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In a linear regression with about 100 observations we have the following estimate for $\beta$: $\hat{\beta}=0.23$ with $\text{SE}(\hat{\beta})=0.08$. Does data suggest a correlation between these two variables?

I think there is a positive correlation because $\hat{\beta} > 0$. But I do not know how to show that $\hat{\beta}$ is significantly different from 0.

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It does show you a positive correlation, since $\hat{\beta} = cov(y,x) / var(x)$ and var(x) can never be negative. (Replace x with a partialled out x in case you have a multivariate linear regression.)

I heavily advise to review the basics of linear regression.

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  • $\begingroup$ This textbook question is looking for a statistical test of the null hypothesis $\beta=0$ against the alternative hypothesis $\beta\ne 0.$ $\endgroup$
    – whuber ♦
    Commented Mar 20, 2020 at 14:18
  • $\begingroup$ But the funny thing is hypothesis testing not curriculum. So that is why I do not know how to do it. $\endgroup$
    – Xenusi
    Commented Mar 20, 2020 at 14:43
  • $\begingroup$ Please have a look a some basic statistcs book. The test statistic is super easy: t = (beta - 0) / se $\endgroup$
    – Jens Friedrich
    Commented Mar 20, 2020 at 16:05
  • $\begingroup$ If hypothesis testing is not in the curriculum, then some equivalent form of statistical reasoning must be on it, for otherwise this would be a useless (and frustrating) question. We are able to answer questions but--unless you tell us explicitly--we are unable to guess what is in your curriculum. $\endgroup$
    – whuber ♦
    Commented Mar 20, 2020 at 16:57

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