23 votes

Can Z values be thought of as the number of standard deviations?

Yes. A Z value of a particular data point tells you how many standard deviations it is from its mean. Z=0 means it has the same value as the population mean, Z=-1 means it is 1std lower than its mean ...
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23 votes

Three questions about the article "Ditch p-values. Use Bootstrap confidence intervals instead"

"Am I right that this is not a p-value (which is the probability to see this or more extreme value of a test statistic)?" Good question! Yes, you're right, it's not a p-value. What's more ...
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  • 11.3k
21 votes
Accepted

Why don’t we calculate the average of an entire given population instead of computing confidence interval to estimate the population mean?

We’d love to calculate population parameters! All of inferential statistics is about inferring. In other words, we are using our data at hand to guess about something greater than the data (e.g., the ...
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  • 32.4k
19 votes
Accepted

Not getting 95% coverage for 95% t-distribution CI

Per @whuber's comment, np.std() provides a biased estimate of the sample standard deviation. Fortunately, the function allows you to correct for that by specifying ...
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  • 2,047
19 votes
Accepted

Is it appropriate to put "error bars" on data when you have the full population?

Error bars show intervals; these intervals must represent something Error bars in a plot show an interval for a particular quantity, and like any element of a plot, these intervals must actually ...
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  • 97.2k
14 votes

Confidence intervals around functions of estimated parameters

Usually we take normality assumption for linear regression models. That is, $y_i\sim N(\beta^Tx_i,\sigma^2)$. From this assumption we derive the asymptotic distribution of $\hat{\beta}$, which is also ...
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  • 2,381
14 votes
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Three questions about the article "Ditch p-values. Use Bootstrap confidence intervals instead"

1 They don’t mean what people think they mean Am I right that this is not a p-value (which is the probability to see this or more extreme value of a test statistic)? Is it a correct procedure for a ...
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13 votes

Not getting 95% coverage for 95% t-distribution CI

In R, using $-notation to pick the 95% CI out of t.test output, I get $0.949 \pm 0.001,$ from 100,000 iterations. ...
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  • 50.8k
13 votes

Three questions about the article "Ditch p-values. Use Bootstrap confidence intervals instead"

The author of the article suffers from not understanding that hypothesis tests and confidence intervals serve different inferential purposes: The confidence interval (bootstrap or otherwise) serves ...
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  • 26.6k
12 votes
Accepted

Confidence intervals around functions of estimated parameters

Two common approaches for this problem are to calculate the non-linear combination of the coefficients directly from the regression or to bootstrap it. The variance in the former is based on the "...
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  • 31.5k
11 votes
Accepted

Can Z values be thought of as the number of standard deviations?

No. The z score is not 'the number of standard deviations'. Instead the z-score of a value is the number of standard deviations that value is above the mean. A z-score of 1.7 is 1.7 standard ...
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  • 261k
10 votes

Is the ratio of two sums normally distributed?

Experiment. The ratio is Cauchy, as noted, if numerator and denominator are both normal distributions centered at zero (not illustrated here). However, the ratio is also often nowhere near normal if ...
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  • 50.8k
10 votes
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Confidence band for simple linear regression - why the curve?

Computing the sample variance of the estimate $\hat{y}$ The estimate for the mean $y$ (as a function of $x$) has the following function in terms of the predictions for coefficients $\alpha$ and $\beta$...
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10 votes

Confidence intervals around functions of estimated parameters

It seems like you are estimating the discriminant of a quadratic function, ie. your function is $$y = \hat{\beta_0} + \hat{\beta_1} X_1 + \hat{\beta_2} X_2 = \hat{\beta_0} + \hat{\beta_1} X + \hat{\...
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10 votes

Three questions about the article "Ditch p-values. Use Bootstrap confidence intervals instead"

I agree that confidence intervals provide a lot more for performing inference than a single p-value for a single hypothesis, but there is no reason to ditch the p-value and no reason to rely solely on ...
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10 votes
Accepted

Confidence interval / p-value duality: don't they use different distributions?

Basically the duality holds, see also this question about the duality: Can we reject a null hypothesis with confidence intervals produced via sampling rather than the null hypothesis? I can think of ...
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9 votes
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How can I obtain a confidence interval for the difference between two probabilities from a binary logistic regression model?

The delta method states $$ \operatorname{Var}(g(X)) = [g'(X)]^2 \operatorname{Var}(X)$$ Because this problem involves two parameters, we can extend this to the multivariate delta method $$ =\nabla g^T ...
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9 votes

Is it appropriate to put "error bars" on data when you have the full population?

@Ben is of course right that there is no "error" if you have the full population. But at least colloquially, "error-bars" are not always referring to uncertainties comming from a ...
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  • 218
8 votes

Why are confidence intervals of hazard ratios not symmetric?

My understanding is that the confidence interval for a hazard ratio should be symmetrical about the mean (the distance between the lower limit and the mean is the same as the distance between the mean ...
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8 votes

What is the history of $p < 0.05$ or 95% confidence?

Fisher suggested the 0.05 level indirectly. He mentioned that two standard deviations is an easy rule for significance, and the 0.05 level is what approximately corresponds to it. From Fisher's 1925 '...
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8 votes

Calculate the standard error of the difference between two independent proportions

In order to understand this material, you should read about the properties of the variance operator. This is a quadratic operator, which operates in a specific way on linear functions of random ...
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  • 97.2k
8 votes

Confidence band for simple linear regression - why the curve?

As you get farther from $\bar x,\bar y$ uncertainty increases. There's fewer and fewer observations when you reach out to distant regions of the domain of your function. The main source of uncertainty ...
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  • 56.6k
8 votes
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Estimating confidence interval of a parameter from the MLE of another parameter

The question, as clarified in comments, is a general one about confidence intervals. It is best framed generally, because the generality strips away irrelevant details to bring out the main idea. So, ...
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  • 290k
7 votes

How can I obtain a confidence interval for the difference between two probabilities from a binary logistic regression model?

There are two principal approaches: Estimate the variance $\hat{\sigma}^2$ of $\theta=p_{x1}-p_{x0}$ and assume that $\theta$ is normally distributed. Then the confidence interval is $\pm z_{1-\alpha/...
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  • 3,111
7 votes
Accepted

How can we know population mean but not variance

$\mathbf{\mu}$ is the theorized value under the null hypothesis. For the situation in general: $$ H_0:\mu = \mu_0\\ H_a:\mu\ne\mu_0 $$ We do our usual fun of calculating the sample mean $\bar x$ and ...
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  • 32.4k
7 votes

Difference between statements about confidence intervals

The sample proportion is different to the population proportion. The first one is a known quantity that you can compute from your sample, whereas the second one is the unknown quantity that you are ...
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  • 97.2k
7 votes
Accepted

Chi-squared confidence interval for variance

Because the chi-squared distribution is skewed, the sample variance is not generally at the center of a 95% CI for the variance (for normal data). You are correct to say that you can often get a ...
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  • 50.8k
7 votes

Chi-squared confidence interval for variance

For univariate continuous asymmetric distributions the highest density region (HDR) can be found by solving a constrained optimisation problem for the boundary points. You are correct that this ...
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  • 97.2k
7 votes
Accepted

Post-hoc power analysis for null results: how to use 95% confidence interval instead?

If your CIs are narrow, then you have an idea of how large the effect is, and you can say with some confidence that the effect is small, and that's why you didn't detect it. If the CIs are wide, then ...
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  • 14.8k
7 votes

Why don’t we calculate the average of an entire given population instead of computing confidence interval to estimate the population mean?

If we're able to observe the entire population of interest then that's exactly what we'd do! In this case we don't require any statistical inference because we directly observe the entire population ...
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  • 97.2k

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