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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Statistics: Hypothesis testing to find p-value [migrated]

I am currently writing a thesis in machine learning and I am trying to use t-test to show that my model is better than the current state of the art. The dataset I used has 16 different tasks and each ...
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Using hypothesis testing to check if a mean of a sample is bigger than the other sample.

Basically I am asked if people over age $35$ spend more money on average than people less or equal to age $35$ with significane level $0.05$. And I have a sample of the population that I can use for ...
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generic constant and Student d.f.

What is a generic constant $k$ in the context of significance of the correlation coefficient ? Please see $(11.4.10)$ the snippet below:
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Chi-square GoF test

Joe likes coin tricks. In one of his tricks, he flips a coin $3$ times. Let $X$ be the number of heads that appear. Joe will repeat this trick so that we have $n$ samples from $X$. Although Joe doesn'...
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Hypothesis testing and critical regions

Suppose $X$ follows a chi-squared distribution with $r$ degrees of freedom, where $r \in \mathbb{Z}^+$ is unknown. Let $X_1, X_2, ..., X_n$ be a random sample of $X$. (i) Suppose we have the following ...
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Hypothesis tests two means

For a random person selected from a population, let $X$ be their Instagram followers and $Y$ be their Facebook followers. We randomly sample the followers of $n$ people, $(X_1, Y_1), ..., (X_n, Y_n)$. ...
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Testing hypotheses proportions

John is an avid soccer player. Last season, he scored a goal $50$ times out of $200$ attempted shots. This season, he will have $200$ attempts. (i) Using John's previous statistics, determine the ...
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Calculating the p value for the sign test

I am utterly confused at calculating the p value for both single sampled and matched pair sign tests. Suppose we have a sample size of 12 and 4 values are below our hypothesized median of $m_x$ and 8 ...
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How to check if a distribution follows a power law?

I have a distribution of the node degrees to their fraction (%) in a network, which may contain 10 to 100 values, all between 1 (max) and near zero. I would like to see if it follows the Power law ...
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Critical value: Testing two normal samples for equality of population variances. [closed]

To test the hypothesis that two population variances are equal, a random sample of size 13 was selected from the first population, and a random sample of size 21 was selected from the second ...
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Hypothesis testing for proportions from three populations

I have following problem : 200 people were surveyed asking about their favorite color. 80 of them said red, 85 blue and 35 said yellow. Verify following hypothesis : proportion of people liking red, ...
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The semantics of rejecting the null hypothesis

My professor said that "You should never accept the null hypothesis, instead you state there is insufficient evidence to reject the null hypothesis" and that led me to a loophole of ...
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Variance of Pearson's chi-squared statistic

Let $\nu=(\nu_1,\ldots,\nu_r)$, $\sum_j \nu_j=n$, be multinomially distributed with parameter vector $p=(p_1,\ldots,p_r)$ and let $$ \chi^2 = \sum_{j=1}^r X_j^2,\qquad X_j:=\frac{(\nu_j - np_j)^2}{...
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Hypothesis test with multiple means?

In a certain genetic experiment, two different varieties of a certain species are crossed and a specific characteristic of the offspring can occur at only three levels A, B, and C, say. According to a ...
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Finding the $p$-value of a test with exponential data

My problem Assume that $X$ is exponentially distributed such that $X\in\operatorname{Exp}(\lambda)$. Let $H_0$ be $\lambda=3$. We want to test $H_0$ against the alternative $H_1$: $\lambda=1$ and ...
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Small sample hypothesis testing with $p$-value

I'm struggling to figure out how to approach this problem: A tailor thinks that about $75\%$ of customers are satisfied with service. A random sample of $15$ customers, $8$ said they were satisfied. ...
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Finding a suitable p-value from a pdf

Consider a distribution defined by a probability density function which has the form A single value y is observed from this distribution. Derive the formula for p, a suitable p-value for testing the ...
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39 views

Which is the rejection region for a hypothesis test with a two-sided alternative and significance α=0.05? [closed]

I need help with this problem, which of a,b,c,d is correct? An influence of a new lubricant is investigated by hypothesis testing. Under the test assumptions, the test statistics are distributed ...
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p-value based on quantile of standard normal distribution for mean inequality hypothesis

Let $z$ be a quantile of standard normal distribution. Find p-value for $z=1.89$ and $H_a : \mu>\mu_0$. The way I tried solving this was by getting the area right of $x=1.89$, which is $0.02938$ ...
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The mean IQ of a sample of 1600 children was 99. Is it likely that this was a sample from a population with mean IQ 100 and standard deviation 15?

I was solving this question: The mean IQ of a sample of 1600 children was 99. Is it likely that this was a sample from a population with mean IQ 100 and standard deviation 15? like this: Here level ...
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P-value changes based on removed variable?

Problem Setting: Assuming that we have $p$ variables $x_1, x_2, ..., x_p$ and a response vector $y$ with length $n$, and the p-value is associated with the F-test with the null hypothesis $H_0: \...
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T-test and F-test in Multiple Linear Regression

In simple linear regression, $$ y = \beta_0 + \beta_1X_1, $$ the T-test for $\hat{\beta_1}$ is $$ H_0: \beta_1 = \beta_1^0 \quad \text{and} \quad H_A: \beta_1 \neq \beta_1^0, $$ where $\beta_1^0 = 0$, ...
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How to show $||Y-\hat{Y}_H||^2 = ||Y-\hat{Y}||^2 + ||\hat{Y} - \hat{Y}_H||^2$

Problem Setting: If we have a testable hypothesis with the form $A\beta = 0$ where $A$ is an $q \times p$ matrix and $\beta$ is an $p \times 1$ vector in a linear model, and we define that $\hat{Y} = ...
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Two-variable hypothesis testing

I have a problem that I am struggling with: Suppose $X$ and $Y$ are independent normally distributed random variables with respective unknown means $\mu _x$ and $\mu _y$ and respective unknown ...
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Computation and Visualisation of Type I and Type II Error

[Undergraduate : Introductory Level Statistics - Hypothesis Testing] There is a general consensus that 80% of all economists believe inflation will increase for 2018. Call this proportion the null ...
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Calculating $(\bar{Y_1}-\bar{Y_2})^T(S_p)^{-1}(\bar{Y}_1-\bar{Y_2})(\frac{1}{n_1}+\frac{1}{n_2})^{-1}$

I am doing an exercise in my book and I can't find where is my mistake. Here is the exercise: Consider a Repeated Measures study with $K=2$ groups and $T=3$ time points, where a random sample with $50$...
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How do we solve this Testing of Hypothesis problem?

To find critical region we can use Lemma-Pearson Lemma $$\frac{L_1}{L_0} > k$$ $$\frac{ \prod_{i=1}^{n} \theta (1-x)^{\theta-1}}{ 1*(1-x)^{1-1}} > k,\space \space\space \theta >1$$ $$\prod_{i=...
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Need for equality in the null hypothesis

In statistical hypothesis testing, the alternate hypothesis is usually chosen to be the statement which bears the burden of proof. Also, the null hypothesis always has an equality ($\leq$ or $\geq$ ...
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210 views

Finding values for some test and power for uniform distribution

I have the random sample $X_1, X_2, \dots, X_n$ drawn from the uniform distribution on $[\varphi, \varphi + 1]$. To test the null hypothesis $H_0 : \varphi = 0$ against the alternative hypothesis $H_1 ...
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Two generators of random points in a disk

I have two ways to generate points in a disk: The first is: $ a, b \sim U[0, 1]. $ Point is generating the next way: $(a \cos(2\pi b), a \sin(2 \pi b))$. The second is: $a, b \sim U[-1, 1]$ and point ...
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Alternative distribution of likelihood ratio test statistic

I am wondering if there is an existing theorem for the distribution of a likelihood ratio test statistic under the alternative. By Wilks theorem, we know the distribution under the null. However, ...
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Likelihood ratio test for distribution with two parameters

We let $X$ and $Y$ be independent and exponentially distributed random variable with $E(X)=\lambda_1$ and $E(Y)=\lambda_2$ where $\lambda=(\lambda_1,\lambda_2) \in {R}_+^2$ and we let $(X_1,Y_1),...,(...
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Statistical test to prove equivalence of two classifiers

I am trying to empirically show equivalence between a logistic regression classifier and another classifier based on a maximum entropy argument. What I have done is used both classifiers on the same ...
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253 views

How can one find $g$ so that the test will have size $\beta$? [closed]

I have the random sample $Y_1, Y_2, \dots, Y_n$ drawn from $\text{Normal}(\mu, \sigma^2)$, where $-\infty < \mu < \infty$ and $0 < \sigma^2 < \infty$. To test the null hypothesis $H_0 : \...
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Pareto distribution: is there a uniformly most powerful test (UMP) at some level $\beta$?

I have that $X_1, X_2, \dots, X_n$ is a random sample from a distribution with density $$f_{\varphi}(x) = \dfrac{3\varphi^3}{x^4} \ \ \ x \ge \varphi > 0$$ Is there a uniformly most powerful test (...
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Component manufacturers - Hypothesis testing [closed]

I need instructions on how to go about solving a) b) and c) for this problem they gave me at university, I've had some sample exercises for hypothesis testing but I have no clue on how to solve this: ...
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Negative Wilks' theorem statistic & zero degrees of freedom

In one of my text examples, we are to use Wilks' theorem to test $H_0 : y \sim \textrm{Bernoulli}(p)$ versus $H_1 : y \sim \textrm{Poisson}(\lambda)$ given $n$ samples and $t = \sum_{i=1}^n y_i$. The ...
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Beta distribution most powerful test over a composite hypothesis

Hello I have a draft of an attempt and am wishing to understand further. The problem is Given one observation from a $\text{Beta}(1,\theta)$ population, to test $H_0:\theta_1\leq \theta \leq \...
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Hypothesis testing for a Poisson distribution

Let $ \mathbf{X}=\left\{X_{1}, X_{2}, \ldots, X_{n}\right\} $ be an independent random sample from the Poisson distribution with parameter $ \theta>0 $. (i) Find the rejection region of the most ...
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“Invariance” of Pearson's Chi-squared test

A company has 100 employees, 40 women and 60 men. At the end of the year, the company decides to give bonuses to 20 people: 16 men and 4 women. I'm asked to find a way to test the presence of a ...
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Sample size formula for linear mixed effects model

I am looking for a sample size formula for a linear mixed effects model to achieve a certain power. I am familiar with the derivation that assumes a two stage formulation, where stage 1 fits the ...
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hypothesis testing and CI

when can't we test the hypothesis testing based on confidence interval and when can we? I just know that one example is since we have used estimator of p to estimate the normal distribution but we use ...
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1 Mean T-Test, Statistics

During a flu vaccine shortage in the United States, it was believed that 45 percent of vaccine-eligible people received flu vaccine. The results of a survey given to a random sample of 2,350 vaccine-...
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Significance Hypothesis Testing

Given an assumed mean of 60, a standard deviation of 6, and a population with a sample size of 50 and a population mean of 62.6, I was able to set up two hypotheses, H0, $\mu=60$ H1, $\mu \ne 60$ I ...
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Why do I have to use a fear as the Alternative-Hypothesis?

Introduction I'd like to understand, why I have to use a fear as the Alternative-Hypothesis and not as the Null-Hypothesis. I'm using my homework as an example for that, which says the following (I ...
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Hypothesis testing - critical region

When finding the critical region, do you choose the x values closest to the significance level even if its bigger then it or does the value have to be smaller then the significance level. So if a ...
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Fisher's exact test calculation

I want to determine whether a new technique for sampling blood cultures causes less contamination of the blood cultures. The group in which i applied the new technique is called the intervention group ...
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How can I determine statistically how well my estimator predicts actual results?

I have a business in which I contract with my clients to perform work for them for an indefinite amount of time. To forecast my profitability, I need to forecast when my existing contracts will end. ...
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Most powerful test. Help using Neyman Pearson

$X_1,X_2,...$ independent continuous random variables with p.d.f $f(x) = \theta x^{\theta-1}$ if $0<x<1 , 0 $ otherwise for $\theta > 0 $ sample size = 1 Use Neyman-Pearson Lemma to drive MP ...
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statistical hypothesis testing.

Vasya Sidorov claims that he goes to the cinema twice as often as to the gym, and to the gym twice as often as to the theater. Over the past six months, he has been to the theater 10 times, to the gym ...

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