How close will $\bar{x}$ be to $\mu$ ?

It depends…

1. …on how consistent the data is $(\sigma_X)$
2. …on how big the sample is $(n)$
3. …on what sample you get

So, we use confidence intervals (ci) to estimate!

Confidence Intervals in general

Estimate $\pm$ [Margin of Error]

(lower bound, upper bound)

Margin of Error is typically based on..

• $\sigma_X$
• $n$
• How confident you want to be that the CI includes what you’re trying to estimate

Interpretation :

• example

Calculations

if $n$ is big enough, the sampling distribution of $\bar{X}$ is NOrmal, and 95% of the time we should get an $\bar{x}$ within $\pm 2 \frac{\sigma_X}{\sqrt{n}}$ of $\mu_X$

(1 - $\alpha$)100% $z$-Based Confidence Interval for $\mu_X$:

$\bar{x} \pm z_{\alpha/2} \frac{\sigma-X}{\sqrt{n}}$

• if we dont know $\sigma_X$, you can use $s_x$ (the sample SD) to estimate

Degrees of Freedom (df)

the t distribution has one parameter: degrees of freedom (df)

df = n-1

Paired Samples

Unpaired (Independent) Samples

Dealing with Categorical Data

In these instances, we might be concerned with the proportion, $p$, of the population that fits a certain category (successes).