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I have made a regression with 2 explanatory variables. The summary of that regression shows that one of my variable has a big p-value (0.705). Should I include that variable when writing the the y hat equation?

This depends on your expected results. In your cases, you have only 2 features and, if you remove one of them, the percentage that you lose important data will really high.

Instead of removing the insignificant feature, you should try to make it better by detecting an anomaly or dropping the outlier. In a common way, plotting covariance matrix to see how relevant btw the features, you can analyze boxplot and adjust the threshold to gain the more reliable data.

If you have enough data, you can split data into training, validation and test set. Then, you can improve your model coefficient by using some voting methods in the validation set.

Finally, you can implement the result coefficient R-square, p-value... and do some test ANOVA testing, AIC score... to compare two cases.

This depends on the goal of your analysis. Have you made a hypothesis that both your explanatory variables affect the dependent variable? In this case you shouldn't remove the variable since you'd be modifying your regression a posteriori (that is after you've collected your data.)

Are you trying to make a descriptive statement about what you're analyzing? For example, are you trying to understand whether education and sex predict income? Similarly, you shouldn't drop a variable since you'll no longer be able to conclude that one of the two variables has no effect.

Finally, are you trying to make a prediction? In this case, it's appropriate to try both models and compare their performance. You can do this using an F-test/ANOVA.