Math 221 – Principles of Statistics

Hypothesis Tests: An Outline

An hypothesis is a claim about a parameter. (Recall that a parameter is a measure of some characteristic of a population.) Usually, hypotheses arise from consideration of theory or anecdotal evidence, or in response to other research.

1) We need two hypotheses for each test. One of them represents “no change,” “no difference,” “no effect,” or something similar. It is called the null hypothesis, and can be thought of as what to expect if things are the way we think they are (or have been).” The other hypothesis represents some alternative to the null hypothesis, (some alternative expectation) and so is called the alternative hypothesis.

2) We establish the standard by which the significance (or lack thereof) of our experimental results will be judged. This standard is the significance level of the test. Its default value is 5% = 0.05, but is typically as low as 1% = 0.01 or as high as 10% = 0.10. Its symbol is the Greek letter .

3) We design the study, describing very carefully the population being studied, the design of the experiment, the sampling method to be used, the measurements we will make and how to make them, organization and training of the research team, etc.

4) We take the sample and measure, keeping very careful records of what we’ve done and the data we’ve obtained.

5) We summarize the data with appropriate descriptive statistics and graphics. Example: , histogram, etc.

6) a) We compute the test statistic. This is a standardized value of a sample statistic.

Example:

b) We use the test statistic to determine how likely it was that we should have gotten the result we got, assuming the null hypothesis (the expectation) is true. This likelihood is called the P-value of the test.

7) We compare the P-value with the significance level alpha. If the P-value , our statistic is contrary to expectation. We say the result is "statistically significant (at the level)" and reject the null hypothesis. If the P-value , our statistic is consistent with our expectation. We say the result is "not statistically significant" and fail to reject the null hypothesis.

8) We interpret our results for appropriate audiences.

Note: In the homework, you will typically carry out only steps 1, 6, 7 and 8. On rare occasions, you may carry out steps 2 and 5. But the homework in this text will not require you to carry out steps 3 and 4. Frankly, you need to learn a lot more before you can do step 3 (no offense) and step 4 requires resources we simply don't have.

David E. Brown

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