Hypotheses for a Population Mean

Every hypothesis test for a mean starts with a claim about the true population average, mu. You can't test that claim directly, so you translate it into two competing statements. The null hypothesis, H0, always says mu equals some specific number: the value nothing-has-changed would predict, or the value the manufacturer advertises. The alternative hypothesis, Ha, says what you'd believe instead if the evidence pushes against the null. It can point one direction, mu less than that number or mu greater than it, or it can go two directions at once, mu not equal to it, depending on exactly what the claim asserts.

To build these from a real claim, find the number being promised, and find the direction of suspicion. A machine that's supposed to fill bottles to 500 mL but is suspected of underfilling gives you H0: mu = 500 against Ha: mu < 500. Notice the null carries the equals sign even though the real-world suspicion is 'less than' — the null is the benchmark you test against, not the claim you expect to win. The alternative carries whatever direction the research question actually cares about: less than, greater than, or simply different from.

This hypothesis-writing step is the opening move of the entire inference process for means. Once mu and its direction are pinned down, the rest of the test — computing a t statistic, finding a p-value from the t-distribution because the population standard deviation is unknown, and comparing that p-value to a significance level — follows the same structure used for proportions. Get the hypotheses wrong and every later calculation answers the wrong question, so this step is where accuracy has to start.