Sampling Distribution of x-bar

The distribution of x-bar

• A sample mean 样本均值 $\bar{x}$ estimates the true population mean $\mu$.
• Its sampling distribution is centered at $\mu$: $\mu_{\bar{x}} = \mu$ (so $\bar{x}$ is unbiased).
• Each sample gives a different $\bar{x}$; the spread shrinks with sample size.
• This is the workhorse statistic for inference about means.

The standard deviation

• The standard deviation of $\bar{x}$ is:
• $$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$$
• The population SD divided by $\sqrt{n}$ — often called the standard error.
• Quadruple $n$ and the spread only halves (because of the square root).

Sample means from a skewed population become approximately normal as n grows

Normality via the CLT

• Is $\bar{x}$ normal? Two ways to justify it:
• If the population is normal, $\bar{x}$ is exactly normal for any $n$.
• Otherwise, the Central Limit Theorem gives approximate normality when $n \ge 30$.
• Always state which justification your problem relies on.

Conditions and probabilities

• Check the 10% condition ($n \le 0.10N$) when sampling without replacement.
• With normality justified, standardize: $z = \dfrac{\bar{x} - \mu}{\sigma/\sqrt{n}}$.
• Then find probabilities for $\bar{x}$ from the standard normal.
• This answers "how likely is a sample mean this far from $\mu$?"

Write from a standard deviation of 3

• The population standard deviation is 12.
• The sample size is 16. The lesson standard deviations 4 and 2 are other settings.