Type I error

A Type I error requires two conditions at once: $H_0$ is true, and the test rejects it anyway. Its long-run rate under a true null is exactly the significance level $\alpha$ (alpha) you chose, which makes it the one error probability you set directly rather than inherit.

Take $H_0: \mu = 100$ against $H_a: \mu > 100$, with $\sigma$ (sigma) known to be 15 and $n = 25$, so the standard error is 3 and an $\alpha = 0.05$ test rejects once $\bar{x}$ (x-bar) clears 104.93. If the true mean really is 100, that happens in 5 percent of samples. Those 5 percent are Type I errors, and there is nothing wrong with the data or the arithmetic in any of them: the sample was unlucky, not mistaken. Drop $\alpha$ to 0.01 and the cutoff moves to 106.98, so the rate falls to 1 percent.

The sentence to retire: "my p-value was 0.03, so there is a 3 percent chance this rejection is a Type I error." Once the decision is made, either $H_0$ is true and you erred or it is false and you did not; the coin has already landed. $\alpha$ describes the procedure across many repetitions, not the single conclusion in front of you, and 0.03 is the p-value, computed under a null that may well be false.

The error is also unavailable when $H_0$ is false, no matter how badly the test performs. Rejecting a false null is the correct decision, and failing to reject it is a Type II error. Which of the two mistakes you could even make depends on a truth you never observe.

Lowering $\alpha$ cuts this error and, with $n$ and the true effect held fixed, raises the other: at $\alpha = 0.01$ the test above has $\beta = 0.7453$ against a true mean of 105, where $\alpha = 0.05$ gave 0.4913. A larger sample cuts both at once. Which error costs more is a judgment about consequences, and AP Statistics topic 3.8, Potential Errors When Performing Tests, asks for that judgment in context.