In a multiple regression model you can test the effect of a single predictor (technically its unique contribution to the model) either via the F test between the full model and the restricted model without that term or with a t test. In this case t^2 = F and the tests should be equivalent. This may not be the case in other models where the model comparisons use different forms of tests.

If you are using any decent statistical software, I'm sure the software will produce these tests easily. Is there something I'm missing?

Sushodhan - 

Hypothesis tests used to seek "yes" or "no" results are problematic.  Generally you are really looking at something where "yes" and "no" are not logical conclusions. You really want to know "how much."  Here you want to estimate how much difference there is between each coefficient (which is itself estimated), and zero.  You could then be better informed as to whether or not it can be dismissed as zero.  The effect size is important.  If you just look at a p-value, you can change your "yes"/"no" 'conclusion' by simply changing the sample size, if you do not adjust the threshold accordingly.

On top of that, 'independent' variables interact, notably through collinearity, but in other ways too.  Thus the coefficients are volatile. 

If you want to know the importance of various independent variables, you can look at them independently, but when used in a multiple regression, there is an interaction.  One may be more or less important in combination with others. 

If you want to find the model which predicts 'best,' you could compare models which are reasonable from a subject matter perspective.  In each case, graphical residual analyses could help determine model suitability, and test data (not used in model selection or estimating coefficients) could be used to avoid overfitting. 

You might find something of use to you at the following link:

 https://www.researchgate.net/publication/313925533_Comparing_Model_Performances_Graphically

I worked with the development of hypothesis testing in the past, but have come to the conclusion that as a matter of practical use, you are often better off without it. 

If you are considering "full" versus "restricted" models, then the restricted model may have substantial "omitted variable bias," so watch for that, as well as for collinearity, which impacts variance, which may occur in the full model.  (Bias and variance are often a tradeoff.) 

At any rate, trying to use hypothesis tests here, rather than checking the performances of alternative models, would, I think, likely lead to as much or more work, with less to show for it.

Cheers - Jim

Your question is fully legitimite, although it seems that not everyone is aware that for an F-test, t-test, likelihood ratio test or whatever, actually both models are being estimated by the software.

If you estimate a regression model, you can assess the size of effects simply by the coefficient table. That way you only have to estimate the full model containing all predictors. Whereas I fully agree with James R Knaub criticism on hypothesis testing. However, sometimes you just need it because it is convention. You can do hypothesis testing on coefficients by looking at the 95% confidence limits. If they contain zero, the effect is not statistically significant on 5% level.