AP Statistics Lectures
by Arnold Kling

A Chart for t-tests and z-tests

Binomial ProportionScalar Mean

These are statistics that come from "either-or" populations. The proportion of people who smoke. The proportion of people who are Democrats, etc. The value of the proportion always falls between zero and one.

• the sample mean is p^ (meaning "p-hat").
• The sample standard error, se, is the square root of [p^(1-p^)/n]

These are statistics that come from populations where the numbers have a scale, such as inches or miles-per-hour or score on a test. The value can be anything--it does not have to fall between zero and one.

• the sample mean is X.
• The sample standard error, se, is the square root of [S(Xi - X)2/(n-1)]
Confidence IntervalsHypothesis TestsConfidence IntervalsHypothesis Tests
• Obtain z* based on confidence level
• Multiply z* by se to get the margin of error, m.
• The confidence interval goes from p^ - m to p^ + m
• If alternative hypothesis is "not equal to" (i.e., a 2-tailed test), then use the significance level, a/2, to select z*. Otherwise, use just a
• In a one-sample test, calculate z by taking the value of p^ minus its value under the null hypothesis and dividing by se. If z is larger than z* (larger could mean more negative if z* is negative), then reject the null hypothesis
• In a two-sample test, then the test statistic looks something like (p^1 - p^2)/[square root of
(se1)2 + (se2)2]
• df = n-1 (degrees of freedom)
• Obtain t* based on confidence level and df
• Multiply t* by se to get the margin of error, m.
• The confidence interval goes from X - m to X + m
• If alternative hypothesis is "not equal to" (i.e., a 2-tailed test), then use the significance level, a/2, to select t* (you also need to use the degrees of freedom) or to compare with the P-value. Otherwise, use just a
• In a one-sample test, calculate t by taking the value of X minus its value under the null hypothesis and dividing by se. If t is larger than t* (larger could mean more negative if z* is negative), then reject the null hypothesis
• In a two-sample test, use a calculator. Since we never obtain t* in this process, we can use the P-value to decide whether or not to accept or reject. When the P-value is lower than a (or a/2 with a two-sided alternative hypothesis), we reject the null hypothesis.