What is Sample?

Example 9.12

A fair coin gives heads with probability $0.5$. Tossing it $100$ times rarely gives exactly $50$ heads: samples fluctuate. Typical runs of $100$ tosses give frequencies like $0.46$, $0.53$, $0.49$ — close to $0.5$, but not equal. Larger samples fluctuate less: with $10\,000$ tosses, observed frequencies are usually within about $0.01$ of $0.5$.

Example 9.14

A factory claims that only $10\%$ of its parts are defective ($p = 0.1$). In a sample of $n = 400$ parts, $64$ are defective: observed frequency $\frac{64}{400} = 0.16$. The fluctuation interval is $\intcc{0.1 - \frac{1}{20}}{0.1 + \frac{1}{20}} = \intcc{0.05}{0.15}$, and $0.16$ falls outside it: the sample casts serious doubt on the claim.