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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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Logic behind hypothesis testing and one-tailed tests

Suppose I have the following: Let's consider a hypothetical experiment to determine whether James Bond can tell the difference between a shaken and a stirred martini. Suppose we gave James Bond a ...
Darren's user avatar
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2 votes
1 answer
66 views

Autocorrelation of p-values after $n$ observations

For $i = 1, 2, \dots$, let $X_i \sim N(0, 1)$ and $Y_i \sim N(0, 1)$, with all observations independent of each other. Suppose that after observing $X_1, \dots, X_n, Y_1, \dots, Y_n$ we calculate the ...
Alex's user avatar
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2 votes
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67 views

Consistency of a hypothesis test

I have given statistical model $((0,1)^n, \mathcal{B}(0,1)^n,\mathcal{P}_n)$, where $\mathcal{P}_n=\{ P_{\theta}^{\otimes n} \ |\ \theta \in (0, \infty) \}$ and each $P_{\theta}$ has density function $...
tychonovs-scholar's user avatar
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2 answers
30 views

How do you determine the critical region/decision rule in hypothesis testing?

I'm currently studying hypothesis testing for one sample t-tests. In a PowerPoint presentation uploaded by my professor, it was stated that the critical region/decision rule for ${H}_{a}$: μ ≠ ${μ}_{0}...
Vee's user avatar
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2 votes
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31 views

UMP test for Bernoulli distribution one sample

Given $\mathcal{X}=\{0,1\}$, $\Theta= \{ 1/2,1/4 \}$ and $\mathcal{P}= \{ B(\theta):\theta \in \Theta \}$, I am trying to construct an UMP test of size $\alpha \in (0,1)$ for $$H_0: \theta=1/4 \,H_1: \...
tychonovs-scholar's user avatar
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28 views

Can two hypothesis tests with opposing null hypothesis be compared to each other?

Suppose I have an estimator $\hat{\beta}$ for some parameter $\beta_0$ and I have two hypothesis tests $$ H_0^{(1)}:\hat{\beta}=\beta_0\quad\text{versus}\quad H_1^{(1)}:\hat{\beta}\neq\beta_0 $$ and $$...
Daan's user avatar
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1 vote
1 answer
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Rejection region in hypothesis test using LRT [closed]

I have $X_i \sim Bi(1, \theta)$ and want to test $H_0: \theta = \theta_0$ vs $H_1: \theta \neq \theta_0$ using the LRT test. I've found that $\lambda = \frac{sup_{\theta \in \Theta_0}L(\theta)}{sup_{\...
Peter Sampodiras's user avatar
2 votes
2 answers
86 views

Which coin are you more confident is biased?

Coin 1 $(c1)$: 550 Heads of 1000 flips. Coin 2 $(c2)$ 60 heads of 100 flips. Let's test the follow hypothesis for each: $$H_0: p = 1/2 \quad H_A: p \neq 1/2$$ We know that: $$\mu = np \quad \sigma = \...
user2330624's user avatar
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1 answer
19 views

Understand computational formula of Cramer von Mises statistic

I've found this question that explains how to get from the integral to computational formula of Cramer von Mises statistic: Computational formula for Cramer-von Mises test statistic. The answer is ...
Édio Renato Fávaro's user avatar
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Search for a statistical test to analyze the "home ground advantage" for rugby matches

I have a database of results from club-level rugby-union matches. It has results from a number of European leagues/competititions over about 15 years. I am trying to do some analysis on the 'home ...
Zélie BESANCENET's user avatar
0 votes
1 answer
22 views

P-value for testing a median to be $M \geq M_0$

I'm trying to solve a question from an introductory textbook on statistics. I am to use the Sign Test and determine if there is significant evidence that the median of a dataset $M$ is "at least&...
fatCat9999's user avatar
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0 answers
35 views

Hypothesis testing on densities of a random variable

I'm solving problems, but my answer doesn't match the answer in the textbook. The problem looks like this: Given a sample $X_1,\ldots, X_n$. Hypotheses about the type of distribution are represented ...
moxeeeem's user avatar
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1 answer
56 views

How do I calculate p-value in this problem?

Random samples of resting heart rates are taken from two groups. Population 1 exercises regularly, and Population 2 does not. The data from these two samples is given below: Population 1: $63, 72, 72, ...
No Go's user avatar
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1 answer
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Interpretation of the one-sided t-test

I am comparing two methods, called A and B, on their performance (the lower the error, the better) I have a set of inputs, and I ran both methods for each input, logging the errors. A consistently ...
Vũ Lê Thế Anh's user avatar
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0 answers
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Hypothesis testing of Precision-Recall curve AUCs

In recent times, I have been about learning classification models (e.g., logistic regression) and how to evaluate them. While learning about the Precision-Recall (PR) curves, it occurred to me that ...
Yat-Hon's user avatar
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Is there a UMP test of $H_0: \theta \le \theta_0$ vs $H_1: \theta > \theta_0$

Let $Y_1, \ldots, Y_n$ be a random sample from a distribution with the density function: $$ fθ(y)=\dfrac{3\theta^3}{y^4} \text{ for } y \ge \theta > 0$$ Is there a UMP test at level α for ...
bthomas28's user avatar
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28 views

Can a Student's t-test be used to identify whether or not a single observed data point should be judged as an outlier from a null distribution?

FYI: This question may be more thematically appropriate for Cross Validated's site, but, from previous experience, there is considerably less traffic there compared to Math, and I still feel that this ...
S.C.'s user avatar
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1 vote
1 answer
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What is the power of the test?

There are $n$ cards in an urn labelled $1$ through $n$, where $1\leq n\leq 5$ is an unknown integer parameter. The null hypothesis $H_0: n=5$ and $H_1: n\leq 4$. We reject $H_0$ if we draw two cards ...
Dave the Sid's user avatar
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0 answers
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Problem about critical value for multiple comparisons

Consider a one-way layout of fixed effect model: $$y_{ij}=\eta+\tau_i+\epsilon_{ij}, \epsilon_{ij} \sim N(0,\sigma^2),i=1,...,k,\ j=1,...,n,$$ $\tau_i$ is the main effect, and we want to test $H_0:\...
eric's user avatar
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0 answers
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Asymptotic Distribution and Describe Sources of Increasing Power in an hypothesis testing problem

I am currently dealing with the following problem in a past exam (with no solution): Suppose $S$ follows the Poisson distribution with mean $2\lambda>0$, here $\lambda$ is a parameter. Another two ...
INvisibLE's user avatar
  • 188
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0 answers
22 views

Using Neyman-Pearson Lemma to Find the Power of Test

Let $X_1, X_2, \ldots, X_n$ be a random sample of size $n$ from a population with a normal distribution $N(\mu, 1)$, where we know that $\mu = \mu_0$ or $\mu_0 + 1$. Consider the hypothesis test $H_0 :...
sasa's user avatar
  • 33
0 votes
1 answer
58 views

Choosing Null Hypothesis

A company producing light bulbs is interested in the average lifetime (in hours) of their bulbs. It is assumed that the lifetime (in hours) of a light bulb follows an exponential distribution. The ...
adisnjo's user avatar
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0 votes
1 answer
135 views

Likelihood ratio test of Poisson distribution

I'm given this problem: Let $X_1,...X_{100}$ be a random sample from a Poisson distribution with mean $\lambda$. Consider testing the hypothesis $H_0$: $\lambda=1$ vs $H_1$: $\lambda<1$. Consider ...
lcthaha's user avatar
0 votes
1 answer
110 views

Constructing most powerful test

Suppose we have a random sample of size $n = 1$ from the probability density function: $$f(x \mid \theta) = \begin{cases} 1 + \theta^2(0.5-x), & 0<x<1\\ 0, & \text{otherwise} \end{cases}$...
adisnjo's user avatar
  • 247
1 vote
1 answer
81 views

Using the critical value from signs test to reject null hypothesis

Question 1 (a) A graduation using $20$ age groups has resulted in $6$ positive and $14$ negative deviations. Carry out the signs test on these data. (b) A graduation covers $20$ age groups. The number ...
Jessie's user avatar
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6 votes
0 answers
109 views

Eigenvalues of likelihood ratio

Consider the following hypothesis testing problem: Under $H_0$: $(X,Y) \sim N(0,1)\times N(0,1)$, i.e. $X$ and $Y$ are independent standard normal. Under $H_1$: $(X,Y) \sim N\left(\begin{pmatrix} 0\\...
efsdfmo12's user avatar
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0 answers
31 views

Hypothesis testing for correlation in bivariate normal sample

Assume that $(X_1, Y_1), . . . , (X_n, Y_n)$ are independent bivariate normal random vectors with parameters $(\mu,\mu, \sigma^2, \sigma^2, \rho)$, where $\mu$ and $\sigma^2$ are known. We are ...
SK25's user avatar
  • 23
2 votes
0 answers
113 views

Deriving the Likelihood Ratio for a Normal Distribution Hypothesis Testing

I'm exploring the likelihood ratio principle in hypothesis testing, specifically within the context of normal distributions, and I've encountered a challenge in deriving a specific likelihood ratio. ...
Johnny Depp's user avatar
0 votes
0 answers
28 views

Confidence interval construction for $f(p)$ of categorical distribution parameter $p$

Let $X_1, \dots, X_n$ be i.i.d. random variables from a categorical distribution with parameter vector $p = (p_1, \dots, p_k)$. Suppose we’re interested in the estimating $\vartheta := f(p)$ for some ...
zxmkn's user avatar
  • 1,550
0 votes
1 answer
36 views

Significance value in Hypothesis testing

A builder claims that heat pumps are installed in $70\%$ of all homes being constructed today in Chennai. Would you agree with this claim if a random survey of new homes in this city showed that $8$ ...
math's user avatar
  • 302
0 votes
0 answers
11 views

Design the problem on testing the null hypothesis $H_0: \pi_{11} \pi_{22}=\pi_{12}\pi_{21} $ under multinomial distribution

Given $(n_{11}, n_{12},n_{21},n_{22}) \sim \text{Mult }(n; \pi_{11}, \pi_{12},\pi_{21},\pi_{22})$, consider the problem of testing the null hypothesis $H_0: \pi_{11} \pi_{22}=\pi_{12}\pi_{21} $ ...
Jonathen's user avatar
  • 1,088
1 vote
1 answer
53 views

Design two independent normal samples to test $H_0: \mu_1=\mu_2$

Given independent random samples $(X_1,...,X_m)$ and $(Y_1,...,Y_n)$, respectively, from the following distributions of $X$ and $Y: X\sim N(\mu_1, \sigma^2)$ and $Y \sim N(\mu_2,\sigma^2)$,consider ...
Jonathen's user avatar
  • 1,088
2 votes
1 answer
53 views

How to design this two sample exponential distribution test?

Given independent random samples $(X_1,...,X_m)$ and $(Y_1,...,Y_n)$, respectively, from the following distributions of $X$ and $Y: X \sim \lambda_1 \exp (-\lambda_1 x) I(x >0)$, and $Y \sim \...
Jonathen's user avatar
  • 1,088
0 votes
0 answers
268 views

Randomized and nonrandomized statistical tests

I've been reading recently about randomized and non-randomized statistical tests and their differences. If I understand the definition correctly, a nonrandomized test for all values of the test ...
gaghan's user avatar
  • 123
1 vote
0 answers
52 views

Intuition behind statistical approach for Hypothesis testing

I have studied Hypothesis testing from thinkstats2 which does simulation to estimate p value. For e.g. In coin toss example, we run the simulation, to generate the test statistic i.e. Difference ...
Gupta's user avatar
  • 121
0 votes
1 answer
37 views

Computing the p-value of one sided hypothesis test

I'm studying this subject on my own and would just like a sanity test to see if I'm doing things correctly. Part a): we consider the test statistic $ \frac{\overline{X} - \mu}{S/\sqrt{n}}$ which at $\...
Bastiza's user avatar
  • 303
0 votes
0 answers
24 views

Continuity of the power function in UMPU testing for two parameter uniform distribution

Suppose $X_1 , \dots, X_n$ are i.i.d. $U(\theta_1, \theta_2)$, with $\theta_1 < \theta_2$. We want to test for $$ H_0: \theta_1 \leq 0 \quad \text{versus} \quad H_1: \theta > 0. $$ It can be ...
ムータンーオ's user avatar
0 votes
0 answers
25 views

Assess if occurences of a binary variale is different from $\alpha \cdot 100 \%$

I have a binary variable, $X_t$, which takes values $1$ or $0$. A value of $1$ indicates that an event happend at time $t$. I would like to statistically test whether we can say that the event happens ...
JKL's user avatar
  • 21
1 vote
0 answers
29 views

Test equality of two covariance matrices

I am trying to test the equality of two covariance matrices based on articles from Srivastava, M.S. and Yanagihara, H., 2010 and Ishii, Yata and Aoshima 2016. I am not a statistician but a programmer. ...
user3204284's user avatar
1 vote
0 answers
17 views

Why do we use the test for a normal mean when variance is unknown, even though we are given the variance?

Take the question: A teacher wants to see the effect of changing how reading is taught to primary school children. The children in Year $4$ take a reading test at the end of the year. In previous ...
billymaths's user avatar
0 votes
0 answers
17 views

Estimation of std error using sample std dev. will result in diff. sample statistic distribution for diff. samples. Won't this affect our decision?

During hypothesis testing (say, comparison of sample mean to a hypothesised population mean), we calculate p-value using sample mean distribution. For constructing this sample mean distribution, we ...
Swapna Sarit's user avatar
0 votes
0 answers
37 views

How to formally derive this formula for Kolmogorov-Smirnov two-sample test D-statistic

I'm trying to wrap my head around Kolmogorov-Smirnov two-sample test. So, Kolmogorov-Smirnov test uses the following D-statistic: $$D_{n_1, n_2} = \sup\limits_{-\infty < x < +\infty} \left|\...
missing_name's user avatar
1 vote
0 answers
80 views

Neyman-Pearson Lemma, critical region

So, a bit of context first, I was studying how to construct the most powerful test using the Nyman-Pearson Lemma. The example went like this: given the normal distribution with known variance $\sigma^...
missing_name's user avatar
1 vote
1 answer
42 views

Proof of Probability of Obtaining Mann Whitney U under the Null. [closed]

I am trying to understand how the Recursion Probability Formula of Mann Whitney U table under the null hypothesis is derived in this Paper, Equation 1 from section 4(given below). $$p_{n,m}(U) = \...
yo wa's user avatar
  • 21
0 votes
0 answers
30 views

For every abelian variety $A$ over a number field $K$ and for a prime number

For every abelian variety $A$ over a number field $K$ and for a prime number $p$ ,$$\begin{equation*} (-1)^{\operatorname{rk}_p (A/K)} = w_{A/K}. \end{equation*}$$ 2-parity conjecture holds for all ...
user avatar
0 votes
0 answers
28 views

What test can we use to compare the sample proportion of multiple dependent samples (> 2) for non-dichotomous data (> 2 categories)?

I am currently studying hypothesis testing for dependent two-sample (proportion). The crux for my question is this, what test does one use to compare the proportion of multiple samples for non-...
Yat-Hon's user avatar
  • 21
2 votes
1 answer
32 views

Can t-tests be 1 sided? This example has confused what I thought I knew.

We first take a random sample of $8$ students, record their final score in $2018$, and check their final scores again in $2019$. $2018: 76, 73, 66, 95, 75, 78, 96, 93$ $2019: 75, 80, 70, 93, 81, 90, ...
Bill Cogn's user avatar
1 vote
0 answers
58 views

Statistical inference for the integral equation

Consider a integral equation $$ \begin{aligned} \mathbb{E} \left[ Y|A \right] &=\mathbb{E} \left[ g\left( W \right) |A \right]\\ \int{yp\left( y|a \right) dy}&=\int{g\left( w \right) p\left( ...
叶心萤's user avatar
0 votes
0 answers
31 views

uniform i.i.d test consistent?

$X_k \sim U(0,L)$ i.i.d; $H_0: L=L_0 ~ H_1: L \ne L_0$; $L,a \in(0,1)$ is a test with the following power function: $P_n(L_0)= P( max_{k=1,..,n}X_k>L_0~\cup~ max_{k=1,..n}X_k<L_0*a^{1/n})$. How ...
VeryGenericUsername's user avatar
0 votes
0 answers
23 views

A hypothesis For $m \geq 1$, $l_j > 0$, and $x_j \in (-1,1)$

(Hypothesis) For integer $m \geq 1$, $l_j > 0$, and $x_j \in (-1,1)$, then the following identity, whether or not to be established: $$ \prod_{j=1}^m \text{Li}_{l_j}\left[ x_j \right] = \sum_{k=0}^...
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