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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Rao Score test and Neyman-Rao test

I've got the Laplace location/scale model where $X_1,...,X_n$ are iid random variables with common density \begin{equation} p_{\theta,\eta}(x)=\frac{1}{2\eta}e^{-|\frac{x-\theta}{\eta}|}. \end{...
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Should I reject the null hypothesis or not?

EDIT: My apologies, I had a coding error. I accidentally used the same standard deviation for both samples. Now that I fixed that, both the normal and Student's confidence intervals are stupidly ...
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Proof of statistical assumptions behind the methods or tests

When reading statistics literature, they provide the assumptions of when specific tests/methodologies can be used. Take factor analysis as an example; you may find this inside a paper: "Generally ...
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Minimum required sample size and confidence interval of the difference

I have calculated a minimum sample size required to compare difference between two independent proportions using various online sources e.g. this one or G*Power software. Here are the details: P1: 0....
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Is my answer to this Poisson distribution story problem correct?

Andreia's secretary makes random errors in his work at an average rate if 1.7 errors every 100 words. (a) Andreia offers the secretary a choice of one of two bonus schemes, based on a sample of 40 ...
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hypothetical test [closed]

Suppose that $X$ is a sample of size 1. Given a significance level $\alpha\le 0.1$, derive a UMP test of the following hypotheses $[H_0: X\sim N(0,1)\ vs.\ H_1: X\sim DE(1/2)]$, where $DE(\lambda)$ ...
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Testing the generation of a sequence of zeros and ones

I am given a sequence of $40$ ones and zeros, and I have to test whether the sequence is random (in the problem this is said to be equivalent to testing the null hypothesis that all ${40 \choose n_1}$ ...
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most powerful test of exponential distribution

It is assumed that X, the lifetime (in hours) of a light bulb, follows an exponential distribution with the following probability density function: f(x; λ) = 1/λ(e^-x/λ) if x > 0 or 0 otherwise (a) ...
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hypothesis testing with central limit theorem

Company A claims that the average number of biscuits in a tin of BiskO shortbread biscuits is $27$. The fair business regulators believes that the Company A’s claims are inaccurate, and that the true ...
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Hypothesis testing for the probability of a graph being conected

Given the probability of every two nodes being connected, write a simulation to estimate the probability of a connected graph. Then use hypothesis testing to justify the result. I can write a Monte ...
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Determine c so hypothesis test has level of significance α= 0.05

I have the solution but can't find an explanation that makes sense to me so I'm hoping someone here can explain it to me. Question: Consider a sample with just one observation $X$ that is assumed to ...
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UMP test-why is it wrong to check it this way?

So, I want to know if a test is a UMP test. for: $ θ_0=3 $ $ θ_1<θ_0 $ using neyman-pearson lemma suppose I got something like this where our estimator for $ θ_0 $ is the mean and for some ...
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How to test randomness of assigning the incoming call to the call center employees?

We have the call center with 6 employees in total. We can't predict when each of them will work, however we would like to ensure that the employees are chosen at random to pick up incoming calls. How ...
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Hypothesis testing on sample mean when observation are correlated

How can I test whether the mean of a sample is significantly larger than the population mean, when observations are not independent, but the exact correlation matrix is known? I have a variable $X$ ...
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Hypothesis testing on Probability of False Alarm using Neyman-Pearson

We have binary hypothesis testing problem as follows: $H_1$ (signal presence) : $y = s + n$ $H_0$ (signal absence) : $y = n$ Without loss of generality, to simplify the problem, let s be a constant ...
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unbiased estimator for supremum of expectation

Can we get an unbiased estimator for $$ \sup_{\theta} E_{X\sim P} [\max\{\theta-X, 0\}] \quad ? $$ Let $X_1$ be a sample from $P$. I tried using the $1$-sample estimator: $$ \sup_\theta \max\{\theta-...
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Confused about interpretation and intuition behind Mcnemar's Test

I've always been under the impression that Mcnemar's Test is for paired categorical data, just like paired t-tests are for paired continuous data. However, I was looking at the Wikipedia article for ...
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How to find p-value when null distribution is of gamma and given observation

Doing a hypothesis question, where $$H_0: \lambda=10$$ $$H_1: \lambda \neq 10$$ null distribution to be ~$\gamma(\alpha=20,\lambda=10)$, where 10 is in rate, and an observed sample =0.8, how do I ...
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What is β in statistics

A simple hypothesis test question. But, while reading the following hypothesis test, I am stuck: H0: β = 0 Ha: β ≠ 0 The answer shows that the null hypothesis represent there is no linear ...
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Hypothesis Testing and Rejection Region Problem

Assume $X_i$, where $i=1,..,n$, is random sample from normal distribution $N(\mu, \sigma^2)$, where both $\mu$ and $\sigma^2$ are unknown. Suppose we use $C = \frac{\bar X_n}{S_n} \gt k$ as the ...
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hypothesis testing and properties(biasedness/unbiasedness, consistency) of the OLS estimator

Consider the following equation * \begin{equation} \label{eq:1} C_{t} = \beta_{1} + \lambda Y_{t} + \epsilon_{t} \end{equation} where, \begin{equation} \label{eq:2} E(\epsilon_{t}\mid Y_{t}) = 0 \...
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Doubt regarding null hypothesis rejection in Z-test

We have been learning about Z-test for null hypothesis rejection. It goes as follows: Suppose $X_1,\ X_2,\ \ldots, X_n$ are samples from $X\sim N(\mu, \sigma^2)$ distribution with sample mean $\...
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Power function for Gamma Distribution

I'm trying to find a power function for a $X_1, ...,X_n$ ~ Exp($\theta$), where $H_0: \theta\geq \theta_0$ vs $H_a: \theta<\theta_0$; where, for test statistic X ~ $\sum X_i$, we would reject the ...
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Power and size of a statistical hypotheses testing

I would like to know here on the page 132 why the pink (the power of the test) is vertically aligned with the blue (the Type I error). What is the intuition behind it ? Also, how the formula (7.2) in ...
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Why do we consider a critical region instead of individual values? Why does the alternative hypothesis determine which tail to consider for rejection?

Setup: Suppose a coin is tossed 8 times and I'm trying to determine whether the coin is biased in favour of landing on heads. Let $X$ be the number of heads in 8 tosses, so $X \sim B(8,p)$. Conducting ...
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Unit-Root Test Alternative Hypothesis - Dickey-Fuller Test

For a simple autoregressive model satisfying ${p_t = \phi_0 + \phi_1 p_{t-1} +\epsilon_t}$, when you want to test whether the series has a unit root (non-stationary) why is the alternative hypothesis ...
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Determining whether a (non normally distributed) population has a smaller mean than other

I am twisting my career towards Data Science and, after some time refreshing my statistics base, I realized that statistics is one of the fields in which people tend to missunderstand more the basis, ...
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Testing first and seconds moments properties of vector generated from N-dim normal distribution

Suppose we have random sample of size T of a real-valued vector $z_t$, which we can assume to have N-dim normal distribution with 0 mean and $I_N$ variance under the null hypothesis. We want to test ...
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Hypothesis Testing in Python: One sample T Test without Standard deviation

Update So the way I solved the question was through taking one sample of the weights array like so sample = np.random.permutation(weights) Then calculate the ...
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Synthetic control

Background: There are 5 product categories for my digital business. And, there are different sets of rules with different weightings of metrics to design product ranking for different category pages ...
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Why Null Hypothesis for contingency tables is 'independent'?

For contingency tables I don't understand why the Null Hypothesis is 'independent'. We calculate Sum(Observed^2 / Expected) - N and compare this with the chi-squared distribution table. Let's say: Sum(...
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Threshold in Neyman-Pearson

Suppose we need to decide between two hypotheses $H_1: P=P_1$ and $H_2: P=P_2$, based on a single sample $X\sim P$ taking values from $\mathbb{X}=\{-1,0,1\}$, where $P_1(-1)=1/2, P_1(0)=1/4, P_1(1)=1/...
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Why don't you calculate individual probabilities in hypothesis testing?

Lets say there is a coin being flipped, and someone thinks that the coin is biased so that the probability of getting heads is less than 1/2. If they flip a coin 10 times, and get heads twice, I ...
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Light Bulb hypothesis testing

One claims that the life time distribution of its Everyday light bulbs is exponential with mean 1000 hours. If you test a random sample of 4 light bulbs and find that the average life time is 900 ...
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What is meant by, "both variables follow a normal distribution" in Pearson's product moment correlation coefficient hypothesis tests?

I am teaching myself further maths A Level Year 1 statistics from the OCR book (A). Chapter $5$ is about Correlation and regression. I get that the ppmcc is given by $r,$ which is just a number with $\...
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Does the population variance equal the variance of a single observation?

According to Wikipedia, the standard error $\sigma^-_x$ of a sample mean can be computed by $\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the standard deviation of a statistical population and $n$ is ...
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Neyman-Pearson and Stein's Lemma

Consider the following problem. Let $X_1,\dots, X_n$ be i.i.d. $\sim P$ taking values from $\mathbb{X}=\{-1,0,1\}$. Consider the hypothesis test $H_1: P=P_1$ vs $H_2: P=P_2$, where $P_1(-1)=1/2, P_1(...
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Is choosing confidence interval bounds after observing data mathematically valid?

Let $D$ be a continuous distribution on the interval $[0,1]$ that is not known to us. We have no prior knowledge about $D$. For a given error tolerance $\delta$ we want to find bounds $a, b$ s.t. $b-a$...
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Sampling trading strategy returns

I want to test annual returns of a stock trading strategy, particularly its skewness (using the D'Agostino's skewness test). I have monthly prices of 20 years, but that means I only have 20 annual ...
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Determining whether two samples are from the same distribution

The context for my question is that we have two sets of data which are of the same family of distributions, both normal, both exponential, etc. If they were both normally distributed, a difference of ...
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A question about the test statistic for testing the difference in two population proportions

For independent samples $X_1,\cdots,X_n $from $\textit{Bernoulli }(p_1)$ and $Y_1,\cdots,Y_m$ from $\textit{Bernoulli }(p_2)$,where $n$ and $m$ are large. Then,the Central Limit Theorem tell us $\frac{...
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Distributed Hypothesis Testing--reference question

If you don't know the correct keyword, you can still miss a key literature search: I have a problem in distributed Bayesian detection with a serial (or tandem) network topology. The probability ...
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What test to use to see the difference between two groupps with the same variance?

I have the data for 3 groups A,B,C with the times of run of 5 kilometers. $ \begin{matrix} Group A & Group B & Group C \\ 27.5 & 35.3 & 45.8 \\ 30.6 & 40.2 & 42.6 \\ 28.5 &...
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How to find Type 2 error without the Normal distribution?

If I have an example as follows with 280 cases studied of a new test developed on animals: true state of the animal non-infected by virus X infected by virus X non-infected by virus X 131 15 ...
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LR-test for a Pareto$(1,\beta)$-distribution

Let $X_1,\ldots,X_n$ be a sample from a Pareto$(1,\beta)$-distribution with density $$f(x \mid \beta) = \begin{cases} \beta x^{-(\beta+1)} & x \ge 1 \\ 0 & \, \text{else} \end{cases}$$ We test ...
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Hypothesis testing for Third and Fourth Moment

Is there any way to test for the null hypothesis that the third $\left(\frac{\sum_{i=1}^n(Y_i-\bar{Y})^3}{n}\right)$ and the fourth $\left(\frac{\sum_{i=1}^n(Y_i-\bar{Y})^4}{n}\right)$ moments are 0? ...
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What is the common $t$-test for $H_0: \mu = \mu_0 \ \text{ and } \ H_A:\mu > \mu_0$?

The following remark came up in my lecture: Let $X_1,X_2,\ldots,X_n$ be normally distributed with known variance $\sigma^2 > 0$. For testing $$H_0: \mu = \mu_0 \ \text{ and } \ H_A:\mu > \mu_0.$...
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How to determine power of a test of the difference of two binomial proportions?

Consider a sample of 800 adults with wrist fracture where 400 are provided an operative treatment and 400 are provided physiotherapy only. Outcome of interest is whether the wrist is fully healed ...
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Statistic Mann-Whitney Table how to calculate the table

In this link there's a tutorial about Mann-Whitney: https://www.real-statistics.com/non-parametric-tests/mann-whitney-test/ . It uses critical values of U test that is given in a table. However, it ...
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Information Theoretic View of Stein's Lemma

Suppose $P$ and $Q$ are two probability distributions on a, say finite alphabet set $\mathbb{X}$. Suppose that a Statistician has to decide between $P$ and $Q$ based on an i.i.d. sample $X_1,X_2,\dots,...
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