18
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Why use 95% confidence interval?
From Wikipedia article 1.96 :
The use of this number in applied statistics can be traced to the
influence of Ronald Fisher's classic textbook, Statistical Methods for
Research Workers, first ...
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15
votes
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Probability vs Confidence
Your question is a natural one and the answer is controversial, lying at heart
of a decades-long debate between frequentist and Bayesian statisticians. Statistical
inference is not mathematical ...
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12
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What is the (fully rigorous) definition of a confidence interval?
Let $$T_1=g_1(X_1,\dots,X_n)$$ $$T_2=g_2(X_1,\dots,X_n)$$ be two statistics where $$X_1,\dots,X_n\sim F_\theta$$ for some unknown parameter $\theta \in \Theta$. Then, $[T_1,T_2]$ is called a ...
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Why use 95% confidence interval?
$95\%$ is just the conventionally accepted boundary for "reasonably certain" in general cases. It has nothing to do with any specific formulas, and is rather an arbitrary choice that statisticians ...
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8
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quant interview: (mathematical modelling) linear regression and statistical significance
This is a classic case of hypothesis testing. Here, our null hypothesis is that there is no significant relationship between $X$ and $Y$ in the simple linear regression model $Y = \beta X + \epsilon$:
...
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5
votes
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Find a pivotal quantity and use it to approximate a 95% confidence interval
First note that $Z_{0.025}=-Z_{0.975}=-1.96$ and solve the inequaalities
$$-1.96 \le \frac{(\overline{Y} - 1.5\theta)\sqrt{12n}}{5\theta} \le 1.96$$
Multiplying by $5\theta$ and dividing by $\sqrt{12n}...
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5
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One tailed confidence interval $1 - 2\alpha $ rationale
First we have to be clear on the definition of $\alpha.$ That may be the
nub of your problem. So I will talk about 95% confidence intervals (CIs).
Two sided CIs:
95% t CI for normal mean $\mu,$ ...
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Why does $2$ appear in $95\%$ confidence intervals?
It's because $$2\approx 1.96 \approx \Phi(0.975),$$
where $\Phi$ is the cumulative distribution function of the standard normal distribution $Z\sim N(0,1)$, that is,
$$\mathbb{P}(Z<1.96)\approx 0....
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What is the (fully rigorous) definition of a confidence interval?
Here is a slightly more general notion of coinfidence set.
At issue is that statements such as $P[\theta \in C(X)]$ are not really probabilistic statements about $\theta$, since in the classical (...
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4
votes
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Is a $90\%$ confidence interval really $90\%$ confident?
Traditional Wald Confidence Interval. You are asking about the 'coverage probability' if traditional
(sometimes called 'Wald') confidence interval (CI) for binomial success
probability $\pi,$ based on ...
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4
votes
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Confidence interval interpretation difficulty
For an analogy, consider the following game. Alice pays Bob five dollars to flip a fair coin. If the coin lands heads, Alice wins ten dollars; if the coin lands tails, Alice wins nothing. Let $W$ be ...
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Statistics. How are standard error and confidence intervals useful without knowing population size?
Here's the thing. Qualitatively, we all 'grock' the Law of the Big Numbers: we all understand the intuitive idea that as the sample size increases, the observed percentage is more likely to ...
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4
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Why use 95% confidence interval?
I don't think it is arbitrary because given a normal distribution
...
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quant interview: (mathematical modelling) linear regression and statistical significance
You can use the $F$-test in order to calculate statistically significance, given you hypotheis you have that
$$
F_n= \frac{\rho^2}{1-\rho^2}*(n-2)
$$
Hence you obtain the following $F_{100} = 0.0098$...
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votes
Can we rely on Confidence Intervals?
Only one specific $95 \%$ confidence interval $(7.6, 8.4)$ is usually not sufficient to derive the wanted information about the statistical parameter of interest. Nevertheless it provides more ...
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4
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Can A Probability Ever Be Outside of $0$ and $1$?
A statistical model is the triad $(\Omega, \mathscr{A}, \mathbb{P})$ where $\Omega$ is a state space with $\omega \in \Omega$; $\mathscr{A}$ is a collection of interesting events, called $\sigma-$...
3
votes
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Confidence in sample mean, given sample variance?
if you look carefully at the construction of a confidence interval:
$$ \mu +/- Z_{\alpha} * \sqrt{VAR(\mu)} $$
$$ VAR[\mu] = VAR[\sum{x_i}/n] = n * VAR[ {x_i}/n] = VAR[X] / n $$
so you just have ...
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Why does change in the alternative hypothesis (with null-hypothesis being the same) influence the p-value?
You're doing a one-sided test, so you're not testing "is mean one different from mean two" you're testing "is mean one less than mean two" and separately "is mean one greater than mean two". If the ...
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What is the radius of a "Gaussian" sphere such that approx. all the population lie within?
The distribution of $\|X\|^2/\sigma^2$ is $\chi^2_k$. Since for $k=1$
$$
P(|x| \le 3 \sigma) = .9973002
$$
I interpret this as the question for which $r$ in the case of $k = 3$
$$
P(\|X\| \le r \...
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votes
Calculating the Confidence Interval
You are correct to use $H_0: p = .5$ versus $H_1: p \ne .5.$ for
your hypothesis testing.
In the US until several years ago, the traditional 95% ci for $p$ is to use the point estimate $\hat p = X/n =...
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votes
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Are the statements about the confidence interval correct?
The first is a bit tricky. It's hard to figure out what the probability that the true value falls within the interval is, and notably this number is not $0.90$. The correct interpretation of the ...
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Statistics simple theory question
You need to look very carefully at the exact wording in your text where
confidence intervals are described.
If yours is a traditional frequentist
text, then 'A' is the only "correct" answer. A ...
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3
votes
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how to interpret the variance of a variance?
Perhaps the difficulty is that the word "variance" is being used in two distinct senses in that phrase. The second instance refers to a sample variance or variance estimator. So the more precise ...
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Confidence interval interpretation difficulty
I think a better way to conceptualize confidence intervals (in the frequentist sense) is to first go back to point estimates.
Suppose we calculate a point estimate $W$ for a fixed but unknown ...
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votes
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Large sample confidence interval
See more details in this answer for the definition of a Clopper Pearson confidence interval, and problems with approximations in confidence intervals (the approach you have used). I've considered the ...
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votes
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Inference about standard deviation of normal sample
First, I checked your computation of the sample variance $S^2$ using R statistical software.
...
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votes
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What's the best estimate from very few observations?
You need to figure out (a) whether differences among A, B, and C are due to
random variation among independent, identically distributed measurements or
(b) whether one or more of A, B, and C is ...
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votes
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Finding sample size given confidence interval and standard deviation
Yes, "how many data?" refers to the sample size $n$.
The key is that the width of your $95\%$ confidence interval will be roughly proportional to $1/\sqrt{n}$ (in fact here, since the model is a ...
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How many trials of flipping a coin are needed to be confident in getting very close to the same number of H/T?
If you flip $n$ coins, the number of Tails follows a binomial distribution that you can approach with a normal distribution of parameters $\mu=\frac n2$ and $\sigma=\frac{\sqrt{n}}{2}$.
With this ...
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votes
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Find the 99% confidence interval (Interval and test for proportion)
Your computation is correct for the type of confidence interval you are using.
However, this kind of confidence interval is known not to provide the
promised level of confidence, in your case 99%.
An ...
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Only top scored, non community-wiki answers of a minimum length are eligible