# What is Hypothesis testing: Definition and 86 Discussions

A statistical hypothesis is a hypothesis that is testable on the basis of observed data modelled as the realised values taken by a collection of random variables. A set of data is modelled as being realised values of a collection of random variables having a joint probability distribution in some set of possible joint distributions. The hypothesis being tested is exactly that set of possible probability distributions. A statistical hypothesis test is a method of statistical inference. An alternative hypothesis is proposed for the probability distribution of the data, either explicitly or only informally. The comparison of the two models is deemed statistically significant if, according to a threshold probability—the significance level—the data would be unlikely to occur if the null hypothesis were true. A hypothesis test specifies which outcomes of a study may lead to a rejection of the null hypothesis at a pre-specified level of significance, while using a pre-chosen measure of deviation from that hypothesis (the test statistic, or goodness-of-fit measure). The pre-chosen level of significance is the maximal allowed "false positive rate". One wants to control the risk of incorrectly rejecting a true null hypothesis.
The process of distinguishing between the null hypothesis and the alternative hypothesis is aided by considering two conceptual types of errors. The first type of error occurs when the null hypothesis is wrongly rejected. The second type of error occurs when the null hypothesis is wrongly not rejected. (The two types are known as type 1 and type 2 errors.)
Hypothesis tests based on statistical significance are another way of expressing confidence intervals (more precisely, confidence sets). In other words, every hypothesis test based on significance can be obtained via a confidence interval, and every confidence interval can be obtained via a hypothesis test based on significance.Significance-based hypothesis testing is the most common framework for statistical hypothesis testing. An alternative framework for statistical hypothesis testing is to specify a set of statistical models, one for each candidate hypothesis, and then use model selection techniques to choose the most appropriate model. The most common selection techniques are based on either Akaike information criterion (=AIC) or Bayesian information criterion (=BIC).

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1. ### What should I do with my spacetime model?

I created a model of spacetime based on an extension I added to the principle of relativity. I then derived the coordinate transformations which preserve the speed of light in all frames, which are different from the Lorentz transformation. I worked out the formulae for energy and momentum, and...
2. ### A Presentation: Venus hot break hypothesis

The idea will be presented at the ancient Venus conference https://www.hou.usra.edu/meetings/ancientvenus2022/pdf/ancientvenus2022_program.htm Houston 25 Jul. The abstract: https://www.hou.usra.edu/meetings/ancientvenus2022/pdf/2024.pdf Some of the earliest members may remember a thread about...
3. ### Solve the problem that involves Hypothesis testing and Bin (n,p)

##p(x≤11)=1-[p(x=12)+p(x=13)+p(x=14)+p(x=15)]## ##p(x≤11)=1-[0.128505+0.266895+0.3431518+0.205891]=1-0.9444428=0.0556## ##⇒p(x>11)=0.9444## ##p(x≤10)=1-[0.042835+0.128505+0.266895+0.3431518+0.205891]=1-0.9872778=0.0127222## Since, ##p(x≤11)=0.0556> 0.05## then it falls on the Accepted region...
4. ### I Rovelli on Quantum Gravity

Rovelli points to three pieces of existing observational evidence that should guide future quantum gravity research. Bottom line: * abandon Lorentz invariance violating quantum gravity theories, * abandon supergravity and string theory, and * stop working on the anti-deSitter/conformal field...

6. ### B Rejection region of hypothesis testing

From z-table, I get the critical value of z is -1.282 Will the rejection region be z < -1.282 or z ≤ -1.282? If I calculate the test statistics and it has value of -1.282, would I reject or accept null hypothesis? Thanks
7. ### How to Determine Whether to Accept or Reject a Hypothesis?

Summary:: Confused on correction for contunuity I'm confused about one step for questions like this where you have to determine to accept or reject the null hypothesis. Sample size: 50 Number of successes: 23 Significance level: 10% Null hypothesis: p = 0.4 Accepted hypothesis: p > 0.4...
8. ### B Stats: Determining significance level in problem

Question: In finance, the strong form of the efficient market hypothesis states that studying financial information about stocks is a waste of time since all public and private information that might affect the stock price is already reflected in the price of the stock. However, a study of 450...
9. ### B Statistics Help : Hypothesis Testing

Answer : I understnad why x(< or = ) 2 but I do not understand why we use 16 instead of 17 for the second range? When P(X>=16) > 0.005(which is the level of significance). Thank you for all the help given :)
10. ### I Hypothesis Testing: Comparing Gaussian Distributions

Hi, I have some set of data and I want to use Hypothesis Testing to discriminate between two hypotheses: H0: My data follows a Gaussian distribution with a given mean and a given std (the actual values are ugly, so let's say mean = 0 and std = 1). H1: My data follows a Gaussian distribution with...
11. ### Showing Rejection Region Equality with Fisher Distribution

Homework Statement [/B] For reference: Book: Mathematical Statistics with Applications, 7th Ed., by Wackerly, Mendenhall, and Scheaffer. Problem: 10.81 From two normal populations with respective variances ##\sigma_1^2## and ##\sigma_2^2##, we observe independent sample variances ##S_1^2## and...
12. ### I Multiple hypothesis testing for radar tracking in clutter

Hello All, The goal is to formulate a multiple hypothesis test for a radar tracking problem when false alarms are occurring and to apply a particle filter on this update step, however I first need to come to/understand the multiple hypothesis formulation in this problem. Say we are interested...
13. ### I Can Bayesian Hypothesis Testing Determine the Authenticity of Cables?

My example question is as below. The question is wordy and hard to wrap your head around but I completely understand what it is asking, I just have no idea how to go about it, or where to start. I have spent hours scratching my head on this one! We have been taught two methods of Bayesian...
14. ### I Why does normal distribution turn into t distribution when variance is unknown?

Suppose ##X## ~ N(##\mu##,##\sigma^2##). Then ##\bar{X}## ~ N(##\mu##,##\frac{\sigma^2}{n}##), where ##\bar{X}## is the random variable for sample mean for samples of size ##n##. But when the population variance ##\sigma^2## is unknown and the sample size ##n## is small, ##\bar{X}## no longer...
15. ### MHB Hypothesis testing - Winning a game

Hey! :o A and B play 100 games of squash; A wins E times. B claims that both of them have the same probability of winning a game. We consider that the games are independent. (a) Formulate the null hypothesis and the alternative hypothesis. (b) For which values of $E$ will the null...
16. ### MHB Performance of students - Hypothesis testing

Hey! :o A teacher wants to find out if the order of the exam tasks has an impact on the performance of the students. Therefore, he creates two versions ($X$ and $Y$) of an exam in which the exam tasks are arranged differently. The versions are randomly distributed so that $n$ students...
17. ### Sampling Distribution Question

I've added a screenshot (https://imgur.com/a/isQXZ) and the text below for your convenience. Please show steps if possible to help my understanding. Thank you. Consider a random variable that is Normally distributed, with population mean mu = E [X] and population variance sigma^2 = var [X]...
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### B Postulates and pseudoscientific hypothesis difference?

A postulate is a statement that is taken to be true, to serve as a premise or starting point for further theories. (e.g. Bohr's postulates) Hypothesis is a theory which can after testing be accepted or rejected. But a postulate is something that is assumed to be true without proof. Among other...
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### Is the claim of 18mg of nicotine per cigarette accurate?

Homework Statement An advertisement for a certain brand of cigarettes claimed that on average there's no more than 18mg of nicotine per cigarettes . A test of 12 cigarettes gave a sample mean of 19.1 . Assuming varience is 4 , test the claim with a significance level of α = 0.05 Homework...
20. ### MHB Calculating Listeners Tuning in 8-9am: Least/Greatest #s

a radio station claims 55% of all listeners tune into their program between 8 and 9 am. To check this claim a competing radio station does a survey of a random sample of listeners in the 8 to 9 am time slot. the survey indicates the claim can not be rejected. if the sample size is 1000 people...
21. ### Conceptual Question regarding hypothesis testing regression

Homework Statement Hi, I had a question regarding testing a regression models coefficients. Say there is a regression model that has the form: y = b0 + b1x1 + b2x2 + b3x3 + b4x4 + e For the sake of simplicity let: e be the random error, x1 is age, x2 is severity, and x3 is anxiety. y is...
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### How Do You Test a Hypothesis Without Sample Variance?

Homework Statement Given ##X_1,\dots,X_{100}##, test ##H_0: \lambda=1## against ##H_a: \lambda=4##. The mean ##\bar{X_{100}}=1.5## (1) Take the decision on 3% level. (2) Find the p-value Homework Equations ##t=\frac{\bar{x}-\mu}{s/\sqrt{n}}## The Attempt at a Solution The level of...
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### Minimize the sum of Type I and Type II errors

Homework Statement Given X_1,\dots,X_n a simple random sample with normal variables (\mu, \sigma^2). We assume \mu is known but \sigma^2 is unknown. The hypothesis is \begin{cases} H_0: & \mu=\mu_0 \\ H_1: & \mu=\mu_1 > \mu_0 \end{cases} Determine the rejection region R...
24. ### I Algorithm to create a composite score

Hi everyone! This is an application question. I would like to get some advice about how to calculate a score based on a set of individual scores in a way that makes most sense. CONTEXT: I am going over some criteria for judging usability of hypotheses. I came up with a whole bunch about a...
25. ### Confusion between z and t values

Homework Statement Hi, Alright so I have some confusion on when to use specific tests and the z vs t test. Given this example (not my homework) could someone please clarify. Alright say you have a random sample of size 200. You find the sample mean to be 10 and the sample standard deviation...
26. ### MHB P-value: 0.0098Area of the critical region: 0.01Critical value(s): ±2.326

Use the following information to answer the questions below: A 0.01 significance level is used for a hypothesis test of the claim that when parents use the XSORT method of gender selection, the proportion of baby girls is different than 0.5. Assume that sample data consist of 62 girls born in...
27. ### I How do you implement the Dickey-Fuller test?

Hi there, I've recently start learning methods for determining whether or not time series are stationary. The first method I'm trying to learn is the 'Dickey-Fuller Test'. This test uses a time series modeled by an AR(1) process. The key is to find whether or not this process contains a unit...
28. ### Neurologist: What P-values should I be expecting?

Inexperienced data analyst here with a real-world example, I have attached a zip-file with screenshots and p-values of the following data. The "reference regions" are Cerebellum White, Cerebellum Gray, and Temporal Cortex. The top-most graphs depict the curves in the indicated region for young...
29. ### Statistics, testing hypothesis

Homework Statement http://prntscr.com/8ea2rk Homework Equations In statistics, when testing hypothesis for both sides (left and right), is it possible that Xaverage - MUo = 0 (upper part of the equation I provided)? If so how to solve this kind of task? The Attempt at a Solution If I get...
30. R

### What is the Bell Hypothesis in Quantum Mechanics?

Hello, I'm a new member on this site and I have a hypothesis that struck me one day. I've been sitting on it for four years but I'm hoping to explore it further and possibly with the help of others. I really hope this site is the place to do it. I'm willing to do the homework required because...
31. ### Hypothesis testing, P-value, β-error, sample size for β<=0.1

Homework Statement The life in hours of a battery is known to be normally distributed, with standard deviation of 1.5 hours. A random sample of 12 batteries has a sample mean life time of 50 hours. a) Test the hypothesis that the mean battery life is 50.5 hours (by using α = 0.05). b) What is...
32. ### Confidence interval of two sample tests

Homework Statement Some car tires can develop what is known as "heel and toe" wear if not rotated after a certain mileage. To assess this issue, a consumer group investigated the tire wear on two brands of tire, A and B, say. Fifteen cars were fitted with new brand A tires and thirteen with...
33. ### Sampling for hypothesis testing

Hi guys. I'm not a statistician although I use it enough that I'm surprised something is bothering me. I'm doing hypothesis testing on a population >100,000. What I'm wondering is whether there is any difference whatsoever between performing multiple tests on several samples or just doing one...
34. ### Finding the P-Value for Hypothesis Testing

Homework Statement Suppose a mutual fund qualifies as having moderate risk if the standard deviation of its monthly rate of return is less than 6%. A mutual-fund rating agency randomly selects 28 months and determines the rate of return for a certain fund. The standard deviation of the rate of...
35. ### Hypothesis Testing: Comparing Birth Weights of Chinese and Caucasian Babies

The problem: A simple random sample of 1862 births of Chinese babies resulted in a mean birth weight of 3171 g and a standard deviation of 428 g (based on “Comparison of Birth Weight Distributions between Chinese and Caucasian Infants,” by Wen et al., American Journal of Epidemiology, Vol.172...
36. ### Can Hypothesis Testing for Proportions Simplify Design Variable Levels?

I read that I can use the hypothesis testing technique for proportions to decide whether the number of levels for a design variable could be reduced. In literature the following typical test was used: Ho: pi=0 Ha: pi \neq0 where pi is the proportion of class i (level i ) within the...
37. ### MHB Which Should You Use in Hypothesis Testing: P-Value or TCV?

how do I tell if my answer should be a P-Value or TCV?
38. ### How Do You Determine Direction in Hypothesis Testing?

Homework Statement For example in my notes in class the prof went over an example with the following null and alternative hypothesis: I feel like you can switch them around but I am not too sure. I have been trying to figure out when to use greater than or less than for the null...
39. ### Hypothesis testing for std. deviation (SD)

Hi, What I know: In a hypothesis test for the mean, we compare a sample mean with a hypothetical sampling distribution of means. And depending on how far it is away from the mean of the sampling distribution, we attribute it the probability of getting that value purely by chance. What I...
40. ### Hypothesis Testing: 95% Confidence of Over 60% Liking Chocolate Ice Cream

In a sample of 100 people. 50 people like ice cream, of those 50 people, 25 like chocolate. Calculate with 95% confidence, that over 60% of people like his chocolate ice cream. H_0= mu > 0.60 H_1=mu < 0.60 25/50=0.50% of people like chocolate ice cream (0.50-0.60)/...
41. ### Multiple Linear Regression - Hypothesis Testing

Homework Statement I'm looking through some example problems that my professor posted and this bit doesn't make sense How do you come up with the values underlined? Homework Equations The Attempt at a Solution Upon researching it, I find that you should use α/2 for both...
42. ### Hypothesis testing of a complex deterministic model

What is a good test for hypothesis testing of a complex deterministic model? What I'm doing: I have a complex system of nonlinear differential equations. I must solve them numerically. There are about 22 input parameters that I have rough physiological ranges for and I use a genetic...
43. ### Hypothesis Testing: Factorial ANOVA

Hello! So I had to propose a research design as an assignment and I chose to look at cognitive function in an exercise group vs. a control group over the period of 10 years. -subjects are tested at the beginning and end of each year The groups were not randomized on perfectly (I used two...
44. ### Statistics - Joint PMF / Hypothesis Testing

Homework Statement We sample a population 50 times with replacement, with all individual sampled equally likely. We survey the gender and quality of life. The counts are: Male: 13 high quality, 11 low | female: 18 high, 8 low. Let P_M, P_F, P_H, P_L to denote these population proportions for...
45. ### Conducting Hypothesis Testing: Level of Significance and P-Value Calculation

Homework Statement I have a few questions i have to solve. They are quite similar. 1. Given the following, draw conclusions about the 10% and 5% levels of significance. Then find a p-value. n = 10, mean: 8.179, std. dev: 0.02, |mean - μ| = 0.021, z_{0.05} = 1.645, z_{0.025} = 1.96...
46. ### Hypothesis Testing (Conceptual Problem)

Homework Statement This is not a homework problem, rather it is a conceptual question. I am wondering if the below is an accurate illustration to explain why I reject the null hypothesis. Such that H_{0}: \mu=\mu_{0} H_{a}: \mu\ne\mu_{0} By relating the reasoning with the diagram...
47. ### Hypothesis Testing Stats: Making a Test w/ Unknown SD & Tables

Statistics doesn't come to me as naturally as math. I'm curious as to how to make a hypothesis test under the assumptions that the population standard deviation is unknown and using tables only. Here is my understanding. Given Suppose: H_0: \mu = \mu_0. Suppose also that: \bar{x}, s is the...
48. ### A question about hypothesis testing

The significance level of a test is the probability that the null hypothesis is rejected when it is true, right? And the critical region is the region that we reject the null hypothesis...so can the significance level be calculated by finding the probability of being in the critical region...
49. ### A Question about Hypothesis Testing

In this link: http://www.math.harvard.edu/~phorn/362/362assn3-solns.pdf Why isn't the significane level for number 5 just 0.01? Because we know that if the mean is 30,000 (the same as the mean for the null hypothesis), the value of the power function is 0.01... Thanks in advance
50. ### Some questions about hypothesis testing

I think I’m a bit confused about what \alpha is. For the definition that is given, what theta really means when they say [tex]\alpha = max_{\theta \in w_0} P_{\theta} [(X_1, … , X_n) \in C]. I just think that it’s not really clear in my head, so is it ok if anybody explains/gives and example...