PICM105 & STATISTICS FOR MANAGEMENT...SRIT / PICM105 – SFM / Non – Parametric Test SRIT / M & H / M. Vijaya Kumar 4 Mon 1124 1120 16 0.014 Tue 1125 1120 25 0.022 Wed 1110 1120

SRIT / PICM105 – SFM / Non – Parametric Test

SRIT / M & H / M. Vijaya Kumar 1

SRI RAMAKRISHNA INSTITUTE OF TECHNOLOGY

(AN AUTONOMOUS INSTITUTION)

COIMBATORE- 641010

PICM105 & STATISTICS FOR MANAGEMENT

Unit IV

Non – parametric test

Introduction:

In general, non-parametric tests:

make few or no assumptions about the distribution of the data

reduce the effect of outliers and heterogeneity of variance

can often be used even for ordinal, and sometimes even nominal, data

Since non-parametric tests do not estimate population parameters, in general there are

no estimates of variance/variability

no confidence intervals

fewer measures of effect size

Also non-parametric tests are generally not as powerful as parametric alternatives when

the assumptions of the parametric tests are met.

Chi –Square Test of Goodness of Fit

Introduction

Chi-Square test of goodness of fit test is a non-parametric test that is used to find out

how the observed value of a given phenomenon is significantly different from the expected

value. In Chi-Square goodness of fit test, the term goodness of fit is used to compare the

observed sample distribution with the expected probability distribution. Chi-Square

goodness of fit test determines how well theoretical distribution (such as normal, binomial,

or Poisson) fits the empirical distribution. In Chi-Square goodness of fit test, sample data is

divided into intervals. Then the numbers of points that fall into the interval are compared,

with the expected numbers of points in each interval.



Definition:

If is a set of observed frequencies and is the

corresponding se of expected frequencies, then

∑

follows chi-square distribution with degrees of freedom.

Conditions for the validity of test:

1. The experimental data (samples) must be independent to each other.

2. The total frequency (no. of observations in the sample) must be large, say .

3. All the individual data’s should be greater than 5.

4. The no. of classes must lies in .

Problem: 1

The following table gives the number of air accidents that occur during the various days of

a week. Find whether the accidents are uniformly distributed over the week.

Days Sun Mon Tue Wed Thu Fri Sat

No. of Accidents 14 16 8 12 11 9 14

Answer:

Null Hypothesis : The accidents are uniformly distributed over the week.

Alternative Hypothesis : The accidents are not uniformly distributed over the week.

The test statistic is

∑

∑



Day Observed

freq

Expected

freq

Sun 14

Mon 16

Tue 8

Wed 12

Thu 11

Fri 9

Sat 14

84 84

Calculated value:

Table value:

Table value of with d.o.f is 12.59.

Decision rule:

so we accept .

Conclusion:

The accidents are uniformly distributed over the week.

Problem: 2

The demand for a particular spare part in a factory was found to vary from day-to-day.

In a sample study the following information was obtained.

Days: Mon Tue Wed Thu Fri sat

No. of spare parts demanded: 1124 1125 1110 1120 1126 1115

Test the hypothesis that the number of parts demanded does not depend on the day of the

week.

Answer:

: The number of parts demanded does not depend on the day of the week.

: The number of parts demanded depends on the day of the week.



∑

∑

Mon 1124 1120 16 0.014

Tue 1125 1120 25 0.022

Wed 1110 1120 100 0.089

Thu 1120 1120 0 0

Fri 1126 1120 36 0.032

Sat 1115 1120 25 0.022

6720

Calculated value:

Table value:


Decision rule:

so we accept .

Conclusion:

The number of parts demanded does not depend on the day of the week.



Problem: 3

The following figures show the distribution of digits in numbers chosen at random

from the following directory:

Digits 0 1 2 3 4 5 6 7 8 9 Total

Frequency 1026 1107 997 966 1075 933 1107 972 964 853 10000

Test whether the digits may be taken to occur equally frequently in the directory.

Answer:

: The digits are taken to occur equally frequently in the directory.

: The digits are not taken to occur equally frequently in the directory.

∑

∑

0 1026 1000 676 0.68

1 1107 1000 11449 11.45

2 997 1000 9 0.01

3 966 1000 1156 1.16

4 1075 1000 5625 5.63

5 933 1000 4489 4.49

6 1107 1000 11449 11.45

7 972 1000 784 0.78

8 964 1000 1296 1.30

9 853 1000 21609 21.61

10000 10000



Calculated value:

Table value:


Decision rule:

so we reject .

Conclusion:

The digits are not taken to occur equally frequently in the directory.

TEST OF INDEPENDENCE OF ATTRIBUTES

Under the null hypothesis : the attributes A and B are independent and we

calculate the expected frequencies for various cells using the following formula.

To conduct the test, we compute

∑∑( )

where number of rows and number of columns.

Remark:

For the contingency table with the cell frequencies and , the value is

given by

Problem: 4

Two researchers adopted different sampling techniques while investigating the same

group of students to find the number of students falling in different intelligence levels. The

results are as follows.

Researches Below Average Average Above average Genius

86 60 44 10 200

40 33 25 2 100

126 93 69 12 300

Would you say that the sampling techniques adopted by the 2 researches are independent?



Answer:

The sampling techniques adopted by the two researchers are independent.

The sampling techniques adopted by the two researchers are not independent.

The test statistic is given by

∑∑( )

The expected frequencies are

( )

( )

86 84 4 0.048

60 62 4 0.065

44 46 4 0.087

10 8 4 0.5

40 42 4 0.095

33 31 4 0.129

25 23 4 0.174

2 4 4 1

Calculated value:

Table value:




Decision rule:

so we accept .

Conclusion:

The sampling techniques adopted by the two researchers are independent

Problem: 5

Test of fidelity and selectivity of 190 radio receivers produced the results shown in

the following table.

Fidelity Selectivity Low Average High Total

Low 6 12 32 50

Average 33 61 18 112

High 13 15 0 28

Total 52 88 50 190

Use the 0.01 level of significance to test whether there is a relationship between fidelity

and selectivity.

Answer:

There is no relationship between fidelity and selectivity.

There is some relationship between fidelity and selectivity.


∑∑( )




( )

( )

Calculated value:

Table value:


Decision rule:

so we reject .

Conclusion:


Problem: 6

Using the data given in the following table to test at 1% level of significance whether

a person’s ability in Mathematics is independent of his / her interest in Statistics.

Ability in Mathematics

Low Average High

Interest

in

Statistics

Low 63 42 15

Average 58 61 31

High 14 47 29

Answer:



There is significant different between ability in Mathematics and interest in Statistics.

There is significant different between ability in Mathematics and interest in Statistics.


Low Average High Total

Interest

in

Statistics

Low 63 42 15 120

Average 58 61 31 150

High 14 47 29 90

Total 135 150 75 360


∑∑( )


( )

( )

63 45 324 7.20

42 50 64 1.28

15 25 100 4.00

58 56.25 3.06 0.05

61 62.5 2.25 0.04

31 31.25 0.06 0.00

14 33.75 390.06 11.56

47 37.5 90.25 2.41

29 18.75 105.06 5.60



Calculated value:

Table value:


Decision rule:

so we reject .

Conclusion:


Problem: 7

An automobile company gives you the following information about age groups and the

liking for particular model of car which it plans to introduce. On the basic of this data can it

be concluded that the model appeal is independent of the age group.

Persons who: Below 20 20-39 40-59 60 and above

Liked the car: 140 80 40 20

Disliked the car: 60 50 30 80

Answer:

There is no significant difference between models and age groups.

There is some significant difference between models and age groups.


∑∑( )

Persons who: Below 20 20-39 40-59 60 and above Total

Liked the car: 140 80 40 20 280

Disliked the car: 60 50 30 80 220

Total 200 130 70 100 500




( )

( )

Calculated value:

Table value:


Decision rule:

so we reject .

Conclusion:

There is some significant difference between models and age groups.



Problem: 8

A sample of 200 persons with a particular disease was selected. Out of these, 100

were given a drug and the others were not given any drug. The results are as follows.

No. of persons Drug No drug Total

Cured 65 55 120

Not cured 35 45 80

Total 100 100 200

Test whether the drug is effective or not.

Answer:

The drugs are effective.

The drugs are not effective.

The test statistic for the contingency table with the cell frequencies and

, the value is given by


Cured

Not cured

Total

Calculated value:

Table value:


Decision rule:

so accept .

Conclusion:

The drugs are effective.



Problem: 9

1000 students at college level were graded according to their I.Q and their economic

conditions. What conclusion can you draw from the following data:

Economic conditions I.Q level

High Low

Rich 460 140

Poor 240 160

Answer:

The I.Q level of students does not differ significantly.

The I.Q level of students differs significantly.



High Low Total

Rich

Poor

Total

Calculated value:

Table value:


Decision rule:

so reject .

Conclusion:

The I.Q level of students differs significantly.



Problem: 10

Find if there is any association between extravagance in fathers and extravagance in

sons from the following data. Determine the coefficient of association also

Extravagant

father

Miserly

father

Extra. Sons 327 741

Misser. Sons 545 234

Answer:

There is no association between extravagance in fathers and extravagance in sons.

There is some association between extravagance in fathers and extravagance in sons.



Extravagant

father Miserly father Total

Rich

Poor

Total

Calculated value:

Table value:


Decision rule:

so reject .

Conclusion:

There is some association between extravagance in fathers and extravagance in sons.



Problem: 11 Out of 8000 graduates in a town 800 are females; out of 1600 graduate employees 120 are

females. Use to determine if any distinction is made in appointment on the basis of sex.

Use 5% level of significance.

Answer:

First we form the table as follows Employment Un employment Total

Males 1480 5720 7200

Females 120 680 800

Total 1600 6400 8000

There is no distinction is made in appointment on the basis of sex.

There is some distinction is made in appointment on the basis of sex.



Extravagant father Miserly father Total

Rich

Poor

Total

Calculated value:

Table value:


Decision rule:

so reject .

Conclusion:

There is some distinction is made in appointment on the basis of sex.



The sign test for paired data

The Sign test is a non-parametric test that is used to test whether or not two groups

are equally sized. The sign test is used when dependent samples are ordered in pairs,

where the bivariate random variables are mutually independent It is based on the

direction of the plus and minus sign of the observation, and not on their numerical

magnitude. It is also called the binominal sign test, with .. The sign test is

considered a weaker test, because it tests the pair value below or above the median and it

does not measure the pair difference.

Test statistic

√

total no.of plus sign.

total number of and signs

Problem: 12

An automotive engineer is investigating 2 different types of metering devices for an

electronic fuel injection system to determine whether they differ in their fuel mileage

performance. The system is installed on 12 different cars and a test is run with each

metering device on each car. The observed fuel mileage performance data are given in the

following table. Use the sign test to determine whether the median fuel mileage

performance is the same for both devices using 5% LOS.

Car: 1 2 3 4 5 6 7 8 9 10 11 12

Device I: 17.6 19.4 19.5 17.1 15.3 15.9 16.3 18.4 17.3 19.1 17.8 18.2

Device II: 16.8 20 18.2 16.4 16 15.4 16.5 18 16.4 20.1 16.7 17.9

Answer:

Car: 1 2 3 4 5 6 7 8 9 10 11 12 Device I: 17.6 19.4 19.5 17.1 15.3 15.9 16.3 18.4 17.3 19.1 17.8 18.2

Device II: 16.8 20 18.2 16.4 16 15.4 16.5 18 16.4 20.1 16.7 17.9

Signs





Null Hypothesis :

That is the median fuel mileage performance is the same for both brands.

Alternative Hypothesis :

That is the median fuel mileage performance is not the same for both brands.

Test statistic

√

√

Table value:

at 5% significance level is 1.96.

Decision rule:

| |

| | so we accept .

Conclusion:

Hence the median fuel mileage is same

for both brands.

Problem: 13

The following data shows that the employee’s rates of defective work before and after a

change in the wage incentive plan. Compare the following two sets of data to see whether

the change lowered the defective units produced. Using the sign test with .

Rejection Region

Acceptance Region

Rejection Region

𝑧



Before 8 7 6 9 7 1 8 6 5 8 10 8

After 6 5 8 6 9 8 10 7 5 6 9 8

Answer:

Null Hypothesis :

There is no change before and after a change in the wage incentive plan.

Alternative Hypothesis : (Left Tailed Test)

There is a lower defective after a change in the wage incentive plan.

From the given data, we have

Before 8 7 6 9 7 1 8 6 5 8 10 8

After 6 5 8 6 9 8 10 7 5 6 9 8

Signs



Test statistic

√

√

Table value:

at 1% significance level is .

Decision rule:

We accept .

Conclusion:

There is no change before and after a

change in the wage incentive plan.

Acceptance Region

Rejection Region

𝑧



One sample sign test

Problem: 14

The following data represent the number of hours that a rechargeable hedge trimmer

operates before a recharge is required.

1.5 2.2 0.9 1.3 2 1.6 1.8 1.5 2 1.2 1.7

Hypothesis of the 0.05 LOS that this particular trimmer operates with a median of 1.8

hours before requiring a recharge.

Answer:

Null Hypothesis:

Null Hypothesis: (Two tailed Test)

Given data is

1.5 2.2 0.9 1.3 2 1.6 1.8 1.5 2 1.2 1.7

Assign for greater than 1.8.

Assign for less than 1.8.

Assign if it is equal to 1.8, we have



Test statistic

√

√



Table value:


Decision rule:

We accept .

Conclusion:

Hence this particular trimmer operates

with a median of 1.8 hours before requiring a

recharge.

Problem: 15

The following are the measurements of breaking strength of a certain kind of 2-inch cotton

ribbon in pounds.

163 165 160 189 161 171 158 151 169 162 163 139 172 165 148

166 172 163 187 173. Use the sign test to test whether the mean breaking strength

of a given kind of ribbon exceeds 160 pounds

Answer:

Null Hypothesis:

Null Hypothesis: (Right tailed Test)

Assign for greater than 160.

Assign for less than 160.

Assign if it is equal to 160, we have



Rejection Region

Acceptance Region

Rejection Region

𝑧



Test statistic

√

√

Table value:


Decision rule:

We reject . Accept .

Conclusion:

Hence the mean breaking strength of a given

kind of ribbon exceeds 160 pounds.

RANK SUM TEST

The test is the standard test for testing that the difference between population

means for two non-paired samples are equal. If the populations are non-normal,

particularly for small samples, then the t-test may not be valid. The rank sum test is an

alternative that can be applied when distributional assumptions are suspect. However, it is

not as powerful as the t-test when the distributional assumptions are in fact valid.

Mann-Whitney U-test

The Mann-Whitney U test is used to compare differences between two independent

groups when the dependent variable is either ordinal or continuous, but not normally

distributed. For example, you could use the Mann-Whitney U test to understand whether

attitudes towards pay discrimination, where attitudes are measured on an ordinal scale,

differ based on gender (i.e., your dependent variable would be "attitudes towards pay

discrimination" and your independent variable would be "gender", which has two groups:

"male" and "female"). Alternately, you could use the Mann-Whitney U test to understand

Rejection Region

Acceptance Region

𝑧

https://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/t_test.htm



whether salaries, measured on a continuous scale, differed based on educational level (i.e.,

your dependent variable would be "salary" and your independent variable would be

"educational level", which has two groups: "high school" and "university"). The Mann-

Whitney U test is often considered the nonparametric alternative to the independent t-test

although this is not always the case.

Test Statistic

√

[ ]

[ ]

and are sum of ranks of each groups.

Problem: 16

A new approach to prenatal care is proposed for pregnant women living in a rural

community. The new program involves in-home visits during the course of pregnancy in

addition to the usual or regularly scheduled visits. A pilot randomized trial with 15

pregnant women is designed to evaluate whether women who participate in the program

deliver healthier babies than women receiving usual care. The outcome is the APGAR score

measured 5 minutes after birth. Recall that APGAR scores range from 0 to 10 with scores of

7 or higher considered normal (healthy), 4-6 low and 0-3 critically low. The data are shown

below.

Usual Care 8 7 6 2 5 8 7 3

New Program 9 9 7 8 10 9 6

Is there statistical evidence of a difference in APGAR scores in women receiving the new

and enhanced versus usual prenatal care? We run the test using the five-step approach.

Answer:

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Null Hypothesis:

There is some difference in usual care and new program.


There is no difference in usual care and new program.

Calculate and :

Here . So take which is smaller value.

Take .

Method I Rank I Method II Rank II

8 10 9 13

7 7 9 13

6 4.5 7 7

2 1 8 10

5 3 10 15

8 10 9 13

7 7 6 4.5

3 2



√

√

Test statistic

Table value:


Decision rule:

| |

| | so we accept .

Conclusion:

There is no difference in usual care and

new program.

Problem: 17

Two methods of instruction to apprentices is to be evaluated. A director assigns 15

randomly selected trainers to each of the two methods. Due to drop outs, 14 complete in

batch 1 and 12 complete in batch 2. An achievement test was given to these successful

candidates. Their scores are as follows.

Method I 70 90 82 64 86 77 84 79 82 89 73 81 83 66

Method II 86 78 90 82 65 87 80 88 65 85 76 94

Test whether the two methods have the significant difference in effectiveness. Use Mann-

Whitney test at 5% LOS.

Rejection Region

Acceptance Region

Rejection Region

𝑧



Answer:

Null Hypothesis:

There is a significant difference in effectiveness between the two brands.


There is no difference in effectiveness between the two brands.


70 4 86 18.5

90 23.5 78 8

82 13 90 23.5

64 1 82 13

86 18.5 65 2

77 7 87 20

84 16 80 10

79 9 88 21

82 13 95 26

89 22 85 17

73 5 76 6

81 11 94 25

83 15

66 3

Calculate and :




Take .

√

√

Test statistic

Table value:


Decision rule:

| |

| | so we accept .

Conclusion:

There is a significant difference in

effectiveness between the two brands.

Rejection Region

Acceptance Region

Rejection Region

𝑧



Problem: 18

The nicotine content of two brands of cigarettes measured in milligrams was found to be as

follows.

Brand A 2.1 4 6.3 5.4 4.8 3.7 6.1 3.3

Brand B 4.1 0.6 3.1 2.5 4 6.2 1.6 2.2 1.9 5.4

Use the rank-sum test; test the hypothesis, at 0.05 LOS, that the average nicotine contents

of the two brands are equal against the alternative that they are equal.

Answer:

Null Hypothesis:

The average nicotine content are equal.


The average nicotine content are unequal.


2.1 4 4.1 12

4 10.5 0.6 1

6.3 18 3.1 7

5.4 14.5 2.5 6

4.8 13 4.0 10.5

3.7 9 6.2 17

6.1 16 1.6 2

3.3 8 2.2 5

1.9 3

5.4 14.5

Calculate and :




Take .

√

√

Test statistic

Table value:


Decision rule:

| |

| | so we accept .

Conclusion:

The average nicotine content are equal.

Rejection Region

Acceptance Region

Rejection Region

𝑧



Problem: 19

Twelve children one each selected from 12 sets of identical twins, were trained by a certain

method A and the remaining 12 children were trained by method B. at the end of the year,

the following I.Q scores were obtained.

Pair 1 2 3 4 5 6 7 8 9 10 11 12

Method A 124 118 127 120 135 130 140 128 140 126 130 126

Method B 131 127 135 128 137 131 132 125 141 118 132 129

Is this sufficient evidence to indicate a difference in the average IQ scores of the two

groups?

Answer:

Null Hypothesis:

The two methods IQ are equal.


The two methods IQ are unequal.

124 4 131 15.5

118 1.5 127 8.5

127 8.5 135 19.5

120 3 128 10.5

135 19.5 137 21

130 13.5 131 15.5

140 22.5 132 17.5

128 10.5 125 5

140 22.5 141 24

126 6.5 118 1.5

130 13.5 132 17.5

126 6.5 129 12



Calculate and :


Take .

√

√

Test statistic



Table value:


Decision rule:

| |

| | so we accept .

Conclusion:

The two methods IQ are equal.

Problem: 20

Two samples are as follows

Values of 1 2 3 5 7 9 11 18

Values of 4 6 8 10 12 13 14 15 19

Test whether the two samples come from the same population at level by using

Mann-Whitney U test.

Answer:

Null Hypothesis:

The samples came from same population.


The samples not came from same population.

1 1 4 4

2 2 6 6

3 3 8 8

5 5 10 10

7 7 12 12

9 9 13 13

11 11 14 14

18 16 15 15

19 17

Rejection Region

Acceptance Region

Rejection Region

𝑧



Calculate and :


Take .

√

√

Test statistic



Table value:


Decision rule:

| |

| | so we accept .

Conclusion:

The samples are came from same population.

Problem: 21

A clinical trial is run to assess the effectiveness of a new anti-retroviral therapy for patients

with HIV. Patients are randomized to receive a standard anti-retroviral therapy (usual

care) or the new anti-retroviral therapy and are monitored for 3 months. The primary

outcome is viral load which represents the number of HIV copies per milliliter of blood. A

total of 30 participants are randomized and the data are shown below.

Standard

Therapy 7500 8000 2000 550 1250 1000 2250 6800 3400 6300 9100 970 1040 670 400

New

therapy 400 250 800 1400 8000 7400 1020 6000 920 1420 2700 4200 5200 4100 undetectable

Is there statistical evidence of a difference in viral load in patients receiving the standard

versus the new anti-retroviral therapy?

Answer:

Null Hypothesis:

There is some difference in effectiveness between the old and new therapy.


There is no difference in effectiveness between the old and new therapy.


7500 26 400 2.5

8000 27.5 250 1

2000 15 800 6

550 4 1400 13

1250 12 8000 27.5

Rejection Region

Acceptance Region

Rejection Region

𝑧



Calculate and :


Take .

1000 9 7400 25

2250 16 1020 10

6800 24 6000 22

3400 18 920 7

6300 23 1420 14

9100 29 2700 17

970 8 4200 20

1040 11 5200 21

670 5 4100 19

400 2.5



√

√

Test statistic

Table value:


Decision rule:

| |

| | so we accept .

Conclusion:

There is some difference in

effectiveness between the old and new

therapy.

Kruskal-Wallis test or H-test

The Kruskal-Wallis H test (sometimes also called the "one-way ANOVA on ranks") is

a rank-based nonparametric test that can be used to determine if there are statistically

significant differences between two or more groups of an independent variable on a

continuous or ordinal dependent variable. It is considered the nonparametric alternative to

the one-way ANOVA, and an extension of the Mann-Whitney U test to allow the comparison

of more than two independent groups.

Rejection Region

Acceptance Region

Rejection Region

𝑧

https://statistics.laerd.com/spss-tutorials/one-way-anova-using-spss-statistics.phphttps://statistics.laerd.com/spss-tutorials/mann-whitney-u-test-using-spss-statistics.php



For example, you could use a Kruskal-Wallis H test to understand whether exam

performance, measured on a continuous scale from 0-100, differed based on test anxiety

levels (i.e., your dependent variable would be "exam performance" and your independent

variable would be "test anxiety level", which has three independent groups: students with

"low", "medium" and "high" test anxiety levels). Alternately, you could use the Kruskal-

Wallis H test to understand whether attitudes towards pay discrimination, where attitudes

are measured on an ordinal scale, differed based on job position (i.e., your dependent

variable would be "attitudes towards pay discrimination", measured on a 5-point scale

from "strongly agree" to "strongly disagree", and your independent variable would be "job

description", which has three independent groups: "shop floor", "middle management" and

"boardroom").

Test Statistic

[

]

Table value : with degrees of freedom.

Total sample groups.

Problem: 22

Use Kruskal-Wallis test to test for difference in mean among the 3 samples. If ,

what are your conclusions.

Sample I 95 97 99 98 99 99 99 94 95 98

Sample II 104 102 102 105 99 102 111 103 100 103

Sample III 119 130 132 136 141 172 145 150 144 135

Answer:

Null Hypothesis:

There is no difference in mean among three samples.


There is some difference in mean among three samples.



95 2.5 104 18 119 21

97 4 102 14 130 22

99 9 102 14 132 23

98 5.5 105 19 136 25

99 9 99 9 141 26

99 9 102 14 172 30

99 9 111 20 145 28

94 1 103 16.5 150 29

95 2.5 100 12 144 27

98 5.5 103 16.5 135 24

; (No. of sample groups)

Test statistic

[

]

[

]

Table value:

The value at LOS with d.o.f is .

Decision rule:

| |

| | so we reject .

Conclusion:

There is some difference in mean

among three samples.

Rejection Region

Acceptance Region

Rejection Region

𝑧



Problem: 23

An experiment designed to compute three preventive methods against corrosion yield the

following maximum depths of pits (in thousand’s of an inch) in pieces of wire subjected to

the respective treatments.

Method A 77 54 67 74 71 66

Method B 60 41 59 65 62 64 52

Method C 49 52 69 47 56

Use the 0.05 level of significance to test the three samples come from identical population

using kruskal-wallis test.

Answer:

Null Hypothesis:

The three samples come from identical population.


The three samples not come from identical population

Method A 77 18 60 9 49 3

54 6 41 1 52 4.5

67 14 59 8 69 15

74 17 65 12 47 2

71 16 62 10 56 7

66 13 64 11

52 4.5


Test statistic

[

]

[

]



Table value:


Decision rule:

| |

| | so we reject .

Conclusion:

There is some difference in mean

among three samples.

Problem: 24

The production volume of units assembled by three different operations during 9 shifts is

summarized in below table. Check whether there is significant difference between the

production volumes of units, assembled by the three operators using Kruskal wallis test at

significant level of 0.05.

Operator I: 29 34 34 20 32 45 42 24 35

Operator II: 30 21 23 25 44 37 34 19 38

Operator III: 26 36 41 48 27 39 28 46 13

Answer:

Null Hypothesis:

There is significant difference between the production volumes of units.


There is no significant difference between the production volumes of units.

Operator I Operator II Operator III

29 11 30 12 26 8

34 15 21 4 36 18

34 15 23 5 41 22

20 3 25 7 48 27

32 13 44 24 27 9

45 25 37 19 39 21

Rejection Region

Acceptance Region

Rejection Region

𝑧



42 23 34 15 28 10

24 6 19 2 46 26

35 17 38 20 13 1


Test statistic

[

]

[

]

Table value:


Decision rule:

| |

| | so we accept .

Conclusion:

There is significant difference between

the production volumes of units.

Problem: 25

A research company has designed three different systems to clean up oil spills. The

following table contains the results, measured by how much surface area (I square meters)

is cleaned in one hour. The data were found by testing each method in several trails. Are

three systems equally effective? Use 5% LOS.

System A 55 60 63 56 59 55

System B 57 53 64 49 62

System C 66 52 61 57

Answer:

Rejection Region

Acceptance Region

Rejection Region

𝑧



Null Hypothesis:

The three systems are equally effective.


The three systems are not equally effective.

System A Syatem B System C

55 4.5 57 7.5 66 15

60 10 53 3 52 2

63 13 64 14 61 11

56 6 49 1 57 7.5

59 9 62 12

55 4.5


Test statistic

[

]

[

]

Table value:


Decision rule:

| |

| | so we accept .

Conclusion:

The three systems are equally effective.

Rejection Region

Acceptance Region

Rejection Region

𝑧



One sample run test:

The one sample run test is used to test whether a series of binary events can be

considered as randomly distributed or not.

A run is a sequence of identical events, preceded and succeeded by different or no

events. The runs test used here applies to binomial variables only. For example, in

ABBABBB, we have 4 runs (A, BB, A, BBB).

One sample runs test hypothesis

In the case of the two-tailed (or two-sided) test, the null and alternative

hypotheses are:

Data are randomly distributed.

Data are not randomly distributed.

In the one-tailed case, you need to distinguish the left-tailed (or lower-tailed or lower

one-sided) test and the right-tailed (or upper-tailed or upper one-sided) test. In the left-

tailed test, the following hypotheses are used:

: Data are randomly distributed.

: There is repulsion between the two types of events.

In the right-tailed test, the following hypotheses are used:

: Data are randomly distributed.

: The two types of events are alternating.

Test statistic

and sample sizes for each groups.

√



Problem: 26

A technician is asked to analyze the results of 22 items made in a preparation run. Each

item has been measured and compared to engineering specifications. The order of

acceptance and rejections is . Determine whether it is a

random sample or not. Use .

Answer:

Given data’s are

Here and

(total no. of runs )

√

√

√

Null Hypothesis:

: The samples are randomly chosen.

Null Hypothesis:

: The samples are not randomly chosen. (Two Tailed)

Test statistic



Table value:


Decision rule:

| |

| | so we accept .

Conclusion:

The samples are randomly chosen .

Problem: 27

40 people were selected at random in the following order

. Assuming the population has 50% men and

50% women, is true that the people were selected at random?

Answer:

Given data’s are

Here and


√

√

Rejection Region

Acceptance Region

Rejection Region

𝑧



√

Null Hypothesis:

: The peoples were selected at random.

Null Hypothesis:

: The peoples were not selected at random. (Two Tailed)

Test statistic

Table value:


Decision rule:

| |

| | so we accept .

Conclusion:

The peoples were selected at random.

Problem: 28

Jack wants an 0.05 level test to determine whether the gender of the people walking into

her store is a random event. The gender data was collected from Jack’s Saturday morning

customers. Runs have been underlined. ,

.

Answer:

Given data’s are

Rejection Region

Acceptance Region

Rejection Region

𝑧



Here and


√

√

√

Null Hypothesis:

: The gender of the people walking into her store is a random event.

Null Hypothesis:

: The gender of the people walking into her store is not a random event. (Two Tailed)

Test statistic

Table value: at 5% significance level is 1.96.

Decision rule:

| |

| | so we reject .

Conclusion:

The gender of the people walking into

her store is not a random event.

Rejection Region

Acceptance Region

Rejection Region

𝑧



Problem: 29

In 30 tosses of a coin the following sequence of heads and tails is obtained.

. Determine the number of runs

and test at the 5% LOS whether the sequence is random.

Answer:

Given data’s are

Here and


√

√

√

Null Hypothesis:

: The sequence is random.

Null Hypothesis:

: The sequence is not random. (Two Tailed)

Test statistic



Table value:


Decision rule:

| |

| | so we reject .

Conclusion:

The sequence is not random.

Problem: 30

In an industrial production line items are inspected periodically for defectives. The

following is a sequence of defective items (D) and non-defective items (N) produced by

these production line. DD NNN D NN DD NNNNN DDD NN D NNNN D N D. Test whether the

defectives are occurring at random or not at 5% LOS.

Answer:

Given data’s are

Here and


√

√

√

Rejection Region

Acceptance Region

Rejection Region

𝑧



Null Hypothesis:

: Defectives are occurring at random.

Null Hypothesis:

: Defectives are not occurring at random. (Two Tailed)

Test statistic

Table value:


Decision rule:

| |

| | so we accept .

Conclusion:

The defectives are occurring at random.

Two Marks

1. Define Chi-Square test for goodness of fit.

Answer:

It is used to observe that the closeness of a sample matches a population. The Chi-

square test statistic is,

∑

with degrees of freedom, where is the observed count, is categories, and is

the expected counts.

2. State the application of Chi-square test.

Answer:

The Chi-square is used most commonly to compare the incidence (or proportion) of a

characteristic in one group to the incidence (or proportion) of a characteristic in other

group(s).

Rejection Region

Acceptance Region

Rejection Region

𝑧



Eg: You might use the Chi-Square test to compare the incidence PONV between patients

that received ondansetron, patients that received droperidol, and patients that received a

placebo.

3. Write the formula for the Chi-square test of goodness of fit of a random sample to

a hypothetical distribution.

Answer:

∑

4. Give the formula for the test of independence for

a b

c d

Answer:

For a 2 x 2 contingency table the Chi Square statistic is calculated by the formula:

5. State the conditions for applying test.

Answer:

The sample observations should be independent.

Total frequency should be at least 50.

No theoretical frequency should be less than 5.

6. What are the advantages of non-parametric methods?

Answer:

If sample sizes as small as N=6 are used, there is no alternative to using a

nonparametric test.

Easier to learn and apply than parametric tests

Based on a model that specifies very general conditions.

No specific form of the distribution from which the sample was drawn.

Hence nonparametric tests are also known as distribution free tests.



7. What are the disadvantages of non-parametric methods?

Answer:

Losing precision/wasteful of data

Low power

False sense of security

Lack of software

Testing distributions only

Higher-ordered interactions not dealt with

Parametric models are more efficient if data permit.

It is difficult to compute by hand for large samples

Tables are not widely available

8. What is the test statistic for Mann-Whitney U-test?

Answer:

Test Statistic

√

[ ]

[ ]

and are sum of ranks of each groups.

9. What is the test statistic for Kruskal-Wallis test or H-test?

Answer:

[

]



Table value : with degrees of freedom.

Total sample groups.

10. What is the test statistic for One sample run test?

Answer:

√

11. What is meant by run test?

Answer:

A runs-test is a statistical procedure that examines whether a string of data is occurring

randomly from a specific distribution. The runs test analyzes the occurrence of similar

events that are separated by events that are different.

12. What is the application of test and test?

Answer:

A z-test is a statistical test used to determine whether two population means are

different when the variances are known and the sample size is large.

A t-test or students t- test used to determine whether two population means are

different when the variances are known and the sample size is small.

13. What is the use of Mann Whitney u test?

Answer:

The Mann-Whitney U test is used to compare differences between two independent groups

when the dependent variable is either ordinal or continuous, but not normally distributed.

14. What is the use of kruskal wallis test?

Answer:



The Kruskal-Wallis H test (sometimes also called the "one-way ANOVA on ranks") is a

rank-based nonparametric test that can be used to determine if there are statistically

significant differences between two or more groups of an independent variable on a

continuous or ordinal dependent variable.

15. What is the use of run test?

Answer:

A runs-test is a statistical procedure that examines whether a string of data is occurring

randomly from a specific distribution. The runs test analyzes the occurrence of similar

events that are separated by events that are different.

16. What is importance of Kolmogorov Simonov test?

Answer:

The Kolmogorov-Smirnov test (Chakravart, Laha, and Roy, 1967) is used to decide if a

sample comes from a population with a specific distribution. ... An attractive feature of this

test is that the distribution of the K-S test statistic itself does not depend on the underlying

cumulative distribution function being

17. List any two non-parametric test.

Answer:

Mann-Whitney U Test

The Kruskal-Wallis Test

Chi-squared Test



Problem: 1

The following table gives the number of air accidents that occur during the various days of

a week. Find whether the accidents are uniformly distributed over the week.

Days Sun Mon Tue Wed Thu Fri Sat

No. of Accidents 14 16 8 12 11 9 14

Problem: 2

The demand for a particular spare part in a factory was found to vary from day-to-day.

In a sample study the following information was obtained.

Days: Mon Tue Wed Thu Fri sat

No. of spare parts demanded: 1124 1125 1110 1120 1126 1115

Test the hypothesis that the number of parts demanded does not depend on the day of the

week.

Problem: 3

The following figures show the distribution of digits in numbers chosen at random

from the following directory:

Digits 0 1 2 3 4 5 6 7 8 9 Total

Frequency 1026 1107 997 966 1075 933 1107 972 964 853 10000

Test whether the digits may be taken to occur equally frequently in the directory.

Problem: 4

Two researchers adopted different sampling techniques while investigating the same

group of students to find the number of students falling in different intelligence levels. The

results are as follows.

Researches Below Average Average Above average Genius

86 60 44 10 200

40 33 25 2 100

126 93 69 12 300

Would you say that the sampling techniques adopted by the 2 researches are independent?

Problem: 5

Test of fidelity and selectivity of 190 radio receivers produced the results shown in

the following table.



Fidelity Selectivity Low Average High Total

Low 6 12 32 50

Average 33 61 18 112

High 13 15 0 28

Total 52 88 50 190

Use the 0.01 level of significance to test whether there is a relationship between fidelity

and selectivity.

Problem: 6

Using the data given in the following table to test at 1% level of significance whether

a person’s ability in Mathematics is independent of his / her interest in Statistics.


Low Average High

Interest

in

Statistics

Low 63 42 15

Average 58 61 31

High 14 47 29

Problem: 7

An automobile company gives you the following information about age groups and the

liking for particular model of car which it plans to introduce. On the basic of this data can it

be concluded that the model appeal is independent of the age group.

Persons who: Below 20 20-39 40-59 60 and above

Liked the car: 140 80 40 20

Disliked the car: 60 50 30 80

Problem: 8

A sample of 200 persons with a particular disease was selected. Out of these, 100

were given a drug and the others were not given any drug. The results are as follows.


Cured 65 55 120

Not cured 35 45 80

Total 100 100 200

Test whether the drug is effective or not.



Problem: 9

1000 students at college level were graded according to their I.Q and their economic

conditions. What conclusion can you draw from the following data:

Economic conditions I.Q level

High Low

Rich 460 140

Poor 240 160

Problem: 10

Find if there is any association between extravagance in fathers and extravagance in

sons from the following data. Determine the coefficient of association also

Extravagant

father

Miserly

father

Extra. Sons 327 741

Misser. Sons 545 234

Problem: 11 Out of 8000 graduates in a town 800 are females; out of 1600 graduate employees 120 are

females. Use to determine if any distinction is made in appointment on the basis of sex.

Use 5% level of significance.

Problem: 12

An automotive engineer is investigating 2 different types of metering devices for an

electronic fuel injection system to determine whether they differ in their fuel mileage

performance. The system is installed on 12 different cars and a test is run with each

metering device on each car. The observed fuel mileage performance data are given in the

following table. Use the sign test to determine whether the median fuel mileage

performance is the same for both devices using 5% LOS.

Car: 1 2 3 4 5 6 7 8 9 10 11 12

Device I: 17.6 19.4 19.5 17.1 15.3 15.9 16.3 18.4 17.3 19.1 17.8 18.2

Device II: 16.8 20 18.2 16.4 16 15.4 16.5 18 16.4 20.1 16.7 17.9

Problem: 13

The following data shows that the employee’s rates of defective work before and after a

change in the wage incentive plan. Compare the following two sets of data to see whether

the change lowered the defective units produced. Using the sign test with .



Before 8 7 6 9 7 1 8 6 5 8 10 8

After 6 5 8 6 9 8 10 7 5 6 9 8

Problem: 14

The following data represent the number of hours that a rechargeable hedge trimmer

operates before a recharge is required.

1.5 2.2 0.9 1.3 2 1.6 1.8 1.5 2 1.2 1.7

Hypothesis of the 0.05 LOS that this particular trimmer operates with a median of 1.8

hours before requiring a recharge.

Problem: 15

The following are the measurements of breaking strength of a certain kind of 2-inch cotton

ribbon in pounds.

163 165 160 189 161 171 158 151 169 162 163 139 172 165 148

166 172 163 187 173. Use the sign test to test whether the mean breaking strength

of a given kind of ribbon exceeds 160 pounds

Problem: 16

A new approach to prenatal care is proposed for pregnant women living in a rural

community. The new program involves in-home visits during the course of pregnancy in

addition to the usual or regularly scheduled visits. A pilot randomized trial with 15

pregnant women is designed to evaluate whether women who participate in the program

deliver healthier babies than women receiving usual care. The outcome is the APGAR score

measured 5 minutes after birth. Recall that APGAR scores range from 0 to 10 with scores of

7 or higher considered normal (healthy), 4-6 low and 0-3 critically low. The data are shown

below.

Usual Care 8 7 6 2 5 8 7 3

New Program 9 9 7 8 10 9 6

Is there statistical evidence of a difference in APGAR scores in women receiving the new

and enhanced versus usual prenatal care? We run the test using the five-step approach.

Problem: 17

Two methods of instruction to apprentices is to be evaluated. A director assigns 15

randomly selected trainers to each of the two methods. Due to drop outs, 14 complete in

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batch 1 and 12 complete in batch 2. An achievement test was given to these successful

candidates. Their scores are as follows.

Method I 70 90 82 64 86 77 84 79 82 89 73 81 83 66

Method II 86 78 90 82 65 87 80 88 65 85 76 94

Test whether the two methods have the significant difference in effectiveness. Use Mann-

Whitney test at 5% LOS.

Problem: 18

The nicotine content of two brands of cigarettes measured in milligrams was found to be as

follows.

Brand A 2.1 4 6.3 5.4 4.8 3.7 6.1 3.3

Brand B 4.1 0.6 3.1 2.5 4 6.2 1.6 2.2 1.9 5.4

Use the rank-sum test; test the hypothesis, at 0.05 LOS, that the average nicotine contents

of the two brands are equal against the alternative that they are equal.

Problem: 19

Twelve children one each selected from 12 sets of identical twins, were trained by a certain

method A and the remaining 12 children were trained by method B. at the end of the year,

the following I.Q scores were obtained.

Pair 1 2 3 4 5 6 7 8 9 10 11 12

Method A 124 118 127 120 135 130 140 128 140 126 130 126

Method B 131 127 135 128 137 131 132 125 141 118 132 129

Is this sufficient evidence to indicate a difference in the average IQ scores of the two

groups?

Problem: 20

Two samples are as follows

Values of 1 2 3 5 7 9 11 18

Values of 4 6 8 10 12 13 14 15 19

Test whether the two samples come from the same population at level by using

Mann-Whitney U test.



Problem: 21

A clinical trial is run to assess the effectiveness of a new anti-retroviral therapy for patients

with HIV. Patients are randomized to receive a standard anti-retroviral therapy (usual

care) or the new anti-retroviral therapy and are monitored for 3 months. The primary

outcome is viral load which represents the number of HIV copies per milliliter of blood. A

total of 30 participants are randomized and the data are shown below.

Standard

Therapy 7500 8000 2000 550 1250 1000 2250 6800 3400 6300 9100 970 1040 670 400

New

therapy 400 250 800 1400 8000 7400 1020 6000 920 1420 2700 4200 5200 4100 undetectable

Is there statistical evidence of a difference in viral load in patients receiving the standard

versus the new anti-retroviral therapy?

Problem: 22

Use Kruskal-Wallis test to test for difference in mean among the 3 samples. If ,

what are your conclusions.

Sample I 95 97 99 98 99 99 99 94 95 98

Sample II 104 102 102 105 99 102 111 103 100 103

Sample III 119 130 132 136 141 172 145 150 144 135

Problem: 23

An experiment designed to compute three preventive methods against corrosion yield the

following maximum depths of pits (in thousand’s of an inch) in pieces of wire subjected to

the respective treatments.

Method A 77 54 67 74 71 66

Method B 60 41 59 65 62 64 52

Method C 49 52 69 47 56

Use the 0.05 level of significance to test the three samples come from identical population

using kruskal-wallis test.

Problem: 24

The production volume of units assembled by three different operations during 9 shifts is

summarized in below table. Check whether there is significant difference between the



production volumes of units, assembled by the three operators using Kruskal wallis test at

significant level of 0.05.

Operator I: 29 34 34 20 32 45 42 24 35

Operator II: 30 21 23 25 44 37 34 19 38

Operator III: 26 36 41 48 27 39 28 46 13

Problem: 25

A research company has designed three different systems to clean up oil spills. The

following table contains the results, measured by how much surface area (I square meters)

is cleaned in one hour. The data were found by testing each method in several trails. Are

three systems equally effective? Use 5% LOS.

System A 55 60 63 56 59 55

System B 57 53 64 49 62

System C 66 52 61 57

Problem: 26

A technician is asked to analyze the results of 22 items made in a preparation run. Each

item has been measured and compared to engineering specifications. The order of

acceptance and rejections is . Determine whether it is a

random sample or not. Use .

Problem: 27

40 people were selected at random in the following order

. Assuming the population has 50% men and

50% women, is true that the people were selected at random?

Problem: 28

Jack wants an 0.05 level test to determine whether the gender of the people walking into

her store is a random event. The gender data was collected from Jack’s Saturday morning

customers. Runs have been underlined. ,

.

Problem: 29

In 30 tosses of a coin the following sequence of heads and tails is obtained.



. Determine the number of runs

and test at the 5% LOS whether the sequence is random.

Problem: 30

In an industrial production line items are inspected periodically for defectives. The

following is a sequence of defective items (D) and non-defective items (N) produced by

these production line. DD NNN D NN DD NNNNN DDD NN D NNNN D N D. Test whether the

defectives are occurring at random or not at 5% LOS.

Values of the Chi-squared distribution

P

DF 0.995 0.975 0.20 0.10 0.05 0.025 0.02 0.01 0.005 0.002 0.001

1 0.0000393 0.000982 1.642 2.706 3.841 5.024 5.412 6.635 7.879 9.550 10.828

2 0.0100 0.0506 3.219 4.605 5.991 7.378 7.824 9.210 10.597 12.429 13.816

3 0.0717 0.216 4.642 6.251 7.815 9.348 9.837 11.345 12.838 14.796 16.266

4 0.207 0.484 5.989 7.779 9.488 11.143 11.668 13.277 14.860 16.924 18.467

5 0.412 0.831 7.289 9.236 11.070 12.833 13.388 15.086 16.750 18.907 20.515

6 0.676 1.237 8.558 10.645 12.592 14.449 15.033 16.812 18.548 20.791 22.458

7 0.989 1.690 9.803 12.017 14.067 16.013 16.622 18.475 20.278 22.601 24.322

8 1.344 2.180 11.030 13.362 15.507 17.535 18.168 20.090 21.955 24.352 26.124

9 1.735 2.700 12.242 14.684 16.919 19.023 19.679 21.666 23.589 26.056 27.877

10 2.156 3.247 13.442 15.987 18.307 20.483 21.161 23.209 25.188 27.722 29.588

11 2.603 3.816 14.631 17.275 19.675 21.920 22.618 24.725 26.757 29.354 31.264

12 3.074 4.404 15.812 18.549 21.026 23.337 24.054 26.217 28.300 30.957 32.909

13 3.565 5.009 16.985 19.812 22.362 24.736 25.472 27.688 29.819 32.535 34.528



14 4.075 5.629 18.151 21.064 23.685 26.119 26.873 29.141 31.319 34.091 36.123

15 4.601 6.262 19.311 22.307 24.996 27.488 28.259 30.578 32.801 35.628 37.697

16 5.142 6.908 20.465 23.542 26.296 28.845 29.633 32.000 34.267 37.146 39.252

17 5.697 7.564 21.615 24.769 27.587 30.191 30.995 33.409 35.718 38.648 40.790

18 6.265 8.231 22.760 25.989 28.869 31.526 32.346 34.805 37.156 40.136 42.312

19 6.844 8.907 23.900 27.204 30.144 32.852 33.687 36.191 38.582 41.610 43.820

20 7.434 9.591 25.038 28.412 31.410 34.170 35.020 37.566 39.997 43.072 45.315

21 8.034 10.283 26.171 29.615 32.671 35.479 36.343 38.932 41.401 44.522 46.797

22 8.643 10.982 27.301 30.813 33.924 36.781 37.659 40.289 42.796 45.962 48.268

23 9.260 11.689 28.429 32.007 35.172 38.076 38.968 41.638 44.181 47.391 49.728

24 9.886 12.401 29.553 33.196 36.415 39.364 40.270 42.980 45.559 48.812 51.179

25 10.520 13.120 30.675 34.382 37.652 40.646 41.566 44.314 46.928 50.223 52.620



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Documents

PICM105 & STATISTICS FOR MANAGEMENT...SRIT / PICM105 – SFM / Non – Parametric Test SRIT / M & H / M. Vijaya Kumar 4 Mon 1124 1120 16 0.014 Tue 1125 1120 25 0.022 Wed 1110 1120