The Normal Distribution, Revisited

The bell curve returns

• Many sampling distributions are well modeled by the normal distribution 正态分布.
• It's the familiar symmetric bell, described by a mean and a standard deviation.
• When a statistic's sampling distribution is normal, we can compute probabilities for it.
• This is why the normal model is the workhorse of inference.

z-scores for a statistic

• A $z$-score 标准分数 says how many standard deviations a value sits from the mean:
• $$z = \frac{\text{statistic} - \text{mean}}{\text{standard deviation}}$$
• For a sample statistic, the "standard deviation" is that of its sampling distribution.
• The $z$-score converts any normal value onto one common scale.

For approximately normal data about 68% lie within 1σ and 95% within 2σ and 99.7% within 3σ

The standard normal

• The standard normal 标准正态 distribution has mean $0$ and standard deviation $1$.
• Every $z$-score lives on it, so one table (or calculator) handles all normal problems.
• Probabilities: area under the curve left/right of a $z$.
• Critical values: the $z$ that cuts off a chosen tail area (e.g. $z^* = 1.96$ for the middle $95\%$).

Why it matters for inference

• Inference asks: "how surprising is my sample, if a claim were true?"
• The normal model turns that into an area (a probability) under the bell.
• A statistic far out in the tail (large $|z|$) is surprising evidence.
• Every confidence interval and test in Units 6–9 runs on this machinery.

Write from a z-score of 3 for a value

• A value is 62.
• The mean is 50 and the standard deviation is 4. The lesson z-score of 2 is another value.