Chapter 7 Nonparametric Statistics

7.1 - Introduction

• Parametric tests– Have a requirement about the distribution of the

population• Nonparametric tests– Have no such requirement

7.2 – The Sign Test

Purpose: To determine if a set of data consisting of + and − signs has an unusually small number of one sign or another. Let– total number of + and − signs in the data– number of times the less frequent sign appears

The test statistic is

29( 0.5) 0.5

for 290.25

for x n

x n

z nn

The Sign Test

Critical value– : See Table C.5– : or

– Reject H0 if test statistic critical value

P-value – is and is

Requirement1. The sample is random

Example 7.2.1

The first row of the table below shows the number of children’s books in 11 randomly selected homes with children in a town. Use the data to test the claim that the median number of books in homes with children in this town is greater than 12.

Example 7.2.1

Parameter: the median number of books

Hypotheses: H0: , H1:

Test statistic:– , – Critical value: 1

P-value:

– Reject H0 : The data support the claim

0 10 0 1 10 110 10( 1) 0.5 (1 0.5) 0.5 (1 0.5) 0.0107

0 1P X

7.3 – The Wilcoxon Signed-Rank Test

Do scores on a writing survey improve from the beginning to the end of the semester?– Test the claim that the median of the population of the

differences of the scores, denoted , is positive– Hypotheses: H0: , H1:

Test Statistic

Calculate the sum of the positive signed ranks and the sum of the negative signed ranks:

Theorem 7.3.1 Assuming the distribution of the differences is symmetric around 0, then the random variable is approximately normally distributed with mean and variance

8 5 3 6 2 9 7 40 and 4 1 5S S

2( 1) ( 1)(2 1)and4 24

n n n n n

The Wilcoxon Signed-Rank Test

Purpose: To test the null hypothesis H0: where m is the population median of the differences of a set of paired data.– Let smaller of and

Test statistic for

f

30( 1) / 4 30

( 1)(2 1)o

4

/ 2rT n nz

n

T n

n nn

The Wilcoxon Signed-Rank Test

Critical value– : See Table C.6– : or

– Reject H0 if test statistic critical value

Requirements1. The sample is random2. The population of differences has a distribution that is

symmetric

( 0.5) ( 1) / 4P-value( 1)(2 1) / 24

T n nP Zn n n

Example

Test statistic:

Critical value: 8P-value:

– Reject H0: The data support the claim

29(9 1) 9(9 1)(2·9 1)22.5, and 71.254 24

(5 0.5) 22.5P-value ( 2.01) 0.022271.25

P Z P Z

7.4 – The Wilcoxon Rank-Sum Test

There are two commonly used techniques for throwing a shot put: glide and rotational. The first and third rows of the table below give the maximum distances (in m) of 12 different athletes using the glide method and 13 athletes using the rotational method at international competitions (data collected by David Meyer, 2010). Use this data to test the claim that there is not a significant difference between these two methods.

The Wilcoxon Rank-Sum Test

Parameters– median of all distances using the glide method– median of all distances using the rotation method

Hypotheses: H0: , H1:Test statistic: Add ranks from the first sample

1 1 2 4 25 129R

Theorem 7.4.1

If there are no ties and both populations have the same continuous distribution, then is an observed value of a random variable with mean and variance

1 1 2 1 2 1 221 1and

2 12R R

n n n n n n n

The Wilcoxon Rank-Sum Test

Purpose: To test the null hypothesis H0: where and are the medians of two independent populations with continuous distributions.Let– and sample sizes where 2

– Rank all the values from 1 to 2

– sum of the ranks from the first sample 1 1 2 1

2R

n n n

The Wilcoxon Rank-Sum Test

Test statistic:

Critical value: See Table C.7– Reject H0 if critical value

P-value: For a one-tail test

– Double this probability for a two-tail testRequirement

1. The samples are random and independent

1 1

1

if 2 otherwise

R

R

R RR

R

1 2 1 2 1( 0.5) wheP-value re 12

RR

R

n n n nRP Z

Example

,

Critical value: 119

– Do not reject H0: There is not a statistically significant difference between the medians of the two methods

1

12 12 13 1 12·13 12 13 1156 and 338

2 12129 156 129

R R

R R

(129 0.5) 156P-value 2 2 ( 1.44) 0.1498338

P Z P Z

7.5 – The Runs Test for Randomness

Purpose: To test the claim that a set of data with two types of values is arranged randomly

Definition 7.5.1 A run is a sequence of data of the same type preceded and followed by data of a different type or by no data at all

Example 7.5.1

A classical music fan has a collection of songs composed by Bach and Vivaldi on her MP3 player which is supposed to randomly choose songs. The order of the composer of the songs played is shown below. Test the claim that the composers are arranged randomly.

– songs by Bach – songs by Vivaldi– Total of runs

Theorem 7.5.1

If the data values are chosen randomly, then is an observed value of a random variable with p.m.f.

for

1 2

1 2

1

1 2 1 2

1 2

1

1 12

/ 2 1 / 2 1if is even

( )1 1 1 1

( 1) / 2 ( 3) / 2 ( 3) / 2 ( 1) / 2if is odd

n nr r

rn n

nf r

n n n nr r r r

rn n

n

Theorem 7.5.1

As , the distribution of approaches a normal distribution with mean and variance

1 2 1 2 1 221 22

1 2 1 2 1 2

2 22 1 and1

R R

n n n n n nn nn n n n n n

The Runs Test for Randomness

Null hypothesis: H0: The data are arranged randomly

Test statistic: Critical values: See Table C.8– Reject H0 if the smaller c.v. or the larger c.v.

P-value: Approximately twice the extreme region under the standard normal bell curve bounded by

R

R

Rz

Example 7.5.1

Hypotheses:H0: The composers are arranged randomlyH1: The composers are not arranged randomly

Test statistic:

Critical values: 7 and 17P-value:

2

2

2 10 12 2 10 12 10 122 10 12 1 11.91 and 5.1510 12 10 12 10 12 1

13 11.91P-value 2 2 ( 0.48) 0.63125.15

R R

Z P Z

Example 7.5.1

Conclusion– Do not reject H0

– It appears that the songs are randomly chosen