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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.

In statistics, a result is called statistically significant if it is unlikely to have occurred by chance alone, according to a pre-determined threshold probability, the significance level. A statistical hypothesis test is a method of inference using data from a study which is used to determine if a measured result differs from a predicted result in a statistically significant way.

The phrase "test of significance" was coined by statistician Ronald Fisher. These tests are used in determining what outcomes of a study would lead to a rejection of a null hypothesis for a pre-specified level of significance. This can help to decide whether results contain enough information to cast doubt on conventional wisdom, given that conventional wisdom has been used to establish the null hypothesis. The critical region of a hypothesis test is the set of all outcomes which cause the null hypothesis to be rejected in favor of the alternative hypothesis.

Statistical hypothesis testing is sometimes called confirmatory data analysis, in contrast to exploratory data analysis, which may not have pre-specified hypotheses.

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What is the theory behind rigorous hypothesis testing?

I understand that hypothesis testing is essentially a statistical form of proof by contradiction. In proof by contradiction, you assume P, then show that it leads to a result Q, which you know to be false. Since we assume logical consistency, Q and…
Daniel Goldman
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Proof and precise formulation of Welch-Satterthwaite equation

In my statistics course notes the Welch-Satterthwaite equation, as used in the derivation of the Welch test, is formulated as follows: Suppose $S_1^2, \ldots, S_n^2$ are sample variances of $n$ samples, where the $k$-th sample is a sample of size…
Bib-lost
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Is $\pi^k$ any closer to its nearest integer than expected?

Particular questions such as Why is $\pi$ so close to $3$? or Why is $\pi^2$ so close to $10$? may be regarded as the first two cases of the question sequence Why is $\pi^k$ so close to its nearest integer? For instance, we may stare in awe in…
Jaume Oliver Lafont
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Proof of Karlin-Rubin's theorem, detail about a real analysis fact.

Although the setting of this question is statistics, the question actually asks for a real analysis fact (monotone functions). Karlin-Rubin's theorem states conditions under which we can find a uniformly most powerful test (UMPT) for a statistical…
user39756
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What is a $\it{nontrivial}\,$ Square Root?

I need to understand the concept behind a non trivial square root. Also how to answer these two questions and how to get to the answer? Give a non-trivial square root of 30 Give a non-trivial square root in the integers for (mod 143)
Bob
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Two definitions of p-value

According to Casella-Berger (2002) the definition of p-value (def. 8.3.26, §8.3.4, p. 397) is: A p-value $p(X)$ is a test statistic satisfying $0 \le p(x) \le 1$ for every sample point $x$. Small values of $p(X)$ give evidence that $H_1$ is true. A…
user1868607
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Finding an upper bound for $\frac{d}{d\theta}\beta^*(\theta)|_{\theta=\theta_0}$

Suppose that a random variable X has a distribution depending on a parameter $\theta$, $\theta \in \Theta$, and consider a test of hypothesis $H_0: \theta = \theta_0$ versus the alternative $H_1: \theta \in \Theta_1$, where $\theta_0 \in \Theta$…
statsguyz
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In statistics, why do you reject the null hypothesis when the p-value is less than the alpha value (the level of significance)

This is a question that I've always wondered in statistics, but never had the guts to ask the professor. The professor would say that if the p-value is less than or equal to the level of significance (denoted by alpha) we reject the null hypothesis…
user111019
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Exponentially weighted infinite sum of Bernoulli variables

Consider the following process. For each integer $i \geq 0$, independently sample a Bernoulli distribution with probability $p = 1/2$, obtaining sample $x_i$. Then calculate $x = \sum_{i=0}^\infty x_i \theta^i,$ where $(\theta < 1)$. What is the…
causative
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Two-tailed hypothesis test; Why do we multiply p-value by two?

I understand that in a two-tailed hypothesis test, we must multiply the p-value by two. i.e. if z=1.95 and it's a one-tailed hypothesis test, our p-value is 0.0256. But, if it's a two-tailed hypothesis test and z=1.95, we must multiply the p-value…
developer01
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Choosing $H_0$ and $H_a$ in hypothesis testing

There seems to be some ambiguity or contradiction in how to correctly choose the null and alternative hypotheses, both online and in my instructor's notes. I'm trying to figure out if this stems merely from my lack of understanding or if there…
Egor
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Chi Squared for Goodness of Fit

Hi, any help is appreciated :) I am trying to teach myself statistics. I've watched the Khan Academy Series on Chi square statistic for hypothesis testing…
dependent somewhere
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Understanding the P-value

I'm having difficulty understanding the p-value. It is said to reject the null hypothesis when the p-value is small. Smaller than the significance level. So does that mean in a hypothesis test, the p-value represents the area of the null hypothesis?…
user
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Uniformly Most Powerful Test for a Uniform Sample

Let $X_{1}, \dots, X_{n}$ be a sample from $U(0,\theta), \theta > 0$ (uniform distribution). Show that the test: $\phi_{1}(x_{1},\dots,x_{n})=\begin{cases} 1 &\mbox{if } \max(x_{1},\dots,x_{n}) > \theta_{0} \quad or \quad \max(x_{1},\dots,x_{n})…
arcbloom
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Likelihood Ratio Test for Linear Regression

I apologize for the image I am posting below. I am new to StackExchange and I am not yet familiar with the MathJaX equations, so I took a screenshot. Here is my question: Let the independent random variables $Y_1, \ldots , Y_n$ have the following…
mrstatguy
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