Sampling Distribution for a Difference of Means
Day 105 - Lesson 10.2
Describe the shape, center, and variability of the sampling distribution of the difference of means.
Activity: Is One Form of the AP Exam Harder?
Print off this AP Scores Template and cut out the cards prior to class. Each pair of students will need a set of cards. In the experiment, the mean AP score for Form A was 0.2 higher than the mean AP score for Form B. While this provides some evidence that one form is harder, does it provide convincing evidence? Students will use a simulation to find out. Assuming the forms are equal difficulty, how likely is it to get a difference of means of 0.20 or higher. Students will shuffle the cards and randomly assign students to the two forms, then find the difference of means that occurs purely by chance due to random assignment. The class will create a dotplot of the differences:
Use this applet (with two groups) to extend this simulation to many, many trials. In the end, students can estimate a P-value and realize that the difference of means of 0.20 is likely to have occurred purely by chance due to the random assignment.
What Does This Dot Represent?
In the debrief, point at one of the dots in the dotplot and ask “What does this dot represent?” The best answer is “A random assignment of the 30 students into two groups and a difference of means calculated from that random assignment”. Point to another dot and ask the same question. The answer is “A different random assignment of the 30 students into two groups and a difference of means calculated from that random assignment”. So our dotplot represents many, many different random assignments of 30 students into two groups and a difference of means calculated for each random assignment.
Is This Really a Sampling Distribution?
A sampling distribution shows the distribution of all the possible statistics based on many, many random samples (not many, many random assignments!). So technically this is not a sampling distribution — it is a randomization distribution. This distinction is not important for AP Statistics students and the reality is that the sampling distribution of the difference of means very closely models the randomization distribution of the difference of means.
Formulas for the Sampling Distribution
We use this document to lead students to describe the shape, center, and variability of the sampling distribution of the difference of means. They must connect it to the shape, center, and variability for the sampling distribution of a single mean from Lesson 7.3. The common mistake here for students is to want to add (or worse to subtract) the standard deviation formulas. Remember that “one does not simply add standard deviations”. Instead, we add variances and then square root.
Note: You will need to buy some chocolate chip cookies tonight in preparation for tomorrow’s lesson.