What we know

• Sampling is randomly chosen $n$ samples in the population again and again.
• For $X\sim N(\mu,\sigma^2)$, we have $\bar X\sim N(\mu,\frac{\sigma^2}{n})$, where $n$ is the count of samples. (If $n\le30$, we should write “Using CLT”)
• Things to care about CLT

Example 1

Analysis

The target is to find $P(\bar X \gt 23)$.

For $X$, we get $\mu=22.3,\ \sigma=4$,

Note: We can’t say $X\sim N(22.3,4^2)$ since the distribution of $X$ is not clear.

Since we don’t know if $X$ is normal, we need to use CLT.

For $\bar X$, we have $n = 64$, so $\bar X\sim N(22.3,\frac{4^2}{64})$.

Solution

Example 2

Analysis

The target is to find $P(\bar X \lt 775)$.

For $X$, we get $\mu=800,\ \sigma=40$, and $X$ is normally distributed.