More formally, the confidence interval for a population mean can be

computed as a sample mean, plus or minus a margin of error.

This is critical value corresponding to the middle whatever you like percent.

So, I have just xx here as a placeholder of

the normal distribution times the standard error of the sampling distribution.

As with the central limit theorem, there are

some conditions that need to be satisfied to

use this formula to construct confidence intervals.

In fact, since this method is based on the

central limit theorem, these are actually the same conditions.

The first condition is independent.

Sampled observations must be independent.

And we talked about this being difficult to confirm.

However, usually we either want a random sample, if we have an

observational study, or a random assignment if we have an experiment and

if were sampling without replacement we want our sample

size to be less then 10% of our population.

The second condition is about the sample size and skew.

We either need n to be greater than or equal

to 30 or larger if the population distribution is very skewed.

And this second condition is actually a little stricter

than what we saw with the central limit theorem.

Because it places

a minimum required sample size requirement.

That's the n greater than or equal to 30.

And we're going to discuss what we do if the

sample size is smaller than 30 in the next unit.

So for now, let's focus on what we call large samples

and these are samples that have at least sample sizes over 30.

Or even larger if the population distribution is very skewed.

So when we're checking our conditions,

we're definitely going to want to see a visualization of the distribution from the

sample that we're going to use as

an indicator for what the population looks like.

Or we're going to need to be told to assume that we're going to

need to be told that perhaps we can assume some normality and proceed.

Earlier we conceptually developed the formula for the confidence interval for

the mean as x bar plus or minus z star times

the standard error.

And we said that the z star for a 95% confidence interval should be

approximately 2, as per the 68, 95, 99.7% rule.

But this rule is simply a rule of thumb, and it's actually not exact.

So how do we find the exact critical value for a 95% confidence interval?

Remember that the confidence level refers

to the middle of the distribution.

So the 95% confidence interval will span the middle 95% of the normal distribution.

So, let's mark that on the normal curve, and we're basically

looking for the cut off values that mark the middle 95%.