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In "An Introduction to Statistical Learning" by James, Witten, Hastie and Tibshirani (http://www.springer.com/gp/book/9781461471370), the t-statistic is defined as

$t = \frac{\hat{\beta}_1 - 0}{SE(\hat{\beta}_1)}$

with $\hat{\beta}_1$ the estimator for $\beta_1$ in linear regression, and $SE(\hat{\beta}_1)$ the standard error for $\hat{\beta}_1$.

The authors than state that t is the number of standard deviations that $\hat{\beta}_1$ is away from 0.

Question: shouldn't it be: "the number of standard errors that $\hat{\beta}_1$ is away from 0"?

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I learned from my statistic courses that for a linear regression model ($Y = X\beta + \varepsilon$), under some assumptions ($\varepsilon$ must be a gaussian white noise), the statistic from the t-test $\displaystyle t = \frac{\beta_1 - \hat\beta_1}{\hat\sigma_{\hat\beta_1}} \sim \mathcal{T}(n-k)$ with n the size of your sample, k, the number of variables and $\hat\sigma_{\hat\beta_1} = \sqrt{\hat\sigma^2(X'X)^{-1}_{(1,1)}}$. That's why I think that the right statement is ""the number of standard deviations that β^1 is away from 0".

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Yes, your re-statement is more strictly correct.

A "standard error" is an estimated standard deviation. In this case, $SE(\hat\beta_1)$ is the estimated standard deviation of $\hat\beta_1$.

The authors' statement is intuitively correct, except that it failed to semantically distinguish the true standard deviation of $\hat\beta_1$ from the estimated standard deviation. The authors of course know very well what $t$ is---they were just trying to write in simple intuitive terms.

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