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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Hypothesis testing - UMP and unbiased tests

I need to get the proof for the following question but could not get the second part done. For the first part, I followed the definiation of UMP unbiased test and got βϕ∗ (θ) > βϕ(θ) for all θ ∈ Θ1 ...
Random Gamer's user avatar
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Detecting the change to a significant level

I have encountered a question about the change of random variables regarding cumulative effect. Here is a simplified version of the question. Suppose a random variable $c(t; \theta_{t})$ is defined ...
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Intuition behind statistical approach for Hypothesis testing

I have studied Hypothesis testing from thinkstats2 which does simulation to estimate p value. For e.g. In coin toss example, we run the simulation, to generate the test statistic i.e. Difference ...
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Computing the p-value of one sided hypothesis test

I'm studying this subject on my own and would just like a sanity test to see if I'm doing things correctly. Part a): we consider the test statistic $ \frac{\overline{X} - \mu}{S/\sqrt{n}}$ which at $\...
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An ideal statistical method to compare two sets of data: a 't-test', '2 sample Kolmogorov Smirnov', or 'cucconi test' [migrated]

I have a set of data points that represent the cardiac function of an animal's heart every quarter of a second, over a certain period. The data points were subjected to ZhangFit for baseline ...
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Continuity of the power function in UMPU testing for two parameter uniform distribution

Suppose $X_1 , \dots, X_n$ are i.i.d. $U(\theta_1, \theta_2)$, with $\theta_1 < \theta_2$. We want to test for $$ H_0: \theta_1 \leq 0 \quad \text{versus} \quad H_1: \theta > 0. $$ It can be ...
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Assess if occurences of a binary variale is different from $\alpha \cdot 100 \%$

I have a binary variable, $X_t$, which takes values $1$ or $0$. A value of $1$ indicates that an event happend at time $t$. I would like to statistically test whether we can say that the event happens ...
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Test equality of two covariance matrices

I am trying to test the equality of two covariance matrices based on articles from Srivastava, M.S. and Yanagihara, H., 2010 and Ishii, Yata and Aoshima 2016. I am not a statistician but a programmer. ...
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Why do we use the test for a normal mean when variance is unknown, even though we are given the variance?

Take the question: A teacher wants to see the effect of changing how reading is taught to primary school children. The children in Year $4$ take a reading test at the end of the year. In previous ...
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Estimation of std error using sample std dev. will result in diff. sample statistic distribution for diff. samples. Won't this affect our decision?

During hypothesis testing (say, comparison of sample mean to a hypothesised population mean), we calculate p-value using sample mean distribution. For constructing this sample mean distribution, we ...
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How to formally derive this formula for Kolmogorov-Smirnov two-sample test D-statistic

I'm trying to wrap my head around Kolmogorov-Smirnov two-sample test. So, Kolmogorov-Smirnov test uses the following D-statistic: $$D_{n_1, n_2} = \sup\limits_{-\infty < x < +\infty} \left|\...
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Neyman-Pearson Lemma, critical region

So, a bit of context first, I was studying how to construct the most powerful test using the Nyman-Pearson Lemma. The example went like this: given the normal distribution with known variance $\sigma^...
missing_name's user avatar
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Proof of Probability of Obtaining Mann Whitney U under the Null. [closed]

I am trying to understand how the Recursion Probability Formula of Mann Whitney U table under the null hypothesis is derived in this Paper, Equation 1 from section 4(given below). $$p_{n,m}(U) = \...
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For every abelian variety $A$ over a number field $K$ and for a prime number

For every abelian variety $A$ over a number field $K$ and for a prime number $p$ ,$$\begin{equation*} (-1)^{\operatorname{rk}_p (A/K)} = w_{A/K}. \end{equation*}$$ 2-parity conjecture holds for all ...
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What test can we use to compare the sample proportion of multiple dependent samples (> 2) for non-dichotomous data (> 2 categories)?

I am currently studying hypothesis testing for dependent two-sample (proportion). The crux for my question is this, what test does one use to compare the proportion of multiple samples for non-...
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Can t-tests be 1 sided? This example has confused what I thought I knew.

We first take a random sample of $8$ students, record their final score in $2018$, and check their final scores again in $2019$. $2018: 76, 73, 66, 95, 75, 78, 96, 93$ $2019: 75, 80, 70, 93, 81, 90, ...
Bill Cogn's user avatar
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Statistical inference for the integral equation

Consider a integral equation $$ \begin{aligned} \mathbb{E} \left[ Y|A \right] &=\mathbb{E} \left[ g\left( W \right) |A \right]\\ \int{yp\left( y|a \right) dy}&=\int{g\left( w \right) p\left( ...
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uniform i.i.d test consistent?

$X_k \sim U(0,L)$ i.i.d; $H_0: L=L_0 ~ H_1: L \ne L_0$; $L,a \in(0,1)$ is a test with the following power function: $P_n(L_0)= P( max_{k=1,..,n}X_k>L_0~\cup~ max_{k=1,..n}X_k<L_0*a^{1/n})$. How ...
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A hypothesis For $m \geq 1$, $l_j > 0$, and $x_j \in (-1,1)$

(Hypothesis) For integer $m \geq 1$, $l_j > 0$, and $x_j \in (-1,1)$, then the following identity, whether or not to be established: $$ \prod_{j=1}^m \text{Li}_{l_j}\left[ x_j \right] = \sum_{k=0}^...
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Trouble understanding a Bayesian update over multiple hypotheses

I have found the following formulation for Bayes' Theorem for multiple hypotheses: $P(h \mid e) = \Large {P(e \mid h) P(h) \over \sum_i P(e \mid h_i) P(h_i)}$ Suppose I have three hypotheses and three ...
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Multiple hypothesis testing and conditional probabilities

Assume we're doing multiple hypothesis testing. That means we have a number of possible ground truth distributions $P_1,..., P_m$ (assume discrete, for convenience), and we're drawing a dataset $X$ of ...
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Trouble finding most powerful test given hypothesis test and density $\theta e^{x - \theta(e^{x} -1)}$

Let $X_{1}, \dots X_{n}$ be a sample from the distribution with density given by: \begin{equation}p_{\theta}(x) = \theta e^{x - \theta(e^{x} - 1)},\end{equation} where $x > 0$ and $0$ otherwise. ...
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Arguing that the distribution is not normal

Assume someone claims that they sampled i.i.d. $x_1, \ldots, x_k$ (let's say $k=10$) from standard normal distribution $\mathcal{N}(0, 1)$. They claim that the sampled values are exactly $x_1=x_2= \...
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Trouble defining hypothesis-testing problem

Consider a person who claims to have favorable chances in a game in the sense that if you randomly draw one card from a set with as many red as black cards, said person has probability 0.6 of naming ...
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Hypothesis testing and equality in distribution

Let $(P,Q)$ and $(P',Q')$ be pairs of random variables and let $\phi, \phi'$ be the most powerful level $\alpha$ tests between $P$ and $Q$ and between $P'$ and $Q'$, respectively. Suppose that $\phi$ ...
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How to Determine if a Claim Should be Rejected in a Binomial Distribution

Context I ran into this problem in the book Fundamentals of Probability with Stochastic Processes (problem 5.1.7): "A manufacturer of nails claims that only 3% of its nails are defective. A ...
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UMP Test for exponential random variable

Let $X_i\sim \text{Exp}(\theta)$ for $i=1,\dots,n$ i.i.d with density \begin{equation*} f(x,\theta) = \begin{cases} \theta \exp(-x\theta ),& x\geq 0\\ 0,& x< 0. \...
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Using p value mathod in hypothesis testing

As we know there are two way test a hypothesis. One is critical value method and the other is p value method. I've discovered that its very difficult to calculate p value in hand besides there are ...
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What's the power of the following Hypothesis test?

This is the question with all information given: "What's the power of a hypothesis test with significance level 0.05, if the probability of picking 5 or more things from a box of 15 is larger ...
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Correct way to formulate and test T-distribution hypotheses

Suppose you have a sample of observations: sample = [-0.38, -0.35, -0.66, -0.45, -0.42, -0.09, -0.50, -0.37, -0.54, -0.32] The correlation between "...
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Test normality from histogram data

I have histogram data that I'd like to formally test for normality but I don't have the actual data points that generated the histogram. The histogram looks something like this: Is there a standard ...
Matías Santurio's user avatar
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Does this problem lead to an approach to hypothesis testing without memorizing test statistics?

I took an online probability course a while back and ran into a textbook example in which a bank needed to calculate what interest rate would yield a $1$% probability of taking a loss on the totality ...
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A question about the definition of p-values

In hypothesis testing, the definition of p value is the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is ...
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Statistical test for convergence of an increasing sequence

Given a data : a finite sequence of 10000 terms of increasing real numbers, each expressed as decimal form, up to 20th decimal place, Also suppose that the data was obtained while testing a conjecture ...
imida k's user avatar
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Two sample Z-Test vs T-Test

A study reports that freshmen at public universities work 10.2 hours a week for pay, on average, and the SD is 8.5 hours; at private universities, the average is 8.1 hours and the SD is 6.9 hours. ...
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Mathematical justification for number of bootstrap simulation reps

I'm currently conducting a bootstrap simulation for a difference of proportions hypothesis test (whether the proportion of Steph Curry's successful basketball shots from beyond the left side of the ...
johnjorgenson's user avatar
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Lack of proof for theorem 5.2 of Srivastava(2005)

Theorem 5.2 in Srivastava(2005) is about the distribution of a test statistic for testing whether the covariance matrix is diagonal or not. In the paper they gave the following result without showing ...
Student's user avatar
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Testing the memoryless property versus independence

I have following sequence of 0-1 values where 1 represents arrival of something, and each 0 and 1 are measured in equal time unit (e.g. every hour). Below is an example of such sequence ...
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$\alpha=\mathbb{P}(\text{test rejects }H_0|H_0)$ (definition of size of test)

From Wikipedia: In statistics, the size $\alpha$ of a test is the probability of falsely rejecting the null hypothesis/making a type $1$ error. For a simple hypothesis, $$\alpha=\mathbb{P}(\text{test ...
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False Acceptance Probability in a binary hypothesis testing

There are two hypotheses about the probability of heads for a given coin: $\theta=0.5$ (hypothesis $H_0$) and $\theta=0.6$ (hypothesis $H_1$). Let $X$ be the number of heads obtained in $n$ tosses, ...
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P value and rejection

Could someone please explain the logic of the reasoning behind why we reject the Ho if the alpha exceeds the p value What I understand is that alpha is the probability of making a type 1 error if Ho ...
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Do you not use the t-table for hypothesis tests with n < 30? Has my professor made a mistakes in the provided answers?

Has my professor supplied incorrect solutions to this Hypothesis Testing excercise? (undergrad statistics class) I have my exam tomorrow, so help would be massively appreciated! Here's the original ...
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Statistical Test - Multiple Reviews of the Same Sample

I work in a field that assess applications for benefit. We are interested in comparing outcomes between client groups to identify potential bias. We plan to anonymizing applications to remove factors ...
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Hypothesis testing in linear regression model for 2 people

I have been given yearly salaries for two people (A and B) for $40$ years. How could I use linear regression to test whether the salary of the first one rises faster than salary of the other one? I ...
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Hypothesis testing involving linear regression model

Suppose I have range of x and y values which I have produced a linear regression model from. How would you answer this question,"Is the mean response (predicted value) for the y value at x = 12 ...
liam song's user avatar
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GLRT for a one-sided composite hypothesis for $N(\mu, \sigma^2)$, $H_0 : \sigma \leq \sigma_0$

This is sort of related to (different hypothesis though): GLRT statistic for composite normal hypothesis, two unknowns GLRT statistic for composite normal hypothesis, two unknowns Problem I am ...
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Best estimate of parameter in a Chi-squared test causes us to reject $H_0,$ but another value of the parameter causes us to accept $H_0$.

For a $\chi^2$ goodness-of-fit test where a parameter is estimated (for example, $p$ is estimated if you're testing to see if the Binomial distribution is a good fit to the data), is it true that if $...
Adam Rubinson's user avatar
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Finding the GLRT for a one-sided hypothesis using uniform distribution

The Problem Let $X_1, \dots, X_n \sim U(\theta, 5)$ where $0 < \theta < 5$ with pdf $$ f(x;\theta) = \dfrac{1}{5-\theta} \quad \theta < x < 5 $$ Find the generalized likelihood ratio test ...
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Convergence of series of random variables - NIST statistical tests

I want to understand NIST Statistical Test Suite tests. Here (https://arxiv.org/pdf/nlin/0401040.pdf) you can read a derivation of spectral test. There are considered two random variables: $$ \begin{...
kamm's user avatar
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What post hoc test to use after Fisher's exact test?

I performed Fisher's exact test on a 4*2 table in SPSS, and I got a significant difference (P= 0.010) and I wonder what is the post hoc test to use following that? is it the adjusted standardized ...
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