What are type I and type II errors, and how does statistical power connect them?

Why

Every test makes a decision under uncertainty, so two kinds of mistake are possible. Rejecting a true null hypothesis is a false positive, and its probability is the significance level alpha, often set at 0.05. Failing to reject a false null is a false negative, with probability beta.

Power, 1 minus beta, is the probability that the study detects an effect of a specified size if that effect is real. Banerjee and colleagues note that many studies set alpha at 0.05 and beta at 0.20, giving 80 percent power. The two errors trade against each other.

Lowering alpha to 0.01 to avoid false positives makes false negatives more likely unless the sample grows. Power depends on four things: the effect size you care about, the variability of the measurement, the sample size and alpha. A small pilot of four animals per group may have little chance of detecting even a meaningful difference, so a non-significant result from it is weak evidence of no effect.

Low power also inflates the effects that do reach significance, because only the luckiest estimates cross the threshold. Power is a property of the design for a stated effect size, computed before the study, not a property of an observed result.