1. Confidence intervals for µ
In general, the best point estimate for µ is
Example: Suppose I know σ=18 and I take a sample of size 36
What values of the sample mean are most likely to occur?
We know that around 95% of the sample
means lie within 2 standard errors
of the mean from checking Z table. It is
also an empirical rule.
Adding these two areas gives
you about This is the α!

2. Confidence intervals for µ
We’re going to use what we just saw to motivate a confidence interval for µ
For any standard error, the bounds are found with
But we don’t know what µ is…let’s just plug in a good guess for it
I claim that these two bounds give an approximate 95% confidence interval for µ

3. Confidence intervals for µ
Let’s go back to the previous example with a standard error of 3
Say I get a sample mean
here
In this case, the interval captured the true µ

4. Confidence intervals for µ
When will we not capture µ?
As soon as my sample mean becomes
less than µ-6 it no longer has µ
The interval will always have this length
since the standard error depends only
on the sample size

5. Confidence intervals for µ
When will we not capture µ?
As soon as my sample mean becomes
greater than µ+6 it no longer has µ
The interval will always have this length
since the standard error depends only
on the sample size
µ captured if sample mean within two standard errors of µ
We just showed this happens 95% of the time!
Our interval estimate is a 95% confidence interval!

6. Confidence intervals for µ
What made the previous interval an approximate 95% confidence interval was the multiple of 2
If we make the multiple larger, then the interval increases and so does our confidence!
Is there a way we can specify a level of confidence and find the multiple from that?

7. Confidence intervals for µ
We’re going to assume that the sampling distribution for the sample mean is normal (might need CLT) and we know σ
The multiple can be found from the standard normal distribution
Step 1: Specify 100*(1-α)% level of confidence
Step 2: Find z value that satisfies
OR…

8. Confidence intervals for µ
The α value is simply a way to find the necessary multiple
: the confidence coefficient (aka the z-value) that satisfies
The 100(1-α)% confidence interval for µ is…
Eg. What are the z-values at confidence level 98%, 95% and 90%?
( , )
Upper bound
Lower bound