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Hypothesis Testing and the p-value.

In this section, you will need to key your students into the precise language being used. While it may not always be completely fair, but the proper language of hypothesis testing is one way that tell the people who have a proper training in statistics apart from the people who are pretending. This is possibly an issue when writing summaries on the open response questions on the AP examination. The use of proper language goes a long ways to promoting the idea that you know what you are talking about. The key phrases are “There is no correlation...”, “There is no difference...” for the start of every null hypothesis, “ Does not reject”, “Does reject” in regards to the results of the test. It might seem strange to students, but this isn’t the place for creativity or fancy language (or good grammar... sort of; many standard math phrases that are used over and over again are grammatically suspect, but we plow ahead anyway).

A good tip for helping students to understand hypothesis testing is to look at a couple of examples for each idea. For instance, when you introduce two-tailed testing, give students some problems where the null hypothesis is already written and they only need to evaluate whether or not they are rejecting in based on the level of significance chosen. If the information is not broken up, then students easily get lost in the details.

Calculating error and knowing the different types may seem trivial. It also isn’t a large part of later classes and work in statistics. However, it is something that students are expected to know for the AP examination, so they should be made aware of that.  

Testing a Proportion Hypothesis

This chapter is about the time that students start having a really tough time keeping all of the test and sample statistics straight. It may be a good use of time to take a break from new information and have students study, re-copy or interact with each of these statistics so they can hopefully keep them straight. The AP examination can be pretty stressful, and if time isn’t taken to review throughout the year, it will be hard for students to recall what they need in the test.

Proportion hypothesis testing is a simple instance of hypothesis testing. There are a couple of things that work slightly differently, like calculating the standard deviation, that should be the focus of this unit.  

Testing a Mean Hypothesis

It is important for students to have an intuitive sense of why the procedures need to change for small samples. The test outlined here should be tested with the chi-squared distribution, but the idea is the same regardless. Everyone is aware that the theoretical probability for a coin flip is .5. Students can then perform experiments with small numbers of observations, say 1 to 10. Discussing all the experiences from the students, they will see that there is great variation in results even though everyone has a fair coin. If all results are combined on one graph, it should be clear, and intuitively so, that the more flips performed resulted in a better match to the theoretical mean. Going one step further, it’s simply impossible to be anywhere close to the theoretical mean with 1,3 or 5 samples. Students should be asked “what are the implications of this?” Likely answers include: the need for more samples (which is not always possible), the problem with using binary or discreet outcomes (get used to it, rarely can we depend on continuous empirical results), and most importantly, the uncertainty that is inherent to small sample sizes. What would it take to be certain that a coin is fair with only 5 experiments? Most students will probably indicate that the they could not reject the hypothesis of the coin being unfair regardless of outcome. Now bumping it up to 10, now there will probably be some outcomes that will not be accepted. This is the conceptual foundation for small population, or non-normal tests.  

Testing a Hypothesis for Dependent and Independent Samples

It is not standard practice among different texts to assign different symbols for the hypothesis depending on if it is a dependent or independent test. However, this is a good idea, especially as it is easy to get confused or forget a step. The key part of this section is understanding the procedure for testing two different populations, especially when testing for growth. This is the first instance where students will have a chance to see and understand how to show a difference outcome based on procedure. This has to take into account the baseline information for each party, making it a dependent sample. The most interesting consequence for students, as outlined in one of the examples, is the utility of this process to examine various practices in school. If students are working on a project for the end of the year, this may become an integral test for answering many of the questions that students might have about their school.

Students may become a little lost with all the tests. There is a chart in a later section outlining when each test is to be used. It is also advisable at this point to have the students create one of their own, with the creation being an exercise to help them remember.  

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