Lesson 15 - 3

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 H0 value)

or – (below H0 value)

– Any values = to H0 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 RandomnessSmall-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 -zα; two-tailed -zα/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 nz0 = --------------------- ½ √n

Left-Tailed Two-Tailed Right-Tailed

H0: M = M0

H1: M < M0

H0: M = M0

H1: M ≠ M0

H0: M = M0

H1: M > M0

k = # of + signs k = 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 H0)

Step 1 Hypotheses: Left-tailedTwo-Tailed Right-Tailed H0: Median = M0 H0: Median = M0 H0: Median = M0

H1: Median < M0 H1: Median ≠ M0 H1: Median > M0

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 -zα; two-

tailed -zα/2). Step 3 Compute Test Statistic:

Step 4 Critical Value Comparison: Reject H0 if Small-Sample Case: k ≤ critical value

Large-Sample Case: z0 < -zα/2 (two tailed) or z0 < -zα (one-tailed)

Step 5 Conclusion: Reject or Fail to Reject

Small-Sample: k Large-Sample: (k + ½ ) – ½ nz0 = --------------------- ½ √n

Small Number ExampleA 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 $1000

+ = 8- = 12k = 8n = 20

CV = 5 (from table X)

Two-Tailed Test: (Med ≠ 300)so k = number of smaller of the signs

We reject H0 if k ≤ critical value (out in the tail). Since 8 > 5, we do not reject H0.

Large Number Example

(k + ½ ) – ½ nz0 = --------------------- ½ √n

285 310 300300 320 308310 293 329293 326 310297 301 315332 305 340242 310 312329 320 300311 286 309292 287 305

A sports reporter claims that the median weight of offensive linemenin 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- = 80 = 3n = 30-3 = 27k = 8

z0 = -13/30 = -1.92

285 310 300300 320 308310 293 329293 326 310297 301 315332 305 340242 310 312329 320 300311 286 309292 287 305

Right-Tailed Test: (Med > 300)so k = number of - signs

Since z0 < zα (-1.28), we would reject the H0 (median = 300) in favor of the alternative, median > 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