Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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

0
votes
1answer
29 views

1st Yr Statistics: Hypothesis testing for the mean number of accidents at the 5 percent sig level

The following table gives the number of fatal accidents of U.S. commercial airline carriers in the $16$ years from 1980 to 1995. [See screenshot below] Do these data disprove, at the $5$ percent level ...
0
votes
1answer
13 views

One sample t test cases

So I want to know when do we accept the null hypothesis and when we don't in t test. Case 1: t cal>t tab and p value >0.05,we accept null hypothesis Case 2: t cal>t tab and p value <0.05 we don't ...
1
vote
0answers
11 views

One sample student t test

My teacher solved it this way H0: μ=140 HA: μ different from 140.Significance level 0.05. Test in two sides. alpha/2=0.05/2=0.025. For this alpha and df=8, we get t(tab)=+-2.896. t cal (using the ...
0
votes
1answer
28 views

Likelihood Ratio Test of exp(λ) distribution with multiple samples

I'm probably really overcomplicating things but I want to specify the likelihood ratio test with significance level $\alpha = 0.05$ I have three random samples (sample sizes: $n_1, n_2$ and $n_3$), ...
1
vote
2answers
47 views

1st Yr Statistics Question: Create an approximate $\alpha$ level test of $H_0 : p_1 = p_2$

Let $X_1$ and $X_2$ be binomial random variables with respective parameters $n_1, p_1$ and $n_2, p_2$. Show that when $n_1$ and $n_2$ are large, an approximate level $\alpha$ test of $H_0 : p_1 = p_2$ ...
0
votes
1answer
24 views

Choosing confidence interval for hypothesis testing

"Suppose that $X_1,...,X_n$ is a sample on $N(\mu, \sigma)$ where $\sigma$ is known. You want to test the null hypothesis $H_0: \mu = 2$ against $H_1 : \mu < 2$ by using a suitable confidence ...
0
votes
0answers
15 views

If n=15 and the standard deviation is known, do I use the t-test or the z-test or chi-squared test?

I am testing my null hypothesis of $\mu=3$ against the alternative of $\mu >3$. $n=16$ and $\sigma=1$ and $\alpha=0.10$. I am conflicted because it is a small sample size and I know the standard ...
0
votes
1answer
11 views

Do you need given standard deviation when determining the critical values of a test statistic?

I am given the following information: $$\alpha = 0.025$$ $$n = 15$$ $$\sigma^2 = 100$$ I need to find the critical values. My attempt: I chose the t-test. $$+/-t_{(n-1,\frac{\alpha}{2})}$$ $$+/-t_{(...
1
vote
0answers
13 views

Test goodness of fit for weighting samples

There are N samples $x_1, x_2, ...,x_N$ for random variable $X$. I know Pearson's chi-squared test can be used to figure whether X follows uniform distribution. First, discretize the range of $X$ ...
0
votes
0answers
19 views

Goodness of fit test

I have the following exercise that shows $n=6$ numbers: $$ 1.40, 1.55, 1.35, 1.50, 1.29, 1.64 $$ Is data normally distributed at the 5% significance level? Surely $\overline{x} = 1.455$, $s=0....
0
votes
0answers
10 views

Test Poisson distribution just from sum of values and sum squared values

We have given number of events over fixed period of time, however the only thing we know are those two sums: $\sum_{i=1}^{50} X_i = 507$ and $\sum_{i=1}^{50} X_i^{2} = 5762$. Now we have to use ...
0
votes
1answer
14 views

Comparing the performance of two algorithms with Wilcoxon Signed-Rank Test

I have two algorithms A and B which have to compute a solution to some problem. Each solution is given some objective value ...
0
votes
1answer
33 views

Find distribution of test statistic under $H_0$

I have a shifted double exponential distribution with density $$f(x;\theta)=\frac{1}{2}e^{-|x-\theta|}$$ Now I have a test statistic given by $$T(x_1,...,x_n;\theta_0) = \sum_{i=0}^n |x_i - \theta_0| -...
0
votes
0answers
29 views

Confidence interval from p value

The question is: given that $H_0: \mu=34, H_a:\mu<34$ gives p-value $p$, find the largest confidence level, $c$, that does not include $34$. The answer does this:$1-2p=c$ But I do t understand ...
0
votes
0answers
13 views

Do we have to assume normality of the data, even when we conduct z-test or t-test with large samples?

I read this lecture note and found that it assumes normality of the data when we conduct z-test or t-test. I can accept that when we have small samples we have to assume normality of the data, ...
0
votes
1answer
20 views

What are the “degrees of freedom” in this Chi Squared test?

I have learnt that the degrees of freedom are the (number of rows - 1) multiplied by (the number of columns - 1). However, I am stuck as to what the degrees of freedom are in the following set-up for ...
0
votes
0answers
23 views

Why the author has taken SD $8$ and $ 7$ as $σ$ and not S

An examination was given to two classes consisting of $40$ and $50$ students, respectively. In the first class, the mean grade was $74$ with a standard deviation of $8$, while in the second class the ...
0
votes
1answer
55 views

Likelihood ratio test for both sided alternative in $U(0,\theta)$ distribution

Suppose $X_1,X_2,...,X_n$ is a random sample from $U(0,\theta)$. We need to construct a likelihood ratio size $\alpha$ test to test $H_0:\theta=\theta_0$ against $H_{1}: \theta \neq \theta_{0}$ My ...
1
vote
2answers
29 views

normal distribution sample mean

Kofi owns a cinema. He wishes to increase attendances and so considers offering customers unlimited amounts of free popcorn and soft drinks. He estimates that the likely increase in attendances would ...
1
vote
1answer
36 views

Need help understanding hypothesis testing

I have the following question: Let $X_{i}$ be Gaussian random variables with $\mu$ = 10 and $\sigma^{2}$ = 1. We decide to use the test statistic $\hat{\mu}$ = $\frac{1}{20}\sum_{i=1}^{20}X_{i}$...
0
votes
0answers
9 views

Degrees of freedom in hypothesis testing with multiple constrained parameters in one constraint

Suppose you are estimating a (multivariate) model with a parameter vector $\theta=(\theta_1,\dots,\theta_p)'$. You have two constraints in the model, and would like to test them with either the ...
1
vote
1answer
98 views

Testing the probability of a Bernouilli variable

I have a Bernouilli variable which is $1$ with probability $p$. I need to test the hypothesis $H_0:p<\theta$ vs. $H_1:p>\theta$, where $\theta$ is a given constant. The question is to find $n$ ...
0
votes
0answers
13 views

What is the relation between standard error and power, and unequal randomization?

If you are doing some kind of hypothesis testing (superiority or equivalence) for the difference of two means/proportions, the standard error is: Square root of (Standard deviation of sample A / nA + ...
0
votes
0answers
15 views

Chi squared test - free parameters

I know that for a Chi squared test where we have $\Theta_0$ free parameters under the null hypothesis $H_0$ and $\Theta_1$ free parameters under the alternative hypothesis $H_1$, that $$2\log\Lambda \...
0
votes
0answers
20 views

Chi-Squared test - degrees of freedom

I have some stream of numbers which I use to compute an empirical distribution the number received. I want to test whether or not each number of sent independently of the previous numbers sent. My ...
1
vote
0answers
53 views

Karlin Rubin Theorem UMP (Uniformly most powerful test ) Is it wrong?

Suppose $X_1, X_2, X_3,\ldots, X_n$ are i.i.d. random variables with a common Poisson$(\lambda)$ distribution. $$X=(X_1, X_2, X_3,\ldots, X_n)$$ and $g(λ)=\lambda(1 - e^{-λ})$ , with $(λ>0)$ ...
1
vote
0answers
29 views

Null hypothesis rejection with Monte Carlo simulation, Prime factorization

I made a list of size 6236, with randomly distributed numbers, and 294 out of 6236 numbers, have been purposefully adjusted, that they all have prime factors of nineteen and that their sum is 19 x ...
1
vote
0answers
7 views

Kolmogorov-Smirnov-Test - which values are compared with each other?

The Kolmogorov-Smirnov-Test can be used to test if a given sample-distribution equals a reference distribution. I'm sorry to give you the link to the German wikipedia, but the english article ...
1
vote
0answers
26 views

Find supremum of Type II error in Neyman-Pearson framework

Let $X_1,\dots,X_n$ be an iid sample from an $N(\theta,1)$ distribution. We want to test $H_0:\:\theta=0$ against the alternative $H_1\:\theta \neq 0$ using the test statistic $$T_n(X_1,\dots,X_n) = \...
0
votes
0answers
9 views

Devising a hypothesis test for machine failure rate

I'm trying to devise a hypothesis test for failure rate data of machines. The gist is that there are some machines in a factory that run all the time. They fail from time to time and are promptly ...
0
votes
0answers
4 views

Testing levels of significance with unknowns

Suppose we want to test the null hypothesis that a population mean is 10, vs the twosided alternative that it does not equal 10. We test a small sample of 19 trials (so n = 19). This gives us: t = (...
0
votes
2answers
35 views

When can we Central Limit Theorem approximation with good approximation?

I think we an use it when n(no. of trials) is large. But my textbook used this approx. by stating that since the expectation is large, we use the approx. I'm unable to understand this, would ...
0
votes
1answer
40 views

Hypothesis Test when unknown mean and unknown variance

let have a random variable X which follows a normal distribution of mean $\mu$ and variance $\sigma$. we want to carry out the following hypothesis test: $H_0: \mu=\mu_0$ against $H_1: \mu\neq\mu_0$ ...
1
vote
0answers
33 views

Die roll and hypothesis testing

First of all this is not a homework or an assignment; it's an exam question that I couldn't solve. "There is a $4$-sided die that has the same probability of showing up for each face when rolled, ...
0
votes
0answers
40 views

Correct interpretation of confidence interval

I think this question may have been asked in one form or another, but I did not find an answer that I understood or thought was satisfactory...so I apologize if it has been explained. My ...
2
votes
1answer
15 views

Correctly Interpreting a Type II Error

I understand that a Type II error that arises from a hypothesis test indicates a failure to reject the null hypothesis $H_0$ when $H_a$ in reality is true. But when I try to interpret a Type II error ...
0
votes
0answers
30 views

Hypothesis Testing with Two-Variable Hypotheses, Arbitrary Significance (Neyman-Pearson Lemma)

I'm trying to catch up in my statistics class before finals and I am posed with this "try at home" question in our notes. It asks me to find the best critical region for testing a null hypothesis vs. ...
3
votes
1answer
102 views

Inference regarding the mean lifetime of a bulb using a new technique

The lifetime in hours of each bulb manufactured by a particular company follows an independent exponential distribution with mean $\lambda$. We need to test the null hypothesis $H_0: \lambda=1000$ ...
1
vote
1answer
50 views

Where he used the information that $\mu \leq \mu_0$?

This example is from casella_statistical inference book. Where he used the information that $\mu \leq \mu_0$? What if $\mu \geq \mu_0$? Thanks.
0
votes
0answers
16 views

How do I show that the MLE for all sides of a die is the same as MLE for 5/6 sides of a die

I have two sets of hypotheses: $H_0$: each of the six sides of a die has probability 1/6 of being rolled, versus $H_a$ being not $H_0$ $H_0$: 5 sides of a six-sided die each has probability 1/6 of ...
0
votes
0answers
22 views

AdaBoost what is the hypothesis

I read the article about AdaBoost on Wikipedia an stumbled across a problem. What is meant by the word hypothesis. In this case I mean that section where they refer to the hypothesis h(x) of a "weak ...
0
votes
0answers
21 views

Explanation of the example: UMPU statistic for a Normal mean estimator

Assume $X$ are generated by $\mathcal{N}(\mu, \sigma^2)$. It is known that the density of the sample vector is $$ f(x) = (2 \pi \sigma^2)^{-n/2} \exp\left \{-\frac{n\mu^2}{2 \sigma^2} - \frac{1}{2 \...
0
votes
0answers
53 views

One sided Wald test

I want to construct the following hypothesis test: $$H_0: \theta < \theta_0 \ \mathrm{vs.} \ H_1: \theta \geq \theta_0.$$ My idea was that the one-sided Wald test is equivalent to testing ...
1
vote
1answer
83 views

On a particular day let $X_1,X_2,X_3$ be the number of boys born

On a particular day let $X_1,X_2,X_3$ be the number of boys born before the first girl is born in hospitals $1,2,3$ respectively.If the observations are $X_1=0$ ,$X_2=3$ and $X_3=2$, find the most ...
0
votes
1answer
37 views

Finding the hypothesis test of $\sigma^2$

I have been working on a problem and this is what is given. $$X_1, ... X_n \sim_{iid} N(\mu_0,\sigma ^2 )$$ I was able to find the MLE of $\sigma ^2 $ which is $$\hat{\sigma^2}_{MLE} = \frac{1}{n} \...
0
votes
0answers
38 views

Hypothesis Testing $X \sim Exp(\text{mean} =\frac{1}{\theta})$: Rejection Region and Power Function

I am working on a problem and I would like to have some advice. The following is the given information. 1), $X$ is exponential with mean $\frac{1}{\theta}$. 2), $H_0: \theta =5$ vs $H_1: \theta<...
0
votes
0answers
13 views

power curve for a hypothesis test in r studio

I want to know how can i plot the power curve for the test : $H_{1} : \sigma ^{2} \neq 1$ with a significance level $\alpha = 10 $% and a Gaussian sample of size n . 1/ For n = 10 , I have to ...
0
votes
1answer
25 views

Unbiased Decision Rule

A decision rule $\delta$ is said to be unbiased if $\mathbb E_\theta[L(\theta^\prime,\delta]\geq\mathbb E_\theta[L(\theta,\delta]$ for all $\theta,\theta^\prime\in\Theta$. In the context of testing ...
2
votes
3answers
118 views

how to check a propriety using r studio

I have to check that this propriety $Z \sim N(0,1)$ and $U\sim \chi ^{2}(10)$ then $ Z/\sqrt{U/10} \sim T(10)$ is true using r studio if anyone can help , much appreciate
0
votes
0answers
11 views

Unnormalized t-test holding level when approximating asymptotic distribution

Let $X$ be some random variable with second moment and denote $\mu = E(X)$ and $\sigma^2 = \textrm{Var}(X)$. Let $X_1, \dots X_n$ be iid copies of $X$ and denote the usual empirical estimate of $\...