p values

I’m guessing that you’re discussing the p-value related to the correlation coefficient.  Of course, the p-value represents the probability of incorrectly rejecting the null hypothesis.  If the p-value is less than some significance level, alpha, (typically practitioners use an alpha of 0.05) then we say that the result is statistically significant (at the 5% level) – i.e. the probability of incorrectly rejecting the null hypothesis is less than 5%. 
For the test I think you’re alluding to, it would indicate that we would reject the null hypothesis that rho (the true correlation coeff) is equal to zero, hence there may be some evidence to suggest that a linear relation is present.  Don’t ignore a scatter diagram though, of course!!!
Hope this helps.

What KMB and Dave are saying is . . .
If the p-value for a correlation coefficient test is less than 0.05, it indicates that the correlation coefficient IS significantly different from zero (either positive or negative) at the alpha = 0.05 level.
This means that there is some significant amount of linear relationship between your two variables of interest.
This test uses a test statistic t0=[r*SQRT(n-2)]/SQRT(1-r^2). It has been proven that IF the true correlation coefficient is equal to zero, then t0 will follow the t-distribution with n-2 degrees of freedom.
Extreme values of this t0 statistic are indicative that t0 does not actually follow a t-distribution with n-2 df, thus it indicates that the true correlation coefficient is NOT equal to zero.
Extreme values of t0 are characterized by small p-values. Thus small p-values indicate that the true correlation coefficient is NOT equal to zero.
Ken K.

The p’value is a way to be sure with the decision, when we are using sample size. The p’value related to the correlation coefficient represents the probability of incorrectly rejecting the null hypotheses ( concept used a lot in advanced statistics analysis ).
For example, results less than 5% ( 0,05) means that the results is statistically significant.Thanks

The basic notion of p value is thisWhat is the probability (p) that the association seen in the data (in this case the correlation) would have been seen by chance (i.e. if in fact there is no relationship between the variables)
Or more accurately what is the probability that you will again get this value for extent of correlation if in fact there is no correlation.If the p-value is 0.05 the common understanding is that the observed relationship would be expected in 5% of measurements if there was no correlation. – You would expect 1 out of 20 samples to sho a correlation when in fact the variables were not correlatedHope this helpsDon

In statistics, you can virtually never get data from an entire population so you have to take samples. A P-value is just an indication that there is a high chance that the factor or data is significant although there are never any black or whites. How sensitively you want to analyze the data (or how thoroughly) is where you set your P-values at for significance. I’m not sure of your exact situation, but it sounds like, the higher you set your P-value, the more thorough you are trying to be, particularly during a DOE when you are trying to measure interaction effects. A P-value of .05 may show no significance. However a P-value of .20 may show significance. Basically, it’s an arbitrary, but well thought out, level to detect significance of your data. The higher, the more thorough.
P-values in normality tests are something a little different. If they are above .05, that means chances are good your data is normal. If it is less than .05, chances are good it’s not and you should look for other options. Again, this is just an inference made from the sample of the population you took. But you better believe, it’s usually dead on.

       The p-value is the smallest alpha level at which you declare significance when in fact there is none. Looking at your nomal curve, if you see that your p-value is more extreme than your alpha value (in other words, it is smaller in area than your alpha value) you reject your null hypothesis and accept your alternative. Makes sense since you already set your alpha to be small, say .01. An even smaller p-value would lead you to accept or declare significance.  
       But you must be careful when testing for rho greater than 0. Remember that it is easy to declare or reject significance if you have a few points sampled. Imagine you have just 3 points and judging from the way they are placed on a scatter plot, they lie in a relatively straight line. Would you be accepting this as having a linear relationship? What if you had a few more points? A good approach is to use a scatter plot in conjunction with your formal test (i.e. t-test).  

JP,
      As previously mentioned, when your P-value is low then the null must go, but the key question is “what is the null?”  If we are talking linear regression the null hypotheis is: Slope = 0 (X is independent of Y).  Changing the input (X) will not change the output (Y).  If P < 0.05, reject the aforementioned statement and conclude as the input changes so does my output.  Note, there is a 1 and 20 chance (p = 0.05) that you have made an error.

I teach my middle school-age son in this way that he can understand
Smaller P ==> More Significant Difference ==> Believe Difference in the Results
Bigger P ==> Less Significant ==> Do Not Trust the Difference in the Results becasue there is big change for random factors to make the results.