“95% CI” means that if we repeated the sampling process many times, 95% of those intervals would contain the true $\mu$. Not “probability that $\mu$ lies in this interval” — $\mu$ is fixed, interval is random.
Where $t^*$ is from the t-distribution with $n-1$ degrees of freedom. Statistics For Dummies
Significance level $\alpha$ = P(Type I error). Power = 1 − P(Type II error). Instead of a single “best guess,” give an interval likely to contain the true parameter. “95% CI” means that if we repeated the
If IQ ~ $N(100,15^2)$, what’s the probability of IQ > 130? $Z = (130-100)/15 = 2.0$, probability ~ 2.5% (from Z-table). 5. Sampling Distributions and the Central Limit Theorem (CLT) The CLT is the most important theorem in statistics for beginners. Central Limit Theorem: If you take many random samples of size $n$ from any population (with mean $\mu$, s.d. $\sigma$), the distribution of sample means $\barx$ will be approximately normal with mean $\mu$ and standard deviation $\frac\sigma\sqrtn$, as $n$ gets large (usually $n \geq 30$). Why this is magic: It doesn’t matter if the original population is weird — the sample mean follows a normal curve. That allows us to make probability statements about $\barx$. Significance level $\alpha$ = P(Type I error)
For population mean $\mu$: $$\barx \pm t^* \cdot \fracs\sqrtn$$