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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What is the null hypothesis?

A person claims that he can differentiate if a medicine is working or a placebo. Therefore we test the null hypothesis $p=0.5$ and $\alpha=0.5$ with $n=50$ with a right handed hypothesis test. Edit: ...
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Hypothesis testing for anti proportional correlation

currently I am a bit in trouble regarding hypothesis testing - definitely my weakest area of math. I was studying it quite a lot in the past days but came across an problem that I don't know how to ...
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Is my Z-Test incorrect? [closed]

Wondering if I could have a second pair of eyes peer review my Z test that I have recently conducted, Hypothesis H0: u1 = u2 H1: u1 > u2 Calculations performed in the image attached: Z-Test ...
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The value of threshold for Hypothesis testing

Is the threshold in Hypothesis testing (I mean the value that we compare the test output with) always between zero and one? and Why? Any help will be much appreciated.
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Transformation to remove column in regression proof

I'm trying to show that the $F$ statistic where the smaller model is just the removal of the $j$th predictor is equivalent to the square of the $t$ statistic for question $3.1$ in Elements of ...
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28 views

Identifying sample size in hypothesis testing question

I have the following question: Find the best size $\alpha$ test of $H_0:\theta=\theta_0$ vs $H_1:\theta =\theta_1$, with $\theta_1>\theta_0$, write down the expression for the power function when $...
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1answer
83 views

Likelihood ratio test question

If $X_i$, $i=1,\ldots,n$, are observations from the normal distribution with known variance $\sigma_i^2$, respectively, and $X_i$'s are mutually independent, construct a test for testing that their ...
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29 views

Likelihood ratio for single observation

Let X be a single observation from the density f(x;$\theta$)= $(2\theta x + 1 - \theta)$ I$_{[0,1]}$(x), where $-1<=\theta<=1$. Find the likelihood ratio test with the significance level $\alpha$...
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Test of hypothesis on the difference between population groups- deciding on the method to use

The problem is given as: In order to investigate the opinion of three regions on the passing of a new policy, random samples were obtained from the regions. The table below shows the results. Use test ...
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2answers
35 views

Statistics hypothesis testing problem - vague question

I am working on revising my statistics knowledge and I came upon an exercise which I don't know how to do. I have been given two sets of data samples: ...
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69 views

Are the null hypothesis and alternative hypothesis mutually exclusive?

I am having a little bit of trouble understanding the concept of p-values. On the one hand, we are saying that the null hypothesis is a hypothesis we set, and the alternative hypothesis is something ...
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1answer
61 views

Karlin-Rubin theorem, normal distribution

Let $X_1,...,X_n \sim N(\mu, \sigma^2)$, where $\mu$ is known and $\sigma^2$ is unknown. We have a pair of hypotheses: $ \begin{cases} H_0: \sigma^2\leq a \\ H_1: \sigma^2 > a \end{cases}$ So I ...
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28 views

What test should I use for this problem? (assessing the significance of a change in a categorical variable between two different sized populations)?

I have 2 high schools, School A and School B. For the first school, I have 5 classes of students; for the second, I have 3 classes (so 8 classes in total). Within each class, I have different ...
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53 views

Why are we always assuming normality?

For a great number of statistical testing (ANOVA or discriminant analysis for example), we suppose that the variables follow a normal law. How many times did I see this supposition ? But I recently ...
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28 views

Two different cases of uniform hypothesis testing

I have two different p-value uniform-distribution problems. I know that the definition of the p-value is: The probability of observing a new $X$ at least as extreme or more extreme than the initial $X$...
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Symmetry test for a coin.

We throw a coin 100 times. We have two hypotheses $H_0$: the coin is symmetrical and $H_1$: probability of head is equal to $\frac{6}{10}$. We used statistics that returned the number of head. Let $\...
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34 views

Finding the p-value between two numbers

I am trying to solve this example in statistics, but my results are different from the ones in the solution and I do not understand why. The example say: A study finds a test statistic $t$-value of $...
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Null Hypothesis -> The No Difference

This company treats advertising costs of over $\$3,000$ per week as high and costs under $\$3,000$ per week as low. Then, the revenue can be modeled as $$Y_{ij} = \mu_i + \varepsilon_{ij}\,,$$ $\...
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Power of point hypothesis test

With $X_i$ i.i.d. $N(\theta,1)$ and $$H_0: \theta=0 vs. H_1: \theta=1$$ I got that the test statistic is $\bar{X_n} > \frac{q_{\alpha}}{\sqrt{n}}+\theta$, where $\theta$ is either 0 for $H_0$ or 1 ...
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Why is hypothesis testing used at all?

I've been reading about different criticisms about the misuse of p values adn hypothesis testing in different fields and I think I understand the main issues with how people wrongly interpret them. In ...
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obtaining the t_P value of the partial correlations

sorry if this sounds stupid, the following picture is from a paper I am reading, I have obtained a matrix of r (the partial correlations of each predictor with the response variable) and I have the ...
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Find the test that minimizes the sum of the type 1 and type 2 error probabilities [closed]

Let we had one observation of $X~ N(\mu,1)$ And consider test : $H_0:\mu=0, H_1:\mu=1$. How to find test that minimizes sum of probablities of type 1 and type 2 errors?
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Test hypotheses using statistic value

I have an exam today, and the solution to this problem that my teacher provided differs from mine: A government department estimates that at most the 12% of the people older than 65 years has a ...
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53 views

Critical region for testing a hypothesis with exponential data.

I have trouble with the following problem: We assume that X has a Exp(λ)-distribution with a unknowns value of λ. We test H0: λ=0.2 versus H1: λ < 0.2. We use X as test statistic. a. ...
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48 views

Hypothesis testing with a exponential distribution

I have the following problem: Given the data $X_1, X_2, \ldots, X_{15}$ which we consider as a sample from a distribution with a probability density of $\exp(-(x-\theta))$ for $x\ge\theta$. ...
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58 views

Statistics: finding an MLE for theta

I have to obtain a likelihood-ratio test for $H_0: \theta =0 $ vs $H_1:\theta \neq 0$. $\theta \in [-1,1]$ I also think the size of the random sample is $n=1$, it says "given an observation of $X$". ...
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Is a 99% confidence level more precise than 95%?

Hi. I'm confused by 'significance level'. If significance level is lower, is the result of test more reliable? I think it's the opposite, so it makes me so confused. Look at the picture and assume ...
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1answer
28 views

Failed Uniformly Most Powerful Test

Suppose $X_1,...,X_n$ is a random sample from $N(\theta,1)$. We want to test $H_0: \theta = \theta'$ vs $H_1: \theta \ne \theta'$. I know that since there are 2 critical region for $H_1$, there is no ...
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1answer
32 views

Creating a confidence interval when sigma is unknown

This is something which has plagued me for months now. And i am not really able to explain it to anyone. I know how to CALCULATE the confidence interval when sigma is unknown . But i don't get how is ...
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Show that a critical region $G$ based on the likelihood ratio is of the form $G=B_{R_1}\cup\overline{B_{R_2}^c}$

From an $N(0,\sigma^2)$- distributed population one draws a sample $X_1,...,X_n$ of size $n$ in order to test the hypothesis $H_0: \sigma=1$ against $H_1:\sigma \neq 1$. Let $B_R$ be the closed ball ...
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1answer
54 views

LR Test in Beta($\theta$, 1) with $H_0 = {\theta_0}$

I'm trying to obtain $\alpha$-level LR Test where $(X_1, ... X_n)$ are from Beta($\theta$, 1) with $H_0 = {\theta_0}$ and $H_0 \neq \theta_0$. I'm looking for $$ \lambda(X) = \frac{\sup_{\theta \in \...
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Wilcoxon test for hypothesis testing in case the null hypothesis uses median as a parameter not mean ( in program R).

I want to test whether I could reject a null hypothesis that the data has a median of 600. I need to use the wilcox test or maybe construct it in R, but I am not sure if the builded in test supports ...
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29 views

Likelihood ratio test: normal for unknown variance

Let $X_1,\dots,X_n\sim N(\mu,\sigma^2)$, $\sigma^2$ is unknow. We want to test $H_0:\mu=5$ vs $H_1:\mu>5 $ $$f_{\theta}(\vec{x})=\left(\frac{1}{\sqrt{2\pi\sigma^2}}\right)^n\exp\...
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Estimation of the Kolmogorov test power [Rstudio].

Let $x$ be a 20-element sample from the T-Student distribution with 4 degree of freedom. I want to do a simulation to estimate the power of the Kolmogorov test for $H_0=F\sim N(0,1)$ if $\alpha=0.01$. ...
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3answers
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Is my six sided dice weighted?

I was playing a game with my son, and it seemed that we rolled two or five about half the time. He was wondering if the die was weighted. So I re-rolled it a bunch of times and jotted down the ...
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LRT statistic on data

I have a dataset with variables date sold, prices, type (house/unit) and number of bedrooms. I have fitted the data with weibull distribution on the variable 'prices' with fitdist(). I would like to ...
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23 views

Is there a proper scoring rule on a general topological space $X$?

I have seen (proper) scoring rules on binary, multiple-choice, and real-valued questions, such as Brier Scores, log scores, and a few others for real-valued examples. Is there any literature on ...
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1answer
23 views

NP Lemma and rejection region having the size.

quick question. I need to find the rejection region of H_0, meaning that I need to find the constant value for which if X=x belongs to the region H_0 is rejected. The pdf is c(1-x)^(c-1) with x ...
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1answer
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Check for distribution of the sample with unknown parameters using ks.test in R.

When I do a ks.test in R for a sample to check from which distribution it is, it gives me a $p$ value less than 0.01 for various distributions and I don't know why. Maybe because of parameters or smth?...
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Minimizing Type I and Type II errors Simultaneously

I am trying to understand why I can't simultaneously minimize Type I error and Type II error simultaneously. A non-randomized Hypothesis test is essentially a deterministic partition of the Range ...
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2answers
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A machine is used to fill packets with rice. The contents of each packet weighs X grams - Normal Distribution

A machine is used to fill packets with rice. The contents of each packet weighs X grams. X is normally distributed with mean μ grams and standard deviation 3.71 grams. The mean weight μ is stated to ...
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Hypothesis testing for the variance of a population

Someone is trying to introduce a new process in the production of a precision instrument for industrial use. The new process keeps the average weight but hopes to reduce the variability, which ...
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1answer
27 views

A research department is investigating the duration of a tire.The department wants to show it exceeds a certain value. Formulate the hypothesis

The research department of a tire manufacturer is investigating the duration of a tire using a new rubber component. 16 tires were produced and the duration was tested. The average duration and ...
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Monotone Likelihood Ratio and UMP Test

I am trying to better understand the scenarios that will ensure a UMP test when a monotone likelihood ratio exists. I know a monotone likelihood ratio as a family of distributions $\mathcal{P}=\{...
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1answer
33 views

What are hypothesis tests for in statistics?

A manufacturer produces a component for car engines. It is known that the diameter of this component is approximately normal with a stdev of 0.0005 milimetres. In a random sample of 15 components ...
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Are these samples enough for a retention analysis? (T-Test)

I think that this is a dumb question but I would like to know if these differences in these KPIs are reliable or I have to change the sample in order to have statistically significant conclusions. ...
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1answer
37 views

Comparing two normals analytically differs from numPy test

Comparing two normal distributions explained, how to find $P(B>A)$ for every $A \sim N(\mu_1,\sigma_1), B \sim N(\mu_2,\sigma_2), $ In my case, there a two binomials - $A \sim Bin(100,0.52)$ and $...
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22 views

how to calculate the Wald test statistic

I am struggling to understand the definition as I am seeing several different definitions on the internet. Lets say I have a question that goes like : Given $Y_1 ...Y_n$ is a sample taken from ...
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1answer
75 views

Most Powerful test

Using Neyman Pearson Lemma, find a Most Powerful test for $H_0 : \sigma = 2$ vs. $H_1 : \sigma = 1 \ $ at level $\alpha$ based on a random sample $X_1, \dots , X_n \ $ from $N(3, \sigma^2 )$. Also ...
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UMP test with two distributions derivation

Suppose X has distribution in $\mathcal{P}=\{B(10, \frac{1}{2}), P(1)\}$ (Binomial and Poisson respectively). Let $H_0 := X \stackrel{D}{=} B(10, \frac{1}{2})$ and $H_1 := X \stackrel{D}{=}P(1)$. ...

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