# 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.

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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 ...
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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 ...
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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 ...
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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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### 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 \\...
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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 ...
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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 ...
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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 ...
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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 ...
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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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### 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 $8$?
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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 ...
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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 ...
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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 ...
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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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### 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 ...
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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 ...
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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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### 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 ...
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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 ...
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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 ...