Alexandru Ioan Cuza University from Ias i [605845]

\Alexandru Ioan Cuza" University from Ias i
Faculty of Mathematics
Tests for normality
Dissertation work
Scientic leader:
Lect. dr. Iulian StoleriuCandidate:
Daniela – Andreea Soltuz
July, 2021
Ia si

Contents
Introduction 2
1 Statistical tests 3
2 Tests for normality 4
2.1 Graphical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Normality tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Testing normality with Matlab . . . . . . . . . . . . . . . . . . . . . 4
3 Aplications 6
Bibliography 7
1

Introduction
2

Chapter 1
Statistical tests
Astatistical test is a method by which statistical decisions are made based col-
lected experimental data. The tests presented below are based on notions of proba-
bility theory. These tests allow us, based on a certain set or sets of experimentally-
selected data, to validate certain parameters estimates of a distribution or even
predict the form of the law of distribution of the considered characteristic.
Statistical test is structure in three types. These types are:
1. Parametric tests
2. Non-parametric tests
3. Goodness-of-t tests.
Type of statistical test is determined by alternative hypotesis ( H1). We have:
left unilateral test, when ( H1) is< 0;
right unilateral test, when ( H1)
3

Chapter 2
Tests for normality
2.1 Graphical methods
2.2 Normality tests
2.3 Testing normality with Matlab
In this section, we will be presented some Matlab functions that are useful in testing
normality of data.
1. The chi2gof (x) function tests, using Pearson's 2, where the vector xcomes
from a normal distribution, with the mean and dispersion estimated using x.
In this case of which data are ungrouped, then the result of the normality test
is obtained from
[h;p;stats ] =chi2gof(X):
In case of which data are groupted, then there have certain particularities
observed, we will have to take into account these particularities. This can be
done bu appealing to the general form of the Matlab function is:
[h;p;stats ] =chi2gof(X;name 1;val1;name 2;val2;:::)
where:
his the result of the test. If h= 1, then the null hypothesis is rejected, if
h= 0, then the null hypothesis cannot be rejected based on the observa-
tions made;
pis the value P (P-value);
the pairsnamei=valuei;8i2Nare optional. Namei variables can be:
number of classes, ` nbins ', a vector of central values of the intervals that
4

Chapter 2. Tests for normality 5
dene the classes, ` ctrs', or a vector with class ends, ` edges '.
Other variables that can be used: ` cdf', `expected ', `nparams ', `emin',
`frequency ', `alpha '.
the memory variable stats displays: chi2stat – statistics2,df- degrees of
freedom, edges – a vector with the ends of the class intervals after sorting,
O- the number of values observed in each class, E- the number of values
expected in each class.
2.Kolmogorov-Smirnov test in MATLAB (single selection)
For the Kolmogorov-Smirnov test for a selection, the MATLAB function is:
[h;p;ksstat ] =kstest (x;F;alpha;type )
where:
h;pare the same with previous test;
In addition to the previous functions, we have the type option. This refers
to how compare the two distribution functions and it can be one of the
following: ` unequal ',`larger ',`smaller '.
ksstat is the same with stats from previous test.
Fis cumulative distribution function for X.
3.Lilliefors Test
4.Anderson-Darling test

Chapter 3
Aplications
6

Bibliography
[1] Jay L. DeVore, Kenneth N. Berk, Modern Mathematical Statistics with Appli-
cations (with CD-ROM), Duxbury Press (2006).
[2] MATLAB and Statistics Toolbox Release 2020a, The MathWorks, Inc., Natick,
Massachusetts, United States.
[3] J. P. Marques de S a, Applied Statistics Using SPSS, STATISTICA, MATLAB
and R , Springer (2013).
[4] I. Stoleriu, Applied Statistics , online lecture notes (2020).
[5] H.C. Thode, Testing For Normality , Marcel Dekker Inc. (2002).
7