Statistics: Hypothesis Testing & Confidence Intervals

H_0 : no effect or no difference (uses =). H_a : the claim you look for evidence of (uses <, > or ≠).

The probability of getting a result at least as extreme as the one observed, assuming H_0 is true.

If p \le \alpha , reject H_0 . If p > \alpha , fail to reject H_0 .

A non-significant result only means there was not enough evidence against H0. It does not prove H0 is true.

A false positive, like convicting an innocent person.

A false negative, like letting a guilty person go.

1 - \beta : the probability of correctly rejecting a false H_0 .

Increase the sample size n Increase the significance level α A larger true effect size (farther from H0)

One-tailed: H_a uses < or >, all of α in one tail. Two-tailed: H_a uses ≠, α split between both tails (so the p-value is doubled).

z = \dfrac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}

t = \dfrac{\bar{x} - \mu_0}{s / \sqrt{n}} with df = n - 1

z = \dfrac{\hat{p} - p_0}{\sqrt{\dfrac{p_0(1 - p_0)}{n}}}

\text{point estimate} \pm (\text{critical value}) \times (\text{standard error})

We are 95% confident the interval captures the true parameter. In repeated sampling, about 95% of intervals built this way would contain it.

Quadruple the sample size, since the margin of error shrinks with the square root of n.

Reject H_0: \mu = \mu_0 exactly when \mu_0 falls outside the 95% interval.

Paired: two measurements on the same subjects (before/after); analyze the differences. Two-sample: two independent groups.

\chi^2 = \sum \dfrac{(O - E)^2}{E}

E = \dfrac{\text{row total} \times \text{column total}}{\text{grand total}}

Whether the means of three or more groups are all equal, using an F statistic (variation between groups vs within groups).

Random: data come from a random sample or randomized experiment Independence: sample is under 10% of the population when sampling without replacement Normal/large sample: CLT applies or success/failure counts are at least 10

A tiny, unimportant effect can be statistically significant with a huge sample. Always ask whether the effect size matters in context.

Since 0.03 < 0.05 , reject H_0 . There is convincing evidence for H_a (stated in context).