# understanding of p-value in multiple linear regression

Regarding the p-value of multiple linear regression analysis, the introduction from Minitab’s website is shown below.

The p-value for each term tests the null hypothesis that the coefficient is equal to zero (no effect). A low p-value (< 0.05) indicates that you can reject the null hypothesis. In other words, a predictor that has a low p-value is likely to be a meaningful addition to your model because changes in the predictor’s value are related to changes in the response variable.

For example, I have a resultant MLR model as
\$
y=0.46753{{X}_{1}}-0.2668{{X}_{2}}+1.6193{{X}_{3}}+4.5424{{X}_{4}}+14.48
\$. and the out put is shown below. Then a \$y\$ can be calculated using this equation.

``````            Estimate      SE        tStat       pValue
________    ______    _________    _________

(Intercept)      14.48     5.0127       2.8886    0.0097836
x1             0.46753     1.2824      0.36458      0.71967
x2             -0.2668     3.3352    -0.079995      0.93712
x3              1.6193     9.0581      0.17877      0.86011
x4              4.5424     2.8565       1.5902       0.1292
``````

Based on the introduction above, the null hypothesis is that the coefficient equals 0. My understanding is that the coefficient, for example the coefficient of \$X_{4}\$, will be set as 0 and another y will be calculated as \$y_{2}=0.46753{{X}_{1}}-0.2668{{X}_{2}}+1.6193{{X}_{3}}+0{{X}_{4}}+14.48\$.
Then a paired t-test is conducted for \$y\$ and \$y_{2}\$, but the p-value of this t-test is 6.9e-12 which does not equal to 0.1292 (p-value of coefficient of \$X_{4}\$.

Can anyone help on the correct understanding? Many thanks!