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Hypothesis Testing and the P-Value

Extend: A Null and Alternative Hypotheses for Any Situation

The nature of the situations that is being tested determines if a right-tailed, left-tailed, or two-tailed hypothesis test is used. These exercises allow the student to consider these types of tests in a different, creative way. They will look at the types of test available, and think of situations where each can be used.

Directions: For each set of null and alternative hypotheses chose a number for the variable and describe a situation where this type of test would be appropriate.

Example:

1. H_0: \mu = a\\H_a: \mu < a

Solution: Let a = 12 \;\mathrm{oz}. This null and alternative hypothesis can be used for testing the mean amount of soda found in all soft-drink cans produced in a specific factory. The consumer is interested in finding out if they are receiving the full amount of soda that they paid for, or if the soda producers are cheating them.

Exercises:

  1. H_0: \mu = a\\Ha: \mu \neq a
  2. H_0: \mu = a\\	H_a: \mu > a
  3. H_0: \mu \le a\\H_a: \mu > a
  4. H_0: \mu \ge a\\Ha: \mu < a

Note: The null hypothesis must have the “equal” part of the two choices.

Testing a Proportion Hypothesis

Project: Hypothesize About Your School

Students will put their new knowledge to work by applying it to their high school. They will also review, and put to use, sampling techniques learned previously in the class. This project will be a fun way to make statistics real, and to make the material relevant, active, and long lasting.

Objective: Test, and construct a confidence interval for, a hypothesis about a proportion that describes the population of your school.

Procedure:

  1. Write a null and alternative hypothesis with a proportion about the students at your school.
  2. Take a sample. Be sure to use proper sampling techniques to get a sample that represents the population. Review chapter six if necessary.
  3. Calculate the sample proportion.
  4. Test the hypothesis at probability level 0.05.
  5. Construct the 95 \% confidence interval for the population proportion using the sample proportion that you found.

Analysis and Conclusion:

Write a report and/or prepare a presentation for the class that explains your null and alternative hypotheses, your sampling method, and the results of your test.

Analyze the results: Were you correct? Why or why not?

How confident are you in the method you used to gather your sample?

If you were to do this over again, what would you change?

How could you improve your accuracy?

Is the hypothesis test or the confidence interval more useful for your purposes?

Is there anything else to take into consideration when interpreting your results?

Testing a Mean Hypothesis

Extension: Comparing a Subgroup to the Whole

The text has addressed two uses for hypothesis tests. The first is to test a claim. A statement is made about the mean of some measurement made on a population, and then that claim is tested by comparing it to a sample from that population. The second is to see if a significant difference can be found between the mean of a subgroup and the mean of the group as a whole. In this second application, both means are known, and are compared with the hypothesis test. In this assignment, students will explore the latter case by writing exercises, and by doing research so they can perform this test with real data.

Part One: Writing Exercises

  1. Think of three situations where it would be useful to compare a subgroup to the whole with a hypothesis test. Each exercise should be from a different context. Think about science, politics, education, social justice, sports, and other areas. Be creative.
  2. Write out an exercise, like what would be presented to a student, for each situation. Choose reasonable numbers for the problem. Be sure to include all the necessary information to complete the test.
  3. Provide complete solutions to these three exercises.

Part Two: Research and Apply

  1. Choose a situation where you can find real data to compare a subgroup to the whole with a hypothesis test. You may have to research a few possibilities to find one where you can get all the necessary data. Make a list of the quantities you will need.
  2. Write out the null and alterative hypotheses. Conduct the test, and interpret the results.
  3. Create a report and/or presentation for the class. Be sure to cite the source(s) of your data.

Include the following:

How did you choose the significance level?

Were these the results you were expecting?

How reliable is the data you collected?

What other tests could be made to expand on what you discovered?

Testing a Hypothesis for Dependent and Independent Samples

Project: Test for Significance in Experiment Results

For this project, students will perform an experiment, thereby actively reviewing proper experimental technique, and analyze the results with a hypothesis test. Because they are working with data that they produced, from a topic that is of interest to them, the learning will be deep and long lasting.

Objective: Conduct a controlled experiment and determine if the results are statistically significant.

Procedure:

  1. Plan, and execute a controlled experiment. Review chapter six to ensure that your experiment meets all the guideline of a clinical trial, except for repetition.
  2. Conduct the proper hypothesis test at the 0.05 significance level to determine if the results of the experiment are statistically significant.

Analysis: Include the following in a written report and/or presentation.

  1. Describe your experiment and how it meets the standards of a clinical trial (without repetition). Was your experiment blind or double blind? Did you use a placebo?
  2. Are the results of your experiment in the form of a mean or a proportion?
  3. Are the two groups dependent or independent?
  4. Did you be use the t distribution or the normal distribution? Why? What is the critical value?
  5. State the null and alternative hypotheses.
  6. Calculate the standard error of the difference between the two groups. Show clear, organized work. Carefully chose the appropriate method.
  7. Calculate the test statistic. Show clear, organized work.
  8. Will you reject or fail to reject the null hypothesis? Why?
  9. Interpret the results of the test in the context of the experiment. What have you learned from the experiment and test?

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