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Questions tagged [hypothesis-testing]

This tag is used for questions regarding how to construct, use a hypothesis test or what is the appropriate test to use for a given case

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Assume the hypothesis $H_0 : u = 59$ by computing the relevant $p$-value

An observed sample of four observations from $N(u, \sigma^2)$ distribution has mean $54.75$ and standard deviation $10.21$. A table of quantiles of the chisquare distribution is given below $(a)$ ...
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20 views

Exponential Test

Given a collection of data points, without any other information I want to test for an exponential distribution. In a different case, I tested for normality using an Accord implementation of the ...
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1answer
15 views

Test if reduction of coefficient of variation is significant

I have the following problem: I want to compare two different quantification processes and I use the coefficient of variation as quality metric. So in principle my data look like: experiment method1 ...
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12 views

Two population hypothesis test sample size

A retailer is trying a new concept in 3 of its 200 stores. The trial period is 10 weeks. The goal is to increse sales with at least 10%. The tricky part is that there is sesonality and campaigns that ...
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1answer
28 views

Hypothesis testing variance using sample mean

I know how to test hypotheses for variance using methods like the chi-square test. However, this problem is asking me to use a rejection region construction in terms of the sum of the sample values (...
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2answers
19 views

Hypothesis testing for sample means within a normal distrubution

The context of the question is that a bakery bakes cakes and the mass of cake is demoted by $X$ such that $X \sim N(300, 40^2)$. A sample of 12 cakes is taken and the mean of the sample is 292g. The ...
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1answer
69 views

Getting a probability $> 1$ in hypothesis test

I have the following hypothesis question. A soft drinks company claims that of all consumers buying their product, $82 \%$ prefer the light version of the drink. To test their claim, data were ...
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1answer
68 views

Hypothesis test concerning a population proportion

The problem states that the national on-time arrival rate for flights in Nov 2018 is 79.66% according to the Bureau of Transportation. Suppose in Nov 2018, we ...
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10 views

Testing if cross-covariance is zero in Hilbert spaces

I'm attempting to generalize a multivariate test of zero cross-covariance between two random varaibles to infinite dimensional Hilbert spaces and I'm looking for some advice / ideas on how to work ...
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8 views

wilcoxon signed rank test interpretation

I try to understand the Wilcoxon signed rank test, but I have not been profoundly successful. I have studied various articles and educational videos, but I have not succeeded yet. The concept I'm ...
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27 views

Conditional probability (Binomial distribution)

I'm doing an exercise of hypothesis testing with $X_1, \ldots X_{12}$, $X \sim Ber(p)$, $H_0\colon p \leq \frac{1}{2}$, $H_1\colon p > \frac{1}{2}$ and a critical region of $C_{\Upsilon}=\{(X_1,\...
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Find the p value following the exponential distribution $\mu=3$

I want to find the $p$-value (manually) of the following Hypothesis testing. $$H_0:\mu\leq 3 \quad \text{vs} \quad H_1:\mu >3$$ The main thing I know is that $$P(\mathrm{Re}\,j \mid \mu \leq 3)=...
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Relatively Straight forward hypothesis test question. [closed]

Screenshot of the question Above I've attached a screenshot of the question I'm trying to answer. Not too sure how to approach it, it is probably very straight forward but I seem to lack the ...
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12 views

Adjusting the level of significance at 6%

What if I used one - way ANOVA set at 0.05 level of significance, and the computed p - value is 0.058, can I adjust the level of significance to 6% instead of 10% so that it will result to significant ...
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Hypothesis testing Type 2 error

Type 2 error Was just wondering i got the answer for all of it but i still do not know why when the true mean is 6.9kg I did not have to do 1-P(type 2 error) to find the answer. The answer when the ...
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Find a UMP test in exponential family - first stage

Consider the exponential family $f(x;\theta) = c(\theta)\exp\{b(\theta) T(x))\}h(x)$ and I would like to find a UMP test for $$H_0: \theta \leq 0\quad \hbox{vs}\quad H_1: \theta > 0.$$ It is ...
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1answer
35 views

Understanding that a hypothesis test does not depend on a certain parameter

I have a very big doubt regarding a test not depending on $\theta_1$. Suppose I have: $$(X_i)_{i=1}^{n}, X_i \stackrel{iid}{\sim} N(\theta, 1), \hbox{ that is } X_i \sim g(x_i, \theta) = (2\pi)^{-1/2}...
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1answer
31 views

Anderson Darling test

I've done an Anderson-Darling test and Python gave me this output ...
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23 views

Probability of significant result is due to chance hypothesis testing

So in principle, a statistically significant result is not attributed to chance. I'm given 70 independent experiments on the number of actions of 500 customers over a course of a month. Each ...
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1answer
15 views

How to calculate test statistic when $H_0$ is not equal

So I'm trying to perform a $t$ test with sample mean $\bar{x}$ and sample variance $s^2$. I know how to do this with $H_0 = a$ and $H_a \ne a$. But now I'm given $H_0 < a$ and $H_a \ge a$. How do I ...
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13 views

Find critical value given test statistic Poisson distribution

I know how to perform $z$ test and $t$ test on a Normal distribution, but now I'm given i.i.d $X_1, X_2,...,X_n$ drawn from a Poisson distribution. Using $\sum_{i=1}^{n} X_i$ as the test statistic, ...
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1answer
12 views

High F-Statistic Value and negative Probability for Granger Causality Result, Interpretation?

I tested the hypothesis: SwissFranc/USD does not Granger Cause S&P500 and received an F-Statistic of 69.1 and a probability of 2E-51. How do I interpret this result and can the probability be 2E-...
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1answer
31 views

What does “more extreme” mean when your test statistic does not “support” either hypothesis?

Consider the following example. My null hypothesis is $p = 0.5.$ My alternative hypothesis is $p > 0.5.$ But when I take a random sample, I the sample proportion is 0.48, with a sample size of ...
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Neyman - Pearson - Test with Poisson distribution

Let $X$ be a Poi($\nu$) distributed rv with unknown $\nu \in (0,\infty)$ and we set $ \theta:=\nu $. We want to test the hypotheses $ H_0:\nu_0=1 $ against $ H_1:\nu_1=\nu_0=2 $ a) We look at the (...
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Neyman-Pearson Lemma for two hypothesis pairs and three parameters

Let $\Theta = \{\theta_0, \theta_1, \theta_2\}$. I want to test the hypotheses $H_0$: $\theta = \theta_0$ vs. $H_1$: $\theta = \theta_1$. Put $\Lambda_0(x) = \frac{\mathcal L(\theta_0|x)}{\mathcal L(\...
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1answer
50 views

Hypothesis Testing under Uniform Distribution Question

The question reads: Let $\theta > 0$ and $X \sim \mathcal{U}[0, \theta]$, i.e. $X$ is uniformly distributed on the interval $[0, \theta]$. Assume that $\theta$ is unknown, but we can observe $X$. ...
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1answer
31 views

Interpretation of the p-value and the test-statistic W of the Shapiro.test in R

Shapiro-Wilk normality test data: Part1 W = 0.14846, p-value = 6.478e-16 Shapiro-Wilk normality test data: Part2 W = 0.47978, p-value < 2.2e-16 Shapiro-Wilk normality test data: Part3 ...
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4 views

Testing cross-covariance in the residuals of a VAR(p) model

Suppose I have a vector autoregressive model of order $p$: $$y_t = c + A_1 y_{t-1} + ... + A_p y_{t-p} + u_t$$, where $y_t$ is a $K\times 1$ vector and $A_i$ are $K\times K$ matrices. We assume the ...
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How do I test if an observed point is expected from a linear regression?

I have a set of $N$ observations on $x$ and $y$, where each observation is characterized as a couple of Gaussian distributions $x\sim N(x_i, {\epsilon_{x,i}}^2), y\sim N(y_i, {\epsilon_{y,i}}^2)$, ...
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How to calculate the critical region in a numerical likelihood ratio test?

Let $N$ be a discrete random variable. Let $H_0$ and $H_1$ be the two hypotheses $$\begin{align}f(n|H_0)=p_n&=\mathbb P(N=n|H_0)\\f(n|H_1)=q_n&=\mathbb P(N=n|H_1)\end{align}$$ where $p_n$ and $...
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1answer
100 views

Likelihood ratio criterion for testing equality of means in 2 normal populations

Let assume we have the following $$ X_{11}, X_{12},...,X_{1n_{1}} \sim N_{p}(\mathbf{\mu_{1}},\mathbf{\Sigma_{1}}) \\ X_{21}, X_{22},...,X_{2n_{2}} \sim N_{p}(\mathbf{\mu_{2}},\mathbf{\Sigma_{2}}) $$ ...
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1answer
42 views

Lemma of Neyman Pearson (how to get critical region)

We have the Random Variable X, which is $\Gamma(p, \lambda)$ distributed with the density: $$f_{p,\lambda}(x)=\frac{\lambda^p}{\Gamma(p)}\cdot x^{p-1} \cdot e^{- \lambda x}$$ with $p = 10$ and $H_0:...
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1answer
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Statistically Ranking Data

We are sampling the rates of $n$ elevators rising up floors in order to see which elevators are the fastest, and which are the slowest. Say we have $k$ time samples from each elevator, $S_1, S_2, ... ,...
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1answer
34 views

Testing whether a scientists finds a cure or not

A scientists claims that he found a drug that in a short time cures depression with a probability of $70\%$. It was tested over a sample of $100$ people and $80$ of them were cured. How can we ...
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Assumptions on test statistics to guarantee uniform asymptotic level

I'm trying to formalize the fundamental ideas behind test statistics in the context of hypothesis testing. In particular, I want to define different concepts of test statistics leading to different ...
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1answer
43 views

Which test statistic is more effective in testing the variance of a normal variable?

Most of the time, to do hypothesis tests is to follow the tedious processes given on statistics textbooks. But recently I find that there are, in general, more than several test statistics that can be ...
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13 views

Distinguish between a paired or two-sample independent t-test

I am trying to determine how to decide whether to use paired t-test or two-sample t-test on this scenario: I want to compare the average amount of dietary fiber (measured in grams) between two ...
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3answers
97 views

Is it possible to eliminate completly the type $1$ error risk?

I was reading here about type $1$ and type $2$ errors and I have a question. Since "No hypothesis test is 100% certain. Because the test is based on probabilities, there is always a chance of making ...
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24 views

Monotone Likelihood ratio and Karlin-Rubin test

I'm studying the Karlin-Rubin test on the book Statistical Inference by Casella and Berger. There the MLR property for a family of pdf is defined as: $\forall \theta_2>\theta_1\:\: \frac{g(t,\...
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1answer
19 views

Changing the null hypothesis in a t-test

Hello. I am working on a problem and am confused. What I know so far is that the two-sided t-test with $\alpha=.05$ will reject the null since the $p-$value is smaller. Here is where I am lost. The ...
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15 views

Calculate confidence-level in A/B-test on revenue

I'm doing a split-test and need to calculate the confidence-level of the result. I need to implement the calculations on my own, not using any online-tools or excel. I managed to do this for binomial ...
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1answer
46 views

Can the Central Limit Theorem be applied here?

My problem statement is to identify in a healthcare organisation, which of it's doctors are lagging in providing proper care to their patients. My Random Variable X is defined as {0 if the patient ...
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1answer
25 views

What does it mean when the exponent is beneath the base number?

I am familiar with the exponential function and the power function. Exponential Function: $y=3^x$ Power of Function: $y=x^3$ But what do the numbers under the theta in this image represent? $h_𝜃(...
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What are the relevant hypothesis?

Octopus Paul in 2010 predicted $8$ matches out of $8$ correctly during the football World Cup. Assume that Paul predicts the winner of a match with probability $p$, assume also that a match can not ...
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How do I pick a rule for decision of a bilateral test?

I have a two random samples normally distribution of length $m = 10$ and $n = 12$. Everything independent. I need to find a rule of decision of level 95% for the hypothesis of equality of the two ...
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Type 1 error in Hypothesis test of One population mean

This is my sceanrio. The hypothesis is that "Polytechnic students are the group who consume more than the average mobile data usage per month in Singapore" This is the claim as it is in the question/...
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20 views

T distribution hypothesis testing

I'm trying to solve the following problem: I don't quite get how to solve the hypotheses testing part. Here is how I tried tackling the problem: $$H_0 : x_t = 0 $$ $$H_1 : x_t \ne 0 $$ I then ...
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How to determine where the data is not linear anymore

I have some data. When I plot these data, the graph looks very linear when $x>x_0$ (In the following figure $x_0$ is shown by dashed lines). One way to determine the exact location where the graph ...
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9 views

Finding Alpha Level from Rejection Rule

I have a question which asks: A researcher wants to evaluate the sensitivity of their assay for measuring urinary mercury. The standard is to have a 99% probability of detecting a sample with ...
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Z test on blocked/partitioned data

I am trying to perform a Z test on a dataset which is partitioned/blocked. Partitioned in a sense that each partition/block has some dissimilarities. The straightforward way can be: get the pooled ...