Confidence Intervals and P Values

If the sampling were repeated $\displaystyle 100$ times and a confidence interval calculated each time, about $\displaystyle 95$ of those intervals would contain the true mean weight of the population.

Zero is a plausible value for the difference, so there is no significant difference between the two means at the $\displaystyle 5\%$ level.

$\displaystyle 0.612 > 0.05$, so there is insufficient evidence to reject the null hypothesis of no difference.
The confidence interval contains zero, which says the same thing: no difference is a plausible value.

$\displaystyle 0.013 < 0.05$, so reject the null hypothesis: there is a significant difference.
The confidence interval does not contain zero, so zero difference is not plausible — the same conclusion.

The probability of obtaining results at least as extreme as those observed, assuming the null hypothesis is true.
A small p-value means the observed data would be unlikely if the null hypothesis were true.

To be more confident of capturing the true value, the interval has to cover a wider range of possibilities.