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 0.05. This is the α!

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

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

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

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!

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?

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…

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