Questions tagged [hypothesis-testing]
This tag is for questions on hypothesis testing in statistics, including questions about constructing or setting up a test, selecting an appropriate test for a particular application, and computing test statistics.
1,579
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About the statistics for the hypothesis test of the mean of two normal population
Let $X = (X_1, ..., X_n)$ be an independent sample from $N(\mu_1, \sigma_1^2)$ and $Y=(Y_1,...,Y_m)$ an independent sample from $N(\mu_2, \sigma_2^2)$. Consider the following hypothesis test given ...
- 1,021
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Probability inequality for a statistic with lower bound constraint, where McDiarmid's inequality is used in proof
I'd like to prove a probability inequality involving a statistic and a random variable as in the following statement. My questions are two-fold:
Is the following proof using McDiarmid's inequality ...
- 311
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30
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The theory behind Tukey's HSD test. [migrated]
I have not been able to find satisfactory explanations for the logic behind Tukey's HSD test.
However, the resources I have been able to find all concern it's use. It's supposed to make multiple ...
- 347
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Confidence Interval and Hypothesis Test Disagree?
I created a problem where I am having my students complete a hypothesis test and confidence interval for proportions to highlight the connection between the two. However, in doing the problem I am ...
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Determining type of statistical test to be used to compare mean hours of sleep for two groups [closed]
I am conducting a study on the sleeping habits of students in social sciences vs science. Specifically, I want to compare the means of hours of sleep between the two groups, both for the whole week ...
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23
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Is the chi squared goodness of fit test valid for the multinomial distribution if the outcomes are unlabeled?
I'm investigating Sketchy Dice Inc. for purportedly selling unfair dice with the following probability distribution:
\begin{align}
p(1)&=0.1 \\
p(2)&=0.3 \\
p(3)&= 0.2 \\
p(4)&= 0.15 \\...
- 474
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28
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Expectation and covariance of Gaussian vector given norm constraint
Let ${\bf{x}}\sim {\cal{N}}({\bf{\mu}},\sigma^2{\bf{I}}) \in \mathbb{R}^n$
I am wondering if I can find an analytical expression for the expectation and covariance of ${\bf{x}}$ given that $\frac{||\...
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Justifying data samples are from different distribution.
Let $x \in \{0,1\}^N$, and
\begin{align}
D &= \begin{bmatrix}
x_{1} \\
x_{2} \\
\vdots \\
x_{M}
\end{bmatrix}
\end{align}
So that $D \in \...
- 31
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Testing hypotheses about parameter function $\lambda^T\beta$
I have a hypothetical situation, where the parameter function $\lambda^T\beta=\alpha_1-\alpha_2$ for the model
$$y_{ij}=\mu+\alpha_i+\epsilon_{ij}, i=1,2, j=1,\cdots n_i$$
is estimated on two ...
- 113
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Hypothesis testing on the existence of a hypothesis (satisfying certain criteria)?
My understanding of probability and statistics is as yet not very good, so please correct me if I get anything wrong, terminologically or conceptually.
Suppose I have a set $X$, and a property (binary ...
- 263
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19
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Statistical test for comparing number of clusters in data
I am performing $K$-means clustering on a dataset consisting of $n$ observations and $d$ variables, and I'm trying to determine the optimal number of clusters. Is there a test that can determine the ...
- 2,984
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What is the distribution specify events happened in two-stage mechanism?
For a discrete random variable representing rare events, we can have $N \sim Poisson(\lambda)$, where $N$ represents the number of successes/failures in a given period. Suppose that there is a ...
- 101
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Inverting UMP tests to get Confidence Interval
Suppose X is an exponential random variable with mean $\mu$, i.e., $X ∼ \mu Exp(1)$.
Invert appropriate UMP tests of $H_0: \mu = \mu_0$ to find a 90% confidence interval for $\mu$ of the form $[∗,\...
- 85
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37
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Deriving Wald Statistic for Gamma Sample with known Alpha parameter.
Let $X_1, X_2, ..., X_n$ be a random sample from a $Gamma(\alpha, \beta)$ population. Assume $\alpha$ is known and $\beta$ is unknown. Consider testing $H_0: \beta = \beta_0$.
I'm trying to derive a ...
- 85
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Python Code for Hypothesis Testing
I was going through some programming assignment for detection and estimation theory.
But I do not know how to proceed.The question is as follows:
Consider the location testing with Gaussian errors ...
- 1
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49
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Show that one combination is more likely than others.
I have 2 variables shape X and size Y.
Shape $X$ has 4 states ($x_{0}, x_1, x_2, x_3$) and size $Y$ has 3 states ($y_0, y_1, y_2$).
The contingency table could look like this:
Y \ X
$x_0$
$x_1$
$x_2$
...
0
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0
answers
10
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Using a proportion test to see if the mean of two data sets are equal or not?
We have a normally distributed data set of size $n = 188$, with mean $\mu = 2913.29$ and standard deviation $\sigma = 697.5$.
The data set is split into two sets A and B of size $n_A$ = 95 and $n_B$ = ...
- 77
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12
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Optimal hyptothesis testing uses sufficient statistics?
In statistical estimation in quadratic risk for a model with a sufficient statistic, in seeking optimal estimators it suffices to consider functions of the sufficient statistic. This is an immediate ...
- 111
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19
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Difference in mean given distribution and true value
Let's say that I know that the theoretical distribution of some metric follows $N(\hat{\mu}, \sigma)$. However, somehow I discover that the true value of the metric is $\mu$. What test can I use to ...
- 253
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14
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When to correct significance level inside statistical tests?
I'm reading about error correction for the significance level when doing multiple tests. However, I don't get the sense behind this.
Let's make an example. I have a cohort of $10k$ people, about $4000$...
- 151
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The equality of variance test
I am facing a strange problem,
Imagine that you want to test whether the standard deviations of two populations are equal. Let us call the standard deviations of the first population $\sigma_1$ and ...
- 140
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13
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Statistical test for overall standard deviation given covariance matrix
I have a model that predicts values and also gives a standard deviation for the prediction. The standard deviation given depends on the input data for the model and thus is different for every ...
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answers
5
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Sampling from either a known or unknown Gaussian
Suppose I have two Gaussian distributions $N(\mu_0, \sigma), N(\mu_1, \sigma)$ with identical variance, where $\mu_0, \sigma$ are known and $\mu_1$ is unknown. Suppose I realize a sample $x$ from one ...
- 187
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18
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T-test for determining which algorithm is experimentally faster
I have written two algorithms to perform the Jacobi iterative method to simulate heat dissipation over a surface. I'd like to see if there is a significant difference between the run times of the two ...
- 167
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vote
1
answer
54
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two samples t-test and ANOVA
Suppose we want to know whether the effectiveness of two methods, methods A and methods B, is comparable. Suppose that the (scientific) criterion for determining whether the methods are identical is ...
- 140
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44
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Is there a way to identify points coming from specific normal distribution from a pool of points belonging to different normal distributions?
Suppose I have a pool of $n$ data points coming from different normal distributions. I know that there are $m_1$ many points generated from $N(\mu_1,\sigma_1^2)$ distribution, $m_2$ many points from $...
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Where could I find the answers of the exercises of "All of statistics" book by Larry Wasserman? [closed]
I am doing my Master 1 in computational and mathematical biology and I have a course "Decisional Statistics" where we study classical and Bayesian statistics.
I am studying from the book &...
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1
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30
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A question regarding the proof of a statistical test for a regression parameter
I have two questions regarding the proof of the following theorem:
Consider the regression model $Y=X\cdot\beta+\epsilon$, where $\epsilon\sim N(0,\sigma^2\cdot I_n)$ and X has full rank. Then, the ...
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Why is n=7 and not 8 in this $\chi^2$ goodness of fit test for Binomial Distribution?
In this problem I clearly see 8 observations. In the solution, in Step 3, why is $n=7$ instead of $8$?
To find $p$ why is the mean divided by $7$? Shouldn't it be divided by $8$?
- 701
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Finding the smallest value of $n$ in order to reject a null hypothesis (t-distribution)
In a t-distribution, the given info in an exercise are $\mu = 300$, $\overline{x} = 303$, the unbiased standard deviation $s_{n-1} = 2$ and the significance level $\alpha = 5\%$. The exercise asks me ...
- 2,367
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Chi-squared test is failing to invalidate null hypothesis for some reason
I'm mostly a coder so apologies in advance if my math notation / explanation isn't ideal. It's been a while since college and statistics also wasn't my strong suite even back then. Would appreciate ...
- 111
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Test between two alternatives for a gamma distributed variable $X$.
Let $X\sim \Gamma(p,\lambda)$, i. e. we have the density:$$f_{p,\lambda}(x)=\frac{\lambda^p}{\Gamma(p)}x^{p-1}\exp(-\lambda x)$$ where $p=2$ and $H_0: \lambda=1$ or $H_1: \lambda=2$. What is the test ...
- 2,122
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Measuring subset of $S^{n-1}$ under Gaussian measure for LLR test
Doing a LLR-test for iid $X_i\sim N(\mu,\sigma^2)$, $\sigma^2$ known, $H_0\colon \mu \leq 0$ vs $H_1\colon \mu >0$ I arrived to the following expression$$\mathbb{P}(\lambda>c\vert \mu)=\mathbb{P}...
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Statistical test used to compare outcomes of patients on 2 different wards
This audit's true numbers and details have been changed to abide by confidentiality standards.
An audit of the patients in a hospital of a certain specialty is carried out to assess whether you are ...
- 268
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28
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How to do Hypothesis testing?
I have two time series data.
Electric vehicle demand at a station (whole number time series) every hour called "Clstr 175"
Weather description (categorical time series) at the station every ...
- 237
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Questions about significance level and Type II error
I am learning tests of statistical hypotheses by myself. The book I used is Probability and Statistical Inference by Robert V. Hogg, etc.
I want to confirm some of my understandings are right or not.
$...
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Are there other controllers similar to p-value, q-value, and power?
According to the Wikipedia entry on q-values...
In statistical hypothesis testing, specifically multiple hypothesis testing, the q-value provides a means to control the positive false discovery rate (...
- 1,554
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Hypothesis Test: Normal vs Poisson distribution
Let's consider $H_0 = \{X\sim\mathcal N_{\mu, \sigma^2}\}$ and $H_1\{X\sim Poi(\lambda)\}$. I need to find a test $\varphi$ that has type $1$ and type $2$ errors of $0$.
I came up with $$\varphi = \...
- 594
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Finding the null & alternative hypothesis
Question: Two different companies (A and B) have applied to provide cable television service in a certain region. Let p denote the proportion of all potential
subscribers who favor the company A over ...
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86
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Chi-square issue
Chi-sq is supposed to be calculated $\sum_{i=1}^R \sum_{j=1}^C \frac{(O_{i,j} - e_{i,j})^2}{e_{i,j}}$. I thought there are two groups and it's even consistent with their chi-square estimate of which ...
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Failing to reject the Ho in normality test
Assume that I am using the Kolmogorov-Smirnov test to test for normality of my data. Then:
Ho: Data is normally distributed
Ha: Data is not normally distributed
In case the p-value is lower than ...
- 1
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Difference between the respondent's cholesterol level and maximum blood pressure. Is this a paired t-test?
This is probably an easy question, but wanted to confirm before I start doing the analysis for my intro stats course...
So a question I have to answer is as below:
Test the difference between the ...
- 3
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20
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How to detect outliers with Gaussian mixture models?
Assuming a linear time-invariant system (LTI):
$$x_t=Ax_{t-1}+Bu_t$$
$$y_t=Cx_t+w_t$$
where $x_t$ is the system output at timestamp $t$, $u_t$ is the input to the system at timestamp $t$, $y_t$ is the ...
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14
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Hypothesis Test for a Constrained Linear Regression
For a constrained linear regression:
\begin{aligned}
&\min_{\beta} \|y-X\beta\|^2_2 \\
& s.t. A\beta=b
\end{aligned}
Is there any way I can exam the significance for each parameters?
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Qeustions about UMP tests and their asyptomically equivalent tests
now I have a question in TSH 3th version: enter image description here
enter image description here
I have already proved that these tests are a.e. However, I feel difficulty to prove the proprtion (1-...
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Building KS table
Stimulated by the curiosity, I wanted to simulate a Kolmogorov-Smirnov table. Initially, I thought it is something that can be done easily in R, but obviously I am missing something. The critical ...
- 21
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Can any sample proportion less than 0.5 be used to support a claim that the population proportion is greater than 0.5?
I am currently reviewing Statistics using Triola's book and I came across the following question:
Can any sample proportion less than 0.5 be used to support a claim that the population proportion is ...
- 2,367
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Use of Neyman-Pearson and Likelihood Ratio
I have a confusion that I would like to help me solve. I understand that Neyman Pearson's lemma is used to find the hypothesis test with the powerful rejection region. However, this only applies when ...
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votes
1
answer
41
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Can I change the statement of null hypothesis?
I am just studying Hypothesis Testing for Difference between Two Population Mean
Here, I found that the Null Hypothesis is assumed that, the mean value of the both populations are same (i.e $\mu_1 = \...
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What is the observed value of the test statistic?
Really struggling with this practice quiz question. Could anybody walk me through how to get an answer?
Thanks!
