Estimation and Hypothesis Testing Faculty of Information
Technology King Mongkuts University of Technology North Bangkok
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Inferential Statistics Inferential statistics is a body of
quantitative techniques that enable the scientist to make
appropriate generalization from limited observations. (Frank &
Althoen, Statistics: Concepts and applications, 1994) 2
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Content Estimation Hypothesis testing Forming hypothesis
Testing population means Testing population variances Testing
categorical data / proportion Hypothesis about many population
means One-way ANOVA Two-way ANOVA 3
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Estimation To infer a parameter of population from a statistic
of sample From x (mean of sample) to (mean of population) From
proportion of sample to p (proportion of population) From SD 2
(sample variance) to 2 (population variance) Point estimation
Interval estimation 4
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Point Estimation A point estimator is a single-valued statistic
that approximates the value of a population parameter. Usually use
the same (unbiased) value as statistics of sample Point estimation
of Point estimation of proportion Point estimation of 2 5
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Interval Estimation A confidence interval is a range of values
that is expected to include the population parameter. Involve
calculating a +-value from the statistics of sample Interval
estimation of mean Interval estimation of proportion Interval
estimation of variance 6
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Hypothesis Testing Hypothesis testing a scientific method is
used for making decision, conclusion or prove of the finding of
research Steps Forming statistical hypothesis from research
hypothesis Define statistical significant level () Usually 0.05
(5%), or 0.01 for research needing higher accuracy Select the
appropriate statistic and calculate SPSS do the calculation
Accept/reject hypothesis Make decision 7
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Hypothesis The expected rational conclusion of statistical
analysis Research Hypothesis Written in text Statistical Hypothesis
Written in mathematical equation using parameter Hypothesis can be
relational and comparative Hypothesis can be directional and
non-directional 8
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Research Hypothesis Examples Handwriting and examination score
are related Relational, non-directional Handwriting and examination
score are positively related Relational, directional Female
students get higher final exam score than male student Comparative,
directional The scores of female and male students are different
Comparative, non-directional 9
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Statistical Hypothesis Mathematical form H 0 : Null hypothesis
(or Test hypothesis) Always non-directional Must have = (also >=
and ,
Statistical Hypothesis Examples Female students get higher
final exam score than male student H 0 : f m Female students get
final exam score higher than or equal to (no less than) male
student H 0 : ? H 1 : ? The scores of female and male students are
different H 0 : f = m H 1 : f != m 11
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Error in Hypothesis Testing Type I Error Error caused by
rejecting H 0 when H 0 is true Probability of type I error is equal
to which is statistical significant level defined in the analysis
Type II Error Error caused by accepting H 0 when H 0 is false
Probability of type I error is equal to 12
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Error in Hypothesis Testing DecisionH 0 : true H 1 : false H 0
: false H 1 : true Accepting H 0 Correct decisionType II Error ()
Rejecting H 0 Type I Error () Correct decision 13
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Hypothesis Testing Directional Test / One-Tailed Test
Right-Tailed H 0 : k Left-Tailed H 0 : >= k H 1 : < k
Non-directional Test / Two-Tailed Test H 0 : = k H 1 : != k : A
population parameter 14
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Testing Population Mean One Sample T-Test Independent Samples
T-Test Paired Sample T-Test SPSS Analyze -> Compare Mean ->
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Steps in Testing Mean Forming statistical hypothesis from
research hypothesis Left-tailed, Right-tailed, Two-tailed Define
statistical significant level () Usually 0.05 (5%), or 0.01 for
research needing higher accuracy Calculate and compare T value to
critical T value from T tableT table Right-tailed: Accept H 0,
reject H 1 if T cal < T table Reject H 0, accept H 1 if T cal
>= T table Left-tailed Accept H 0, reject H 1 if T cal > -T
table Reject H 0, accept H 1 if T cal
One Sample T-Test Test mean of one sample against a test value
Variable is either interval or ratio EX: Test if average total
score is more than 55 H 0 : 55 If the hypothesis is true then we
should reject H 0 and accept H 1 Calculate statistic (use SPSS)
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SPSS Analysis Result SPSS uses Sig.(2-tailed) or p-value to
show test result SPSS only does Two-tailed Divide this p-value by 2
to get one-tailed If p-value is less than (e.g. 0.05) then the test
is significant Reject H 0, accept H 1 Thus the average total score
is more than 55 at significance level 0.05 t: the calculated T
value df: degree of freedom 18
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Independent Samples T-Test Test mean of one sample against
another Assumptions Independent variable consists of two
independent groups. Dependent variable is either interval or ratio
Dependent variable is approximately normally distributed Similar
variances between the two groups (homogeneity of variances) 19
https://statistics.laerd.com/spss-tutorials/independent-t-test-using-spss-statistics.php
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Independent Samples T-Test EX: Test if male students get lower
total score than female students H 0 : m >= f H 1 : m < f If
the hypothesis is true then we should reject H 0 and accept H 1
Calculate statistic (use SPSS) 20
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Levenes Test for Equality of Variances For independent samples
T-Test, the calculation for T value is different when: Both samples
have the same variance ( 1 2 = 2 2 ) AND The variances are
difference ( 1 2 != 2 2 ) Use variance test to determine this See
Levenes Test for Equality of Variances in the table If the value of
Sig. is >= (e.g. 0.05) then the two variance is equal use the
first row of the result If the value of Sig. is >= (e.g. 0.05)
then the two variance is NOT equal use the second row of the result
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Result The p-value (2-tailed) is 0.033 < , thus the average
score of male and female students are different The p-value
(1-tailed) is 0.033/2 = 0.0165 < , thus the result is
significant Check the Group Statistics, female group has higher
mean, thus reject H 0 and accept H 1 - the research hypothesis is
true According to Levenes Test, use the first row (Sig. = 0.530
> ) 22
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Paired Sample T-Test Test means of paired samples against each
other Same sample group (or two dependent samples) Assumptions
Dependent variable is interval or ratio The differences in the
dependent variable between the two related groups are approximately
normally distributed. Independent variable consists of two related
groups or "matched-pairs". No outliers in the differences between
the two related groups. 23
https://statistics.laerd.com/spss-tutorials/dependent-t-test-using-spss-statistics.php
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Paired Sample T-Test EX: Test if final score is not different
from midterm score of the same group of student H 0 : D = 0 H 1 : D
!= 0 If the hypothesis is true then we should accept H 0 and reject
H 1 Calculate statistic (use SPSS) 24
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Result The p-value (2-tailed) is 0.000 < , thus the result
is significant Thus reject H 0 and accept H 1 - the research
hypothesis is false 25
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Testing Categorical Data or Proportion One variable binomial
proportion One variable multiple groups proportion (Goodness of Fit
Test) Two variables Chi-square Test of Independence Two variables
Test of Homogeneity 26
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Binomial Determining the proportion of people in one of two
categories is different from a specified amount H 0 : p D = p 0 H 1
: p D != p 0 SPSS assumes numerical data Recode data into number
e.g. M,F -> 1,2 Analyze->Nonparametric Tests->Legacy
Dialogs->Binomial E.g. the proportion of male student is 0.5 H 0
: p D = 0.5 H 1 : p D != 0.5 27
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Result Careful about the Test Prop. SPSS considers the first
observation (row) as first group Exact Sig. is 0.04 < , the
result is significant, thus reject H 0 and accept H 1 - proportion
of male students is not 0.5 If tested at 0.6 28
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Multiple Groups Goodness of Fit Test Determining the proportion
of groups is different from a specified ratio O: Observed E:
Expected Analyze -> Nonparametric Tests -> Legacy Dialogs
-> Chi-Square E.g. the proportion of sections is 1:2:1:2:1 29
https://statistics.laerd.com/spss-tutorials/chi-square-goodness-of-fit-test-in-spss-statistics.php
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Result The values in the expected values ratio correspond to
groups in order of appearance in the observation row. Asymp.Sig. =
0.000 < , the result is significant, thus reject H 0 and accept
H 1 - the proportion is not 1:2:1:2:1 30 Frequency less than 5
might make the analysis not meaningful
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Chi-square Test of Homogeneity Used to determine whether the
proportion of one variable is similar when grouped by another
variable two or more groups in each variable H 0 : p 1 = p 2 = p 3
= = p n H 1 : p 1 p 2 p 3 p n Data -> Weight Cases -> Weight
cases by -> Do not weight cases SPSS uses proportion of total
population Select frequency variable to test dependency Analyze
-> Descriptive Statistics -> Crosstabs Statistics -> Tick
Chi-square Cells -> Tick Expected (optional) 31
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Result E.g. The proportion of selected of major is similar in
both genders of student? H 0 : p m = p f H 1 : p m p f Pearson
Chi-Square: Asymp.Sig. 0.010 < Reject H 0 and accept H 1 - the
proportion is not similar in each gender 32
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Chi-square Test of Independence Used to determine whether the
effects of one variable depend on the value of another variable (2
variables) H 0 : Variable x and variable y are independent of each
other H 1 : Variable x and variable y are dependent of each other H
0 : (O - E) 2 = 0 H 1 : (O - E) 2 0 Data -> Weight Cases ->
Weight cases by -> Do not weight cases SPSS uses proportion of
total population Select frequency variable to test dependency
Analyze -> Descriptive Statistics -> Crosstabs Statistics
-> Tick Chi-square Cells -> Tick Expected (optional) 33
https://statistics.laerd.com/spss-tutorials/chi-square-test-for-association-using-spss-statistics.php
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Result E.g. Determine if gender and major are independent based
on total score H 0 : gender and major are independent of each other
H 1 : gender and major are dependent of each other Pearson
Chi-Square: Asymp.Sig. 0.00 < Reject H 0 and accept H 1 - the
two variables are dependent of each other based on total score
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Are they the same? Test of Homogeneity and Test of Independence
use the same calculation Test of Homogeneity tells if the
proportion is the same H 0 : Proportion is similar for all groups H
1 : Proportion not similar for some/all groups Test of Independence
tells if two variables are dependent H 0 : Two variables are
independent H 1 : Two variables are dependent 35
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Are they the same? Consider this The proportion of selected
major is the same for any gender That means no matter the gender,
the proportions remain the same That means gender has no effect of
selection of major and therefore the two are independent 36