Hypothesis testing using Pearson's correlation is giving me the wrong hypothesis.

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Say I have the two hypotheses:

  • H0: there is no linear relationship between the two variables.
  • H1: there is a linear relationship between the two variables.

Now, my significance level is 0.05, and I have used Pearson's correlation coefficient to calculate the p-value for my two variables (shown below).

(-0.19047068210886346, 4.588168807135144e-138)

As you can see, my coefficient is -0.19 and my p-value is 4.6.

Therefore, as my p-value is > 0.05, does this mean I have to accept H0, that there is no linear relationship between the two variables?

Though, my correlation coefficient is -0.19, which indicates a very weak linear relationship, so how do I prove this with my hypothesis test?

What has gone wrong? There clearly is a linear relationship here, but the hypothesis test says otherwise.