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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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Identifying all the 1s in a sea of 0s

I have an $N$-length binary string, the total number of 1s in the string is D. Assume N is much larger than D. Also, D is fairly larger than 1. (For example, N=1000, D=40). Now, my aim is to find ...
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Joint confidence interval for regression parameters

assume that we have a model $Y_i = \beta_0 + \beta_1 x_i + \epsilon_i$ where $\epsilon_i$ is normally distributed with mean 0 and variance $\sigma^2$. Assume also that $\sum_{i=1}^{n} x_i = 0$ and $\...
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Is it necessary to keep the regression model linear for checking the Granger-Causality relationships between the variables of a data-set?

For checking the Granger's Causality between two variables of a data-set, lets say to check X granger-causes Y, we create two regression models, a restricted model(containing the lagged values of ...
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Solution verification for a hypothesis testing question

I am posting this question as a solution checking. Let $X_1,...,X_{30}$ be a random sample from the exponential distribution with unknown mean $\mu\in \{1,1/\delta\}$ (where $\delta>1)$. Consider ...
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On a Bayesian hypothesis testing

Let $X_1,...,X_n$ be a random sample and $\lambda >0$ be a parameter, with $X_i |\lambda \sim Poisson (\lambda)$ and $\lambda \sim Gamma(\alpha, \beta) (\lambda)=\dfrac {1}{\Gamma(\alpha) \beta^\...
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How to find the critical value of a test given that the test statistic is derived from a normal distribution (and is an order statistic).

Let $X_1,...,X_n$ be i.i.d from a normal distribution with expectation $\theta$ and variance 1. The test for testing $H_0:\theta=\theta_0 $ v.s. $H_1:\theta=\theta_1$ where $\theta_0<\theta_1$ has ...
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How do I calculate the F value from the Null Hypothesis of equality of means

This question comes from a practice exam, the following contains the question and the answer provided. However my teachers for this exam are non-respondent and I really need someone to please help ...
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Hypothesis testing using the 95% Confidence Interval of Sample Mean

I have a population mean of 120 (mu). I have a sample distribution with a mean of 131.05 and a standard-deviation of 11.00945. I have a sample size of 20, 19 degrees of freedom (n-1). I am performing ...
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Statistics : Hypothesis Testing [closed]

I'm back with another question I can't solve. For this question, I was able to get part of the answer; nk$^{n-1}$(1-k), but I can't seem to see where you'd get the k$^n$ part. Questions Answers
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Tests for log-normal distribution

I am facing a very large dataset (e.g. 100.000 entries) and would like to answer the question if we can apply a lognormal distribution. I used a QQ-Plot in R but the lognormal distribution doesnt ...
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hypothesis testing and confidence interval for a population proportion

I'm trying to understand why a basic undergraduate intro to statistics still uses the normal approximation to do hypothesis tests for population proportions. It used to be that the computations were ...
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Likelihood Ratio Test Variance of Normal Distribution

Let $X_1,...,X_n$ be a random sample from $N(0,\sigma_X^2)$ and let $Y_1,...,Y_m$ be a random sample from $N(0,\sigma_Y^2)$. Define $\alpha := \sigma_Y^2/\sigma_X^2$. Find the level $\alpha$ LRT of $...
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Standardize binomial variable for non-constant meta-population - binomial z-scores

Let's say I'm doing a meta-analysis of a general experimental protocol that has been applied to some experiments (with 1,0 type outcomes) across a variety of experiment-type sub-groups. I want to test ...
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statistical test of normal and non-normal distributed data

I have 2 samples of data. The first sample is the precision of the first algorithm A for 25 test cases and the second sample is precision for 25 test cases of second algorithm named B. I test both ...
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Hypothesis Testing for Probability and Statistics

Question: The present defective rate of a product in a certain factory is 3%. A random sample of 250 of products produced using a new technique which is supposed to improve the quality of production ...
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Testing difference in means for two bimodal distributions?

I have two bimodal distributions of data with two peaks (one around 0 and the other around 1). I have provided an example of one of the distributions. Although their means and variances are different,...
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Neyman-Pearson lemma for the sample of size one

Let $X$ be a continuous random variable on $[0, 1]$ with a density function $$f(\theta |x ) = \frac{x^{\theta}}{\theta + 1}$$ Let's take a sample that consists on the only one element from the given ...
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where does the data for null hypothesis come from?

I have a question on test statistic. We use test statistics to ascertain the hypotheses against the rejection region. My question is from where does the data for null hypothesis come? Was there ...
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Calculate the power of a test with an unknown $\mu$

In order to prevent damages to the wind turbines, the windspeed is monitored every 5 seconds, and if it is determined that the variance of the windspeed is high the turbine switches off to a safe ...
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$X_1$,$X_2$…$X_n$ are iid N($\mu,\sigma^2$). Derive a confidence interval for the parametric function $\mu+\sigma$ and $\mu/\sigma$.

I know how to find confidence interval for each of the parameters $\mu$ and $\sigma^2$ but don't know how to find in the case of parametric function.
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Understanding the method of obtaining an expression for the $p$-value using a test with test statistic $T =\sqrt{n}(\bar{X}_n - \theta_0)/\sigma$

We are in a setting where we $X_1,\ldots,X_n\sim N(\theta, \sigma^2)$ where we assume $\sigma^2$ to be known. We test $H_{0}:\theta\leq\theta_{0}$ versus $H_{1}:\theta>\theta_{0}$. I am aware that ...
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To find the p values in hypothesis testing

The null hypothesis says that at least 20% of college students are left-handed. If we took a sample of 20 college students and let $X$ be the number of lefties in the sample. Calculate the p values if ...
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Showing Rejection Region Equality with Fisher Distribution

I'll preface this with saying that this question is a homework question, but not one that is graded or turned in in any form. Specifically it is from the text Mathematical Statistics with Application ...
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Bernoulli trials hypothesis test

I'm studying about hypothesis test theory and I've reached the next exercise: Let $X\sim Ber(\theta)$. Then, we know that for a sample of $n=10$, we reject $H_0:\theta = 0.5$, and accept $H_1: \theta ...
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calculate sample size for specified type II error probability, comparing 2 proportions

This is from Devore Probability and Statistics for Engineering and Sciences 9th edition, chapter 9, section 9.4, exercise #53 on p.398. The problem involves a hypothesis test comparing two ...
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hypothesis testing, P-value calculation, standard deviation unknown

i have following problem, "The null hypothesis says that a sprinter's reaction time follows a normal distribution with mean at most 0.150 seconds. Six measurements of a sprinter's reaction time show ...
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Combining chi square charecterisitc

I am reading Statistics by David Freedman Suppose the same die had been used to generate the data in tables 4A and 4C (p. 531), rolling it first 60 times for table 4A, and then 600 times for ...
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Hypothesis testing of binomialy distributed data

I haven't done any hypothesis testing for years since I left school and I just wanted to refresh my memory of it. The hypothesis is stated as following: Assume that average high school student has a ...
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Question about Karlin-Rubin theorem related to UMP level-alpha test

As the Karlin-Rubin theorem is presented in my textbook, it applies if our null hypothesis states that $\theta=\theta_0$ (or $\theta<\theta_0$) and our alternate hypothesis states that $\theta>\...
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Deriving a level-alpha likelihood-ratio test for simple null hypothesis versus two-sided composite alternate hypothesis

Let $X_1,...,X_n$ be iid $N(0,\sigma^2)$ where $\sigma>0$ is unknown. With a preassigned $\alpha$ in $(0,1)$, my task is to derive a level-alpha LR test for $H_0: \sigma=\sigma_0$ versus $H_1: \...
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Using Neyman-Pearson Lemma to find UMP level-alpha test with composite null and alternative hypotheses

Let $X_1,...X_n$ be iid with gamma pdf given by $f(x|\delta, b)=b^{-\delta}\Gamma(\delta)^{-1}x^{\delta-1}e^{-x/b}$, where $x>0$, and both parameters are unknown. Suppose we are testing $H_o: (b<...
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GLR Test for 2 samples from exponential distributions

I am trying to solve this complicated Generalised Likelihood Ratio Test problem for 2 samples as part of my independent study course. However, the book that I am using does not talk about 2 samples at ...
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paired vs two sample t-test

It's been awhile since taking Prob. & Stats in college. My question is regarding which test to use in the following situation. I have two data sets that contain values regarding some testing/...
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MAP/MLE/Neyman Pearson Example Problem

I'm currently attempting a practice problem and wanted help checking my work: The random variable X is such that P(X=1) = 2/3 and P(X=0) = 1/3. When X = 1, the random variable Y is exponentially ...
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F test for variance ($H_0 : σ_1^2 ≤ σ_2^2$)

Consider two different normal distributions for which both the means $µ_1$ and $µ_2$ and the variances $σ_1^2$ and $σ_2^2$ are unknown, and suppose that it is desired to test the following hypotheses: ...
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McNemar's test reliability

I am wondering how reliable the statistical significant McNemar test is. I am perplexed by the fact that elements of the main diagonal do not contribute to the outcome of rejection or not of the null ...
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If $X_i\sim U(0,\theta)$, then there exists UMP test for $H_0:\theta=\theta_0$ vs $H_1:\theta>\theta_0$

Let $X_{1},\dots,X_{n} \sim Unif(0,\theta)$ independently. Show there exists a UMP size $\alpha$ test for testing $H_{0}:\theta=\theta_{0}$ vs $H_{1}:\theta>\theta_{0}$. My attempt: I attempted ...
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How to compute a conditional probability related to Type II error?

I would like to know how to compute the following conditional probability. Let $X|\theta \sim \mathcal{U}(0,\theta)$ be a uniform random variable. Given a fixed $\theta = \theta_0 > 0$, one can ...
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If the p-value for a certain test statistic's value is .025, then the null hypothesis will be retained if test statistic is set to α = .02

T/F: If the p-value for a certain test statistic's value is .025, then the null hypothesis will be retained if test statistic is set to α = .02 I know that the z score for α/2 = 2.326 which is ...
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When to reject the null hypothesis given a uniformly most powerful test?

Let $X$ be a random variable that takes values in $\{-1,0,1\}$. We want to test $H_{0}:Pr(X = -1|\Theta = 0) = Pr(X = 0|\Theta = 0) = Pr(X = 1| \Theta = 0) = 1/3$ versus $H_{1}=Pr(X = -1 |\Theta = ...
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Two sample z test: Applicability

The following example exercise is taken from Statistics by Freedman A geography test was given to a simple random sample of 250 high-school students in a certain large school dishict. One ...
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hypothesis testing - gamma distribution

Let W = Y/B0 be a Random variable that has a gamma(2n,1) distribution. [Y has a gamma(2n,B) distribution and W = Y/B]. i) Suppose you want to test H0 : B ≤ B0 against H1 : B > B0 for some B0 > 0. How ...
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Determine whether the following $H_0$ can be accepted or not using hypothesis testing

$H_0 : p=0.5$ $H_1 : p > 0.5 $ where p is the probability of heads from a coin flip. Let $W_1$ be the number of heads from 10 coin flips and and $W_2$ be the number of heads from 1000 coin ...
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Neyman-Pearson lemma with pdf of $X$ vs Neyman-Pearson Lemma with likelihood functions

As written in the subject line, I’m confused about the statement made in the Neyman-Pearson Lemma, largely because one of my two textbooks states the Lemma in terms of the pdf of $X$ while the other ...
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For which $\alpha$-sizes is there a MP level-$\alpha$ test?

Let $X_1,...,X_n$ be idd and uniformly distributed over $(0,\theta)$. Choosing between the null hypothesis which states that $\theta=\theta_0$ and the alternate hypothesis which states that $\theta=\...
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Determining the most powerful test for a simple hypothesis

Let $X_1,...,X_n$ be $N(\mu,\sigma^2)$ where $\mu$ is unknown and $\sigma$ is known. Let the null hypothesis $H_0$ state that $\mu=\mu_0$ and the alternate be that $\mu=\mu_1$ where $\mu_1>|mu_0$. ...
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How to make a Hypothesis test with a Random Variable?

You want to try the hypothesis $H_0:\mu=5$ against $H_1:\mu>5$, using a single observed value of a random variale with density function $f(x;u)=\frac{1}{\mu}e^{-\frac{x}{\mu}},x>0$.Consider the ...
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When does a MP level-alpha test not exist?

I’m currently reading about the Neyman-Pearson Lemma for the MP level-$\alpha$ test. My question is, when does there not exist a MP level-$\alpha$ test? For example, suppose that $X_1,...,X_n$ are ...
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Using Excel to find the P-value of a t-statistic.

I am currently studying probability and statistics, and since my future goal involves some data analysis I am trying to learn basic use of excel. In the moment I am learning t-distributions and ...