Tuesday January 21st: Confidence Intervals with Means

A confidence interval is used when we have a sample and want to estimate the corresponding population parameter. If we have x-bar, we want to estimate mu. If we have p-hat, we want to estimate p. We do this by following this formula (for either means or proportions):

sample +/- critical value(std. dev)

For means, this is x-bar +/- critical value (Sx/root(n))
For proportions, this is p-hat +/- critical value (root(p-pat(1-p-hat))/n).

A critical value is derived from the Normal Curve, and we use it to estimate the interval at a certain level of confidence. On the AP exam, you will typically be asked to calculate the interval using 90%, 95%, or 99% confidence, although we can certainly derive the other critical values using the Normal Curve.

The three critical values that you will need to know are:
90%: 1.645
95%: 1.96
99%: 2.58

When you construct your interval for x-bar, you must follow three important steps.
1) Check Assumptions. They are:
-Assume the data came from a simple random sample
-Assume that the data is independent: that the sample size is less than 10% of the population size
-Assume that the sample size is sufficiently large by the Central Limit Theorem. For x-bar, it must be greater than or equal to 30 observations to roughly resemble a Normal curve.
2) Calculate the interval by using
x-bar +/- critical value (Sx/root(n))
3) Interpret in Context:
"We are (percent) confident that the true population parameter (mu or p) for (whatever your problem is about in context) is between (the lower and upper part of your interval).