TEST OF SIGNIFICANCE

1. TESTS OF SIGNIFICANCE
MODERATOR: PRESENTER:
MR.ARUN GOPI DR.ANCHU R NATH
LECTURER IN BIOSTATISTICS FIRST YEAR PG RESIDENT
DEPT. OF COMMUNITY MEDICINE

2. PLAN OF PRESENTATION:
 HISTORY
 INTRODUCTION
 HYPOTHESIS TESTING
 NULL HYPOTHESIS & ALTERNATIVE HYPOTHESIS
 TYPE I & TYPE II ERROR
 P VALUE
 PARAMETRIC TEST
 NON-PARAMETRIC TEST
 SUMMARY
 REFERENCES
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3. HISTORY
• The term Statistical significance was
coined by the Ronald Fisher (1890-1962)
Father of Modern Statistics.
•Student t-test : William Sealy Gosset
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4. INTRODUCTION
STATISTICAL
ANALYSIS
ESTIMATION
Prevalence or descriptive
study
In single group
Given proportion or
prevalence
Given mean and SD
STATISTICAL
INFERENCE
(HYPOTHESIS TESTING)
Done in two or more
groups.
Given two group mean &
SD
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5. HYPOTHESIS TESTING
• During investigation, there is assumption and presumption, which
subsequently in study must be proved or disproved.
• To test the statistical hypothesis about the population parameter or true
value of universe.
Two Hypothesis are made to draw the inference from the sample value:
1) A null hypothesis or hypothesis of no difference (H0)
2) Alternative hypothesis of significant difference (H1)
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6. CHARACTERISTICS OF HYPOTHESIS:
 Hypothesis should be clear and precise.
 It should be capable of being tested.
 It should state relationship between variables.
It must be specific and stated as simple as possible.
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7.  There is no difference between the statistic of a sample and parameter
of population or between statistics of two samples.
 The observed difference is entirely due to sampling error, i.e., it has occurred
purely by chance.
Example:
There is no difference between the incidence of measles between
vaccinated and non-vaccinated children.
NULL HYPOTHESIS
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8. ALTERNATIVE HYPOTHESIS
 Sample result is different, that is greater or smaller than the hypothetical
value of population.
Example: weight gain or loss due to new feeding regimen.
 Test of significance is performed to accept the null hypothesis or to reject it
and accept the alternative hypothesis.
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9. INTERPRETING THE RESULT OF HYPOTHESIS:
 The null Hypothesis is true – our test accepts it because the result
falls within the Zone of acceptance at 5% level of significance.
 The null hypothesis is false- test rejects it because the estimate
falls in the area of rejection.
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10. ZONE OFACCEPTANCE:
• If the result of a sample falls in the plain area i.e.
within the mean + 1.96 standard error (SE), the
null hypothesis is accepted.
ZONE OF REJECTION:
• If the result of a sample falls in the shaded area,
i.e beyond mean + 1.96 SE , it is significantly
different from the universe value.
• So null hypothesis is rejected
and alternative hypothesis is accepted.
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11. TYPE I AND TYPE II ERROR
When a null hypothesis is tested , there may be four possible outcomes:
Type I error – rejecting the null hypothesis when null hypothesis is true.
It is called ′𝜶 𝒆𝒓𝒓𝒐𝒓 ′
Type II error – accepting null hypothesis when null hypothesis is false .
It is called ′𝜷 𝒆𝒓𝒓𝒐𝒓′
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12. P – VALUE:
 It is the probability of obtaining a result equal to or more extreme than what
was actually observed.
 First introduced by Karl Pearson in his Pearson’s Chi squared test
Choice of cut-off value:
 Arbitrary cut off 0.05 (5% chance of a false positive conclusion)
 If p < 0.05 , statistically significant – Reject H0 , Accept H1
 If p > 0.05 , statistically not significant – Accept H0 , Reject H1
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13. P-value Interpretation:
A p-value measures the strength of evidence against a hypothesis.
• If the p- value is small , then either the null hypothesis is false or we got
a very unlikely sample.
• If the p-value is large , then there is a weak evidence against null
hypothesis , as a result its accepted
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14. Test of Significance ???
 A formal procedure for comparing observed data with a claim(also called a
hypothesis) whose truth we want to assess.
 A significance test uses data to evaluate a hypothesis by comparing sample
point estimates of parameters to values predicted by the hypothesis.
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15. Why Test of Significance???
 Have the observation changed with time / intervention?
 Do two or more groups observations differ from each other?
 Is there an association between different observations?
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16. Stages in performing a Test of Significance:
 A research question
 A null hypothesis (H0) suitable to the problem is set up.
 An alternate hypothesis is defined if necessary.
 A suitable statistical test , using a relevant formula is calculated.
 Then the p value is found out, corresponding to the calculated value
of test.
 If the p value is < 0.05, null hypothesis is rejected.
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17. PARAMETRIC
TEST
• When population distribution is
normal
• commonly used for normally
distributed interval or ratio data.
• More powerful or efficient when
compared
• Can’t be used in small sample.
NON-
PARAMETRIC
TEST
• When population skewed.
• Can be used to analyse data that
are non-normal or are nominal or
ordinal.
• Less powerful and less efficient
when compared.
• can be used in small sample.
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18. TEST OF SIGNIFICANCE
PARAMETRIC TEST NON-PARAMETRIC TEST
 Independent t test
 Paired t test
 ANOVA
 Repeated Measure ANOVA
 Pearson’s Correlation Test
 Mann –Whitney U test
 Wilcoxon signed rank test
 Kruskal Wallis test
 Friedman’s test
 Spearman Correlation test
 Chi square test
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19. PARAMETRIC TEST
Student’s t – Test:
 Developed by Prof. W.S. Gossett in 1908, who published statistical
papers under the pen name of ‘Student’.
T-test
Independent t-test
Paired t-test
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20.  Indication for the test:
1. When samples are small.
2. Population variance are not known.
Assumptions made in the use of t-test
1. Samples are randomly selected.
2. Data utilized is Quantitative.
3. Variable follow normal distribution.
4. Samples size lower than 30.
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21. INDEPENDENT T TEST
We compare the means of two different samples.
Degree of Freedom: number of values in the final calculation of a statistics that
are free to vary.
𝑑𝑓 =degree of freedom
𝑛𝑖 = sample size
𝒅𝒇 = (𝒏𝒊 - 1)
t =
𝒙𝟏−𝒙𝟐
√
𝒔𝟏
𝟐
𝒏𝟏
+
𝒔𝟐
𝟐
𝒏𝟐
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22. Example: The marks of boys and girls are given:
Is there any significant difference between marks
of boys and girls?
Firstly , we will calculate mean, SD, DOF
Boys Girls Girls
N1=9
df = 9-1 = 8
X1= 9.778
S1 = 4.1164
N2 = 10
df = 10-1 = 9
X2= 15.1
S2 = 4.2805
t =
𝒙𝟏−𝒙𝟐
√
𝒔𝟏
𝟐
𝒏𝟏
+
𝒔𝟐
𝟐
𝒏𝟐
= = - 2.758
-2.758 < 2.652
So we have to accept null hypothesis.
i.e, there is no statistical significant difference between the marks of boys and girls.
Marks :Boys Girls
12 21
14 18
10 14
8 20
16 11
5 19
3 8
9 12
11 13
15
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23. 28-06-2023 23

24. PAIRED T-TEST
We compare the means of two related or same group at two different time.
𝒕 =
𝒎
𝒔
𝒏
m = mean of difference between each pair of values
s = SD of difference between each pair of values
n = sample size
Example: BP of 8 patients before and after an antihypertensive drug are
recorded:
Is there any significant difference between BP reading before and after?
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25. Firstly, we find the mean, SD of difference between each pair of values.
Mean (m) = 𝑑= 465 = 58.125
8 8
Before After d(= Before-After)
180 140 40
200 145 55
230 150 80
240 155 85
170 120 50
190 130 60
200 140 60
165 130 35
𝑑 = 465
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26. H0: there is no significant difference between BP before & after the drug
H1: there is significant difference
Let the alpha value is 0.05 , DOF = 8-1 = 7
t value = 2.36
𝒕 =
𝒎
𝒔
𝒏
= 9.38
9.38 > 2.36
So, we have to reject null hypothesis.
i.e. there is significant difference between BP reading before & after drug.
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27. 28-06-2023 27

28. ANOVA (Analysis of Variance)
 Given by Sir Ronald Fischer
 Principle aim of statistical model is to explain the variation in
measurements.
 Test of significance for more than 2 groups independent of each other.
Assumptions for ANOVA
1. Sample population follow normal distribution.
2. Samples are selected randomly and independently.
3. Each group have common variance.
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29. Test statistics for ANOVA is F-test
ANOVA
ONE WAY ANOVA
TWO WAY ANOVA
One way ANOVA Two way ANOVA
One factor or independent
variable more than one factor or
independent variable
Compares 3 or more levels of one compares the effect of
factor multiple levels of 2 factors
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30. ANOVA = Variance between groups
Variance within groups
Variance between >
Variance within Reject H0
Variance between < or =
Variance within Fail to Reject H0
Example: We want to see if three different studying methods can lead to different
mean exam scores or not. To test this , we select 30 students and randomly assign 10
each to use a different studying method.
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31. Sno Method
A
Method
B
Method
C
1. 10 8 9
2. 9 9 8
3. 8 10 7
4. 7.5 8 10
5. 8.5 8.5 9
6. 9 7 8
7. 10 9.5 7
8. 8 9 10
9. 8 7 9
10. 9 10 8
8.7 8.6 8.5
Overall mean = 8.6
Between group variation = 10*(8.7-8.6)^2 +
10*(8.6-8.6)^2 + 10*(8.5-8.6)^2 = 0.2
Within group variation = 𝑋𝑖𝑗 − 𝑋𝑗 2
Method A = (10-8.7)2 + (9-8.7)2 + (8-8.7)2 + (7.5-
8.7)2 + (8.5-8.7)2 + (9-8.7)2 + (10-8.7)2 + (8-8.7)2
= 6.6
Method B = 10.9
Method C = 10.5
Within group variation = 6.6 +10.9+10.5=28
Variance between = 0.2 = 0.0071 <
Variance within groups 28
Accepting the null hypothesis
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32. 28-06-2023 32

33. REPEATED MEASURE ANOVA
Statistically significant differences between three or more dependent samples.
For example, if a sample is drawn of people who have knee surgery,
These people are interviewed for pain perception before surgery , 1 week and
2 weeks after surgery.
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34. Example: Therapy after a slipped disc has an influence on patient’s perception of
pain. Measuring the pain perception before, in the middle and at the end of
therapy.
H1: there is a significant difference
among the dependent groups
H0:there are no significant
difference among the dependent
groups
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35. 28-06-2023 35

36. PEARSON’s CORRELATION TEST
 Test to compare the linear relationship between two quantitatively measured
or continuous variables.
 Eg: Height and weight , temperature and pulse
 The extent of relationship measured by Pearson’s correlation coefficient ‘r’.
𝑥 & 𝑦 – variable samples
𝑥 & 𝑦 mean of values in x & y samples.
Assumptions made in calculation of ‘r’
1. Subjects selected for study with pair of X & Y value are chosen randomly.
2. Both X & Y variables are continuous & follow normal distribution.
𝑟 =
𝑥 − 𝑥 (𝑦 − 𝑦 )
√ 𝑥 − 𝑥2 (𝑦 − 𝑦2)
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37. r = +1 r= -1 r=0
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38. • Each point in the graph represents a single persons paired measurement of height &
weight.
• r = +0.38 ---- positive correlation.
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39. 28-06-2023 39

40. NON PARAMETRIC TEST
MANN-WHITNEY U TEST
• Determine whether two independent samples have been drawn from the
same population.
• Analyses the degree of separation ( or the amount of overlap) between
Experimental & Control groups.
n1n2 : sample sizes
R1 and R2 are sum of ranks assigned to group I & II
To be statistically significant obtained U has to be equal or less than critical
value.
𝑼 = 𝒏𝟏 𝒏𝟐 +
𝒏𝟏(𝒏𝟏+𝟏)
𝟐
- R1 or R2
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41. EXAMPLE : A researcher, while conducting studies on the Biomass of various
trees, wished to determine if there was a difference in the biomass of male and
female Juniper trees. So, he randomly selected 6 tress of each gender from the
field. He dries them to constant moisture, chips them, and then weighs them to
the nearest kg.
•H0: There is no difference between the biomass of
male and female Juniper trees
•H1: There is a difference between the biomass of male
and female Juniper trees
n1= 6 , n2 =6
R1 =23 ,R2 =55
Ucalculated = min (34, 2) = 2
Ucritical = 5
Ucalculated < Ucritical . Hence, we can reject the null hypothesis.
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42. 28-06-2023 42

43. WILCOXON SIGNED RANK TEST
• Used to compare two related samples , matched samples or repeated
measurements.
Assumptions:
1. Data are paired & come from same population.
2. Each paired is chosen randomly & independently.
To be statistically significant , obtained W has to be equal or less than
critical value.
Example: In order to investigate whether adults report verbally presented
material more accurately from their right than from their left ear , a
dichotic listening test was carried out. The data were found to be
positively skewed.
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44. Participant Lt ear Rt ear Difference
(d)
1 25 32 -7
2 29 30 -1
3 10 8 2
4 31 32 -1
5 27 20 -7
6 24 32 -8
7 26 27 -1
8 29 30 -1
9 30 32 2
10 32 32 0
11 20 30 -10
12 5 32 -27
To the rank the difference:
Lowest difference = -1 (1+2+3+4=10/4 = 2.5)
next lowest difference = 2( 5+6=11/2 = 5.5)
Adding the scores with + sign = 13
- sign = 53
Smaller value W = 13
N is the number of differences( omitting 0 difference)
N = 12 -1 = 11
Critical value ( N=11 , p = 0.05 ) = 14
Calculated value 13 < critical value 14
There is a difference between the number of words recalled from the Rt ear & number of
words recalled from Lt ear.
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45. Category Pre test Post Test Z P
Knowledge 21 (4-30) 48 (12-54) 6.56 0.001
Practice 11.2 (2-22) 22 (8- 33) 8.99 0.001
P value <0.05 , there is a statistically significant difference in the knowledge of pre
test & post test of rabies & its prevention.
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46. KRUSKAL WALLIS TEST
• Used to compare three or more independent groups.
• We use sum of the rank of k samples to compare the distribution.
• The test statistic for the Kruskal Wallis test ( denoted as H) is
defined as:
• samples drawn from the same population
T
o test the Ti = rank sum for the ith sample i = 1, 2,…,k of population
medians among groups
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47. EXAMPLE: In a manufacturing unit, 4 teams of operators were randomly
selected and sent to 4 different facilities for machining techniques training. After
the training, the supervisor conducted the exam and recorded the test scores.
At 95% confidence level does the scores are same in all four facilities?
• H0: The distribution of operator scores are same.
• H1: The scores may vary in four facilities
Hcalculated = 9.77 > Hcritical = 7.81
Hence, we reject the null hypotheses
So, there is enough evidence to conclude that difference in test scores exists for four
teaching methods at different facilities.
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48. 28-06-2023 48

49. FRIEDMAN’s TEST
• Non –parametric measure to repeated ANOVA
• To test for differences between groups (three or more paired groups) of the
dependent variable.
Assumptions:
• Samples are not normally distributed
• One group that is measured on three or more different occasions.
• Group is a random sample from the population.
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50. EXAMPLE: Department of Public health and safety monitors whether the
measures taken to clean up drinking water were effective. Trihalomethanes
(THMs) in 12 counties drinking water compared before cleanup, 1 week later,
and 2 weeks after cleanup.
•H0 = the cleanup system had no effect on
the THMs
•H1= the cleanup system effected the THMs
Significance level α=0.05
Qcalculated =20.16 > Qcritical = 6.5
hence reject the null hypotheses.
So, it is concluded that the cleanup system effected the THMs of drinking water.
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51. 28-06-2023 51

52. SPEARMAN’s CORRELATION TEST
• Assess the relationship between two variables.
• Rho ρ – non-parametric measure of statistical dependence between two variables.
d – difference between ranks of each observation.
𝜌 = 1 −
6( 𝑑2
)
𝑛(𝑛2 − 1)
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53. Example: 5 college students having following ranks in maths & science
subjects.
Is there an association between Science & Maths rank?
𝜌 = 1 −
6( 𝑑2
)
𝑛(𝑛2 − 1)
= -0.5
There is negative correlation between the
Science & maths subject rankings
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54. 28-06-2023 54

55. CHI – SQUARE TEST (X2) TEST
• An important continuous probability distribution
• Applied for smaller & larger samples
Prerequisites for Chi-square test:
1. The sample must be a random sample.
2. None of the observed values must be zero.
3. Data should be qualitative categorical.
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56. Steps in calculating (X2) value.
1) Make a contingency table mentioning the frequencies in all cells.
2) Determine the expected value (E) in each cell.
3) Calculate the difference between observed and expected values in each
cell (O-E)
E= row total x column total
Grand total
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57. 4) Calculate X2 value for each cell
5) Sum up X2 value of each table to get X2 value of table
6) Find out the p value from table.
7) If p > 0.05 - difference is not significant – null hypothesis accepted
If p <0.05 - difference is significant – null hypothesis rejected.
X2 of each cell = (O-E)2
E
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58. EXAMPLE: Attack rate among vaccinated & unvaccinated children against
measles.
Group Attacked Not-
Attacked
Total
Vaccinated
(obs)
10 90 100
Unvaccinated
(obs)
26 74 100
Total 36 164 200
Prove protective value of vaccination
by X2 test at 5% level of significance.
Group Attacked Not-
Attacked
Total
Vaccinated
(Exp)
18 82 100
Unvaccinated
(exp)
18 82 100
Total 36 164 200
X2 =Ʃ (O-E)2
E
= 8.67
Calculated value (8.67) > table value
(3.84) for p value 0.05.
Null hypothesis is rejected.
Vaccination is protective.
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59. 28-06-2023 59

60. SUMMARY
PARAMETRIC TEST NON-PARAMETRIC TEST
Independent measures,
2 groups
INDEPENDENT T TEST MANN-WHITNEY TEST
Independent measures,
> 2 groups
ANOVA KRUSKAL WALLIS TEST
Repeated measures,
2 dependent groups
PAIRED T TEST WILCOXON SIGNED RANK TEST
Repeated measures,
> 2 dependent groups
REPEATED MEASURE ANOVA FRIEDMAN TEST
Correlation test PEARSON SPEARMAN
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61. SELECTION OF THE STATISTICAL TEST
OBJECTIVE /
STUDY DESIGN
TYPE OF
OUTCOME
NATURE OF
OUTCOME
• Cohort
• Case control
• Cross-sectional
• Clinical trial
• Qualitative
• Quantitative
• Normal or not
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62. REFERENCES
1. Kadri A M. IAPSM’s Textbook of Community Medicine ,2nd ed. New Delhi: Jaypee
brothers medical publishers (P) Ltd; 2021. Chapter 11 ,Research methodology and
biostatistics; p.186-190.
2. K Park .Park’s Textbook of Preventive and Social Medicine , 27th ed.Jabalpur, M/s
Banarasidas Bhanot: 2022 Chapter 21,Health information and basic medical statistics,
page no: 978-981.
3. Bratati Banerjee.Mahajans methods in Biostatistics for medical students and research
workers ,9th edition , New Delhi :Jaypee brothers medical publishers (P)
Ltd:2018.Chapter 8,SamplingVariability and Significance ,p.183-247.
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al. Melatonin Supplementation and Anthropometric Indices: A Randomized Double-
Blind Controlled Clinical Trial. Biomed Res Int. 2021 Aug 10;2021:3502325.
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Support and Health-Promoting lifestyle in Pregnant Women: A Cross-Sectional Study. J
Caring Sci. 2021 May 24;10(2):96–102.
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63. 6.Adane T, Getaneh Z, Asrie F. Red Blood Cell Parameters and Their Correlation with
Renal Function Tests Among Diabetes Mellitus Patients: A Comparative Cross-Sectional
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https://sixsigmastudyguide.com/1-sample-sign-non-parametric-hypothesis-test/
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