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My absolute t-value is greater than t-critical value. This means that I can reject my null hypothesis which was that $\beta_1\leq 0$. Therefore, $\beta_1\gt 0$ and my alternative hypothesis is correct. However, my data has a negative slope. which tells me that $\beta_1\lt 0$. What am I missing here?

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    $\begingroup$ Can you show your regression output? If you want to provide your data as well that would excellent. What do you mean the data has a negative slope—like the scatterplot of Y on X? $\endgroup$ – Brennan Sep 23 '19 at 5:47
  • $\begingroup$ Also, when one is using doing conventional (Neyman-Pearson) hypothesis tests, one should specify the null hypothesis as an equality, otherwise the "statistics" aren't meaningful as random variables. $\endgroup$ – Student Sep 28 '19 at 9:19
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You seem to have $H_0: \beta_1 \le 0$ and $H_1: \beta_1 > 0$. In this case, you reject $H_0$ in favor of $H_1$ if your t-value (not the absolute t-value, but the t-value itself) is bigger than the one-tailed critical value. In your case, $\hat\beta_1 < 0$ so your t-value is negative and so you do not reject $H_0$ in favor of $H_1$.

Edit

Regarding significant* variables with the wrong signs (negative when expecting positive in your case), Jeffrey Wooldridge (Introductory Econometrics: A Modern Approach, 7e, end of Section 4-2f, p. 134) writes:

"A significant variable that has the unexpected sign and a practically large effect is much more troubling and difficult to resolve. One must usually think more about the model and the nature of the data to solve such problems. Often, a counterintuitive, significant estimate results from the omission of a key variable or from one of the important problems we will discuss in Chapters 9 and 15."

Note *: "Significance" means rejection of the null in favor of the two-sided alternative. "Significance" does not necessary imply the rejection of the null in favor of a one-sided alternative.

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  • $\begingroup$ This does not answer the question, rather just formats what OP has stated already. What is OP missing? This does not answer this $\endgroup$ – Brennan Sep 24 '19 at 6:51
  • $\begingroup$ @Brennan There is nothing is missing. And my answer does not just format what OP has stated already. The null is not rejected in the given situation. OP wrote "I can reject the null hypothesis" but that's not correct. I thought one would understand it if one read the question and my answer carefully. If not, well, that's fine too. $\endgroup$ – chan1142 Sep 24 '19 at 12:04
  • $\begingroup$ Ah I re-read it all and this makes perfect sense, sorry $\endgroup$ – Brennan Sep 24 '19 at 16:42
  • $\begingroup$ This is the kind of answer I was looking for. I though we looked at the absolute T-value as per the decision rule. Could you intuitively explain what a negative t-value means? $\endgroup$ – Rumi Sep 26 '19 at 15:07
  • $\begingroup$ @Rumi Please see if the edit is relevant to your question. $\endgroup$ – chan1142 Sep 27 '19 at 16:07

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