce propositions from the model.

Statistical propositions include—

  • a point estimate, which is a particular value that best approximates some parameter of interest,

  • an interval estimate, for example, a confidence interval (or set estimate), which is an interval constructed using a data set drawn from a population so that, under repeated sampling of such data sets, such intervals would contain the true parameter value with the probability at the stated confidence level,

  • a credible interval, which is a set of values containing, for example, 95% of posterior belief,

  • rejection of a hypothesis, or

  • clustering or classification of data points into groups.

3930 questions
1
vote
2 answers

Statistical Interference

A controller of a store wishes to estimate the average amount spent each month by individuals holding credit cards to within ±€6 of the true amount. Based on previous experience it is known that the standard deviation is €21. Determine the sample…
Tom
  • 95
1
vote
1 answer

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 center. This greater spread is due to the extra…
J. Fischer
  • 11
1
vote
1 answer

Counterexample in Convergence in Distribution

I'm in my statistical inference course, and I've reached a problem related to convergence in distribution that I am slightly stuff on. Consider random variables $X_n,$ $Y_n$ who converge in distribution to $X$ and $Y$ respectively ($X_n…
Terry
  • 13
  • 2
1
vote
1 answer

Reversing Central Limit Theorem?

I have a question like this. A company manufactures light bulbs. The life time of bulbs is assumed to be normally distributed. The CEO claims that an average light bulb lasts $300$ days. A researcher randomly selected $64$ bulbs for testing and…
Padmal
  • 595
1
vote
1 answer

Normal Distribution sample mean and population mean?

Assume that house prices in an area are normally distributed with a standard deviation of \$ 20,000. A random sample of 16 houses is taken. What is the probability that the sample mean differs from the population mean by more than \$ 5,000. I can…
Padmal
  • 595
1
vote
1 answer

if the distribution of a positive random variable $X$ form a scale family, how can I show the distribution of $LogX$ form a location family?

if the distribution of a positive random variable $X$ form a scale family, how can I show the distribution of $LogX$ form a location family? It's obviously true, but I have no idea how to prove it, any help will be appreciated.
user133140
  • 885
1
vote
1 answer

Is the absolute value of $x$ a sufficient statistic to the continuous distribution which is symmetric about $y$ axis

Does the absolute value of $x$ is sufficient statistic to the continuous distribution which is symmetric about $y$ axis? e.g. the standard normal.
Jakoer
  • 315
1
vote
0 answers

comparing 2 datasets which have different distributions

I'm currently analysing two datasets. They report the same information, but in different ways. I am looking to draw comparisons between the way items fail in each of the datasets. In the first dataset we observe raw data: Time between failures of…
Jack Pedley
  • 156
1
vote
1 answer

Small question about the mean not being in its confidence interval

To study a certain characteristic about a population of people we take a sample of $100$ individuals. The $80$ percent confidence interval for the mean is $(0.9,1.1)$. Part I: Find the sample mean and standard deviation (easy). $\bar x = 1$ and…
George
  • 543
1
vote
0 answers

Why isn't the estimator of the square of a parameter the square of the estimator of the parameter?

Let $X_1,...,X_n$ be a sample from a distribution having as a p.d.f: $f(x) = \frac1{\theta} e^{-x/\theta}, x,\theta > 0$ and $0$ elsewhere. The maximum likelihood estimator of $\theta$ is $\bar{X}=\sum_{i=1}^n X_i /n$. Why isn't the maximum…
George
  • 543
1
vote
0 answers

Why is $nS_X ^2/\sigma ^2$ $\chi ^2(n-1)$, while the other is $\chi^2(n)$?

Suppose $X_1,...,X_n$ is a random sample from a distribution having $N(\mu, \sigma^2)$. What is the conceptual difference between: $$ \frac1{n} \sum_{i=1}^n (X_i - \bar{X})^2$$ and $$ \frac1{n} \sum_{i=1}^n (X_i - \mu)^2 ?$$ And why, when multiplied…
George
  • 543
1
vote
1 answer

Maximum likelihood estimator of $\theta$, $f(x;\theta) = 1/2 e^{-|x - \theta|}$

I'm given $f(x;\theta) = \frac12 e^{-|x - \theta|}$, $-\infty < x < \infty$ and $0 < \theta < \infty$. I want to find the maximum likelihood estimator of $\theta$. I found: $$\ln L(\theta; x_1,..., x_n) = -n \ln 2 - \sum |x_i - \theta|$$ Usually I…
George
  • 543
1
vote
1 answer

Statistical Estimation

There is an example problem in my book that doesn't explain how they got to this answer: sample: $217$ sample mean: $132.5$ standard deviation: $10$ "The $95$ part of the $68-95-99.7$ rule for Normal distributions says that $x$ is within $1.4$ cm of…
user246787
  • 11
1
vote
1 answer

Proof that the least square estimators are normally distributed

In my book I have the following proof showing that one of the least square estimators is normally distributed: $\hat\beta_i$ = $\frac {S_{xy}}{S_{xx}}$ = $\frac {1}{S_{xx}}\sum_1^n({x_i}- \bar{x})(Y_i -\bar{Y}) =…
user202542
  • 851
1
vote
0 answers

Statistical Modeling with the combination of two models

I'm having a modeling problem now. Assume we have discrete random variable Y and continuous random variables X and Z. First, we assume a logistic regression between Y and Z.(Assumption One) Also, we assume a regression model X~Y+Z. (Y is used as a…
Bing
  • 23
1 2 3
…
8 9