Interpretation of the p-value of the y-intercept coefficient in a linear regression I am trying to interpret one of the p-values in a one variable linear regression.  Some of the answers I've seen for similar questions were not worded as thoroughly as I would have liked.  My interpretation is deliberately verbose because it will aid my understanding if faults are found within it.
From Microsoft Excel the linear regression formula from 90 samples of (x,y) pairs is 
y = 0.514x + 0.00087 
and the p-value of the first coefficient is 4e-16 (scientific notation) and for the second it is 0.0027.
Would it be correct to say that the interpretation of the p-value of the 0.00087 term is:

Under the assumption that the true value of the y-intercept is zero
  and the first coefficient is 0.514, random sampling of the same number
  of (x,y) pairs, specifically 90, would result in a least squares best
  fit line with a y-intercept at least as extreme as 0.00087, with a
  probability of 0.0027.

If not, then what would be the correct interpretation?
Not so importantly, but just to be complete, I am also inquiring if it would be more accurate and complete to put the relevant phrase as 

"at least as extreme as 0.00087 in the same direction, that is,
  positive".

Edit: The Excel funcion is Tools > Data Analysis > Regression in Office 2003 with service pack 2. Excel regression p-values on coefficients are 2 sided.
Edit: Regarding differentiation from this question here: The most up voted answer there discusses the p-value of a hypothesis, which seems ill defined or at least not specific.  I am not interested in that.  I am interested in the p-value of a coefficient that is not the coefficient of an independent variable. I am being very specific.
 A: Your interpretation is almost right.
A right interpretation should contain the following information:


*

*There are two approaches to interpretating of p-values:


*

*The Frequentist interpretation, which your answer correctly used: The p-value is the probability of observing a value (in your case, the association between y-intercept and response) as extreme or more ('extreme' implies a two-tailed test), if the null hypothesis is true (in your case that is, the association between y-intercept and response is truly absent in the population, i.e. y-intercept = 0. In some tests it can mean the difference is 0);

*or, the probability of obtaining that estimate of the parameter (e.g. intercept; using this statistical approach), or a more extreme value, if the population value for that parameter is 0.  Your definition correctly uses the frequentist form.


*As you can see from point 1, you do not need to assume the other coefficients are correct when interpreting p-values in a regression model... just that the same approach was used.  However, it does assume that those parameters are estimated.  So, your definition lacks in saying that 'first coefficient is 0.514'. All you need to assert was that the first coefficient is being estimated, i.e. '...the true value of the y-intercept is zero, in the presence of x.'. The values of other coefficients are immaterial to the definition of the p-value of any coefficients.

*The y-intecept is referring to the value of y when all xs are zero. You correctly implied this point.


You should also note that your example, in using the frequentist approach, is not free from your wants and subjective beliefs. Specifically, the p-value is tied to the design of the experiment you ran. You acknowledged this when you mention using the same number of sampling pairs.
With regards to your second question, the typical p-value reported for a regression equation is implicitly two-tailed. So, it refers to the absolute value of the parameters obtained. You didn't provide the Excel function you used to calaculate the p-value, but I'd check there to see if Excel is calculating one-tailed (in the same direction) or two-tailed (extreme or more extreme) p-values.
A: Assuming the coefficients to be normally distributed with the mean of 0 and an estimated standard error which you have not mentioned, the p value tells the quantile of how far the calculated value is from the mean. In the given case if you think that the value is significant at 99.73% level, even then the coefficient is different from 0. If the confidence level that you want is higher than this, then you fail to reject the hypothesis that the coefficient is different from 0.
