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A confidence interval is a range of values, derived from sample data, that is likely to contain the value of an unknown population parameter. More specifically, a 95% confidence interval suggests that if we were to take 100 different samples and compute a confidence interval for each sample, then approximately 95 of the 100 confidence intervals will contain the population mean.

In the context of the deflated footballs, constructing a 95% confidence interval for the mean psi of the Patriots' footballs gives us a range of values that we are 95% confident contains the true average psi of all their footballs. If this confidence interval does not contain the league's standard pressure requirement of 12.5 psi, it provides statistical support towards the claim that the Patriots' footballs were indeed deflated.

The one-sample t-test is used to determine whether there is a statistically significant difference between the mean of a single sample and a known population mean. This test is particularly useful when the population standard deviation is unknown and the sample size is relatively small, which often necessitates the use of the t-distribution.

In our exercise, the one-sample t-test compares the average psi of the sample of footballs to the standard psi of 12.5. The formula for the t-test is given by \[t = \frac{\overline{x} - \mu_0}{s / \sqrt{n}}\], where \overline{x} is the sample mean, \mu_0 is the population mean asserted by the null hypothesis (in this case, 12.5 psi), s is the sample standard deviation, and n is the sample size. The resulting t-score and corresponding p-value will help us determine whether the observed average psi is statistically significantly different from the expected 12.5 psi.

Statistical significance is a determination that the observed difference between groups or the relationship between variables is unlikely to have occurred purely by chance, given the null hypothesis is true. Significance is usually determined by the p-value, which, in simplest terms, is the probability of observing your data or something more extreme under the assumption that the null hypothesis is true.

In the case of the deflated footballs problem, a p-value that is less than the chosen significance level of 0.05 (or 5%) would indicate that there is statistically significant evidence to reject the null hypothesis. This would mean there is less than a 5% chance of obtaining a sample mean of 11.5 psi (or lower) if the true population mean is the required 12.5 psi. Therefore, a p-value less than 0.05 would suggest that the footballs were statistically significantly deflated with respect to the league standard.