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I conducted a regression in Excel between variables x and y. The correlation is 0.3. The coefficient for x is 3. The p-value for the coefficient is 0.5.

I have read in some places the p-value of 0.5 can be used to say BOTH the x coefficient and correlation coefficient are insignificant. It seems as if people are saying the p-value for the x coefficient is the same as the p-value for the correlation coefficient.

Is the one p-value next to the independent variable also the p-value for the correlation coefficient? If not, why would the Excel regression output not include a p-value for the correlation? Does it have anything to do with the fact that standardized data will have independent variable coefficients which are equal to their correlation coefficients?

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In a single-covariate model $y=a + b \cdot x$, it will be the case that $b = \rho_{yx} \cdot \frac{\sigma_x}{\sigma_y},$ where $\sigma$ is the standard deviation and $\rho$ is the correlation.

When variables are standardized, their $\sigma=1$, so then $b = \rho_{yx}$. In that case, the coefficient will be equal to the correlation and so the p-value will be the same. It sounds like this is what you have in Excel.

The usual null for the regression coefficients is a two-sided test that each coefficient is zero. You reject the null when the p-value is small. In your case, the p-value is large, so you cannot reject the null that the effect is zero in favor of the alternative that it is not zero. This says that the data is consistent with zero linear relationship between $y$ and $x$. But this does not mean that there is no linear relationship: you just can't tell with the data you have.

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  • $\begingroup$ (+1) Perhaps it's just me, but I prefer "single-predictor model" to "univariate model". (I see two variables there.) $\endgroup$
    – Nick Cox
    May 18, 2021 at 8:52

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