1. Test of a Population Median

2. The Population Median (  ) The population median ( , P 50 ) is defined for population T as the value for which the following statements hold true: Pr{ randomly selected member of T has value <  } =.50 Pr{ randomly selected member of T has value >  } =.50 Population T Population Median 

3. The Sample Median (p 50 ) The sample median (p 50 ) for a random sample S from population T is a value that approximately splits S in half. That is, #{members of S with value > p 50 }/n .50 and #{members of S with value < p 50 }/n .50 sample S For large random samples, p 50  . sample median p 50 proportion of sample over here .50

4. The Null Hypothesis We have a population (T), with unknown population median (  ). The null hypothesis(H 0 ) states that the true value of  is the real number  0. If H 0 is correct, then we expect real random samples to be approximately bisected by the value  0.

5. Alternative Hypothesis: Guess is too Large (  0 >  ) If this alternative hypothesis (H 1 ) is correct, then we expect that the samples will be split “highly” by our guess (  0 ). 00 proportion of sample over here >.50 proportion of sample over here <.50 Under this alternative, we compute a test error as error = #{sample points in S with values <  0 }. If H 0 holds, then we expect to see errors that are approximately n/2. If H 1 holds, then we expect to see errors larger than n/2.

6. Alternative Hypothesis: Guess is too Small (  0 <  ) If this alternative hypothesis (H 1 ) is correct, then we expect that the samples will be split “low” by our guess (  0 ). 00 proportion of sample over here >.50 Proportion of sample over here <.50 Under this alternative, we compute a test error as error = #{sample points in S with values >  0 }. If H 0 holds, then we expect to see errors that are approximately n/2. If H 1 holds, then we expect to see errors larger than n/2.

7. Computation Identify the null median (  0 ). Identify the alternative hypothesis: 1.  <  0 2.  >  0 3.    0 Compute the appropriate error: 1. #{sample points <  0 } 2. #{sample points >  0 } 3. Maximum of #{sample points <  0 } and #{sample points >  0 }

8. Computation Refer error to table for approximate p-value via table look-up: Match sample size and error. Obtain base p-value from matched row. Double base p-value if alternative hypothesis is    0.

9. Interpretive Base Briefly identify the population of interest. Briefly identify the population mean of interest. Briefly describe the family of samples. Briefly describe the family of errors. Apply p-value to the family of errors.

10. Populations, Samples and Families 1 We begin with a population T and a population median . For any fixed sample size, n, the Family of Samples consists of the collection of all possible random samples of size n from T. Each individual member of this Family of Samples is a single random sample of size n from T. The Null Hypothesis claims that the true value for the population median (  ) is the real number  0. A sample error can be computed from each member of the Family of Samples. If we compute a sample error from each member of the Family of Samples, we obtain a Family of Errors. Each member of this Family is a single sample error computed from a member of the Family of Samples.

11. Populations, Samples and Families 2 If the null hypothesis is correct, then the family of errors obeys a particular probability law, and we can obtain a p-value for any member of the family of errors. The p-value for a particular member of the family of errors is the probability of observing an equally bad or worse error, given the correctness of the null hypothesis. p-value(our_error) = Pr{error  our_error|  =  0 }

12. Median Test: Basic Elements of Interpretation Briefly identify the population of interest: “The population consists of …” Briefly identify the population median of interest: “We seek to evaluate the population median …” Briefly describe the family of samples: Each member of the Family of Samples is a single random sample of n=? Members of the population. The FoS consists of every possible random sample of this type …”

13. Median Test: Basic Elements of Interpretation Briefly describe the null (H 0 ) and alternate hypotheses (H 1 ). “The null hypothesis is that the population median … equals … against out alternative hypothesis that our population median …” Briefly describe the error rule and our family of errors. “We compute each sample error by … Computing an error in the same way for every member of the family of sample yields a family of errors.” Apply the p-value to the family of errors. “If the null hypothesis is correct, then the probability of observing errors as bad as or worse than our error is approximately …”

14. Interpreting a P-value In our setting, a p-value is a conditional probability. The prior event is the correctness of the null hypothesis. The event of interest is observing an error more severe than our single computed error.

15. Interpreting a P-value The test is active when we obtain small p-values. As p-values approach 0, the interpretation of our p- value forces us into a Rare Event problem. A p-value “close to 0” forces us to either view our single sample as a rare event, or to question the correctness of our null hypothesis.

16. Interpreting a P-value If an observed p-value falls below.05, we call the test “statistically significant,” and usually decide that the null hypothesis is not supported by the sample. P-values below.01 yield tests that are called “highly significant.” The 5% and 1% standards are based on tradition and professional practice. The general idea is simple. The smaller the p-value, the less faith we place in the null hypothesis.