Lesson Inferences about Measures of Central Tendency

Objectives Conduct a one-sample sign test

Vocabulary One-sample sign test -- requires data converted to plus and minus signs to test a claim regarding the median –Change all data to + (above H 0 value) or – (below H 0 value) –Any values = to H 0 value change to 0

Sign Test ●Like the runs test, the test statistic used depends on the sample size ●In the small sample case, where the number of observations n is 25 or less, we use the number of +’s and the number of –’s directly ●In the large sample case, where the number of observations n is more than 25, we use a normal approximation

Critical Values for a Runs Test for Randomness Small-Sample Case: Use Table VII to find critical value for a one-sample sign test Large-Sample Case: Use Table IV, standard normal table (one-tailed -z α ; two- tailed -z α/2 ). Small-Sample Case: If n ≤ 25, the test statistic in the signs test is k, defined as below. Large-Sample Case: If n > 25 the test statistic is (k + ½ ) – ½ n z 0 = ½ √n Left-TailedTwo-TailedRight-Tailed H 0 : M = M 0 H 1 : M < M 0 H 0 : M = M 0 H 1 : M ≠ M 0 H 0 : M = M 0 H 1 : M > M 0 k = # of + signsk = smaller # of + or - signs k = # of - signs where k = is defined from above and n = number of + and – signs (zeros excluded) Test Statistic Signs Test for Central Tendencies

Hypothesis Tests for Central Tendency Using Signs Test Step 0: Convert all data to +, - or 0 (based on H 0 ) Step 1 Hypotheses: Left-tailedTwo-TailedRight-Tailed H 0 : Median = M 0 H 0 : Median = M 0 H 0 : Median = M 0 H 1 : Median M 0 Step 2 Level of Significance: (level of significance determines critical value) Determine a level of significance, based on the seriousness of making a Type I error Small-sample case: Use Table X. Large-sample case: Use Table IV, standard normal (one-tailed -z α ; two- tailed -z α/2 ). Step 3 Compute Test Statistic: Step 4 Critical Value Comparison: Reject H 0 if Small-Sample Case: k ≤ critical value Large-Sample Case: z 0 < -z α/2 (two tailed) or z 0 < -z α (one-tailed) Step 5 Conclusion: Reject or Fail to Reject Small-Sample: k Large-Sample: (k + ½ ) – ½ n z 0 = ½ √n

Small Number Example A recent article in the school newspaper reported that the typical credit-card debt of a student is $500. Professor McCraith claims that the median credit-card debt of students at Joliet Junior College is different from $500. To test this claim, he obtains a random sample of 20 students enrolled at the college and asks them to disclose their credit-card debt. $6000 $0 $200 $0 $400 $1060 $0 $1200 $200 $250 $250 $580 $1000 $0 $0 $200 $400 $800 $700 $1000 $6000 $0 $200 $0 $400 $1060 $0 $1200 $200 $250 $250 $580 $1000 $0 $0 $200 $400 $800 $700 $ = 8 - = 12 k = 8 n = 20 CV = 5 (from table X) Two-Tailed Test: (Med ≠ 300) so k = number of smaller of the signs We reject H 0 if k ≤ critical value (out in the tail). Since 8 > 5, we do not reject H 0.

Large Number Example (k + ½ ) – ½ n z 0 = ½ √n A sports reporter claims that the median weight of offensive linemen in the NFL is greater than 300 pounds. He obtains a random sample of 30 offensive linemen and obtains the data shown in Table 4. Test the reporter’s claim at the α = 0.1 level of significance. + = 19 - = 8 0 = 3 n = 30-3 = 27 k = 8 z 0 = -13/  30 = Right-Tailed Test: (Med > 300) so k = number of - signs Since z 0 300

Summary and Homework Summary –The sign test is a nonparametric test for the median, a measure of central tendency –This test counts the number of observations higher and lower than the assumed value of the median –The critical values for small samples are given in tables –The critical values for large samples can be approximated by a calculation with the normal distribution Homework –problems 5, 6, 10, 12 from the CD