BİL/CSE 395 APPLIED DATA
ANALYSIS
06-Parametric Tests
Assist. Prof. Dr. E. Burcu MAMAK EKİNCİ
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Parametric Tests
Ø T TEST (STUDENT’S T TEST)
■ One Sample T-Test
■ Two Sample T-Test
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Student’s t-test
■ The Student's t-test is used in statistics to determine whether there is a
significant difference between the means of two groups.
Types of Student’s t-Tests
1. One-Sample t-Test : Compares the mean of a single sample to a known population
mean.
2. Independent (Unpaired) t-Test : Compares the means of two independent groups.
Example: Comparing test scores of two different classes.
3. Paired t-Test : Compares means from the same group at different times. Example:
Measuring weight before and after a diet.
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One Sample t test
■ Assumptions of the one sample t-test:
Ø The data are sampled from a normal distribution.
Ø This actually has more to do with the sampling distribution of
sample means being approximately normal than the actual
population
Ø The sampling distribution of sample means for sufficiently large
sample sizes (𝑛 > 30) will always be normally distributed (Central
Limit Theorem)
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One Sample t test
■ A one-sample t-test is used to determine whether the mean of a single sample is significantly different
from a known population mean.
Hypotheses:
𝐻! : 𝜇 = 𝜇! 𝐻! : 𝜇 = 𝜇! 𝐻! : 𝜇 = 𝜇!
𝐻" : 𝜇 ≠ 𝜇! 𝐻" : 𝜇 < 𝜇! 𝐻" : 𝜇 > 𝜇!
Test statistic is
# ! µ!
$=
" !
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Example 1:
The manager of an online shopping site wants to a research about the time his
customers spend on this shopping site. The manager claims that the mean time
spent on the site is different from 40 minutes. For this, 50 people were randomly
selected from the customers and the time they spent on the website in a single
login was measured. Can it be said that the mean time spent on the website is
different from 40 minutes?
It was shown in the application "Chapter 5" that the spent time variable came from
a population with a normal distribution.
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One sample t test in R
Hypothesis:
𝐻! : 𝜇 = 40
𝐻" : 𝜇 ≠ 40
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Alternative function of one sample t test
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Independent sample t-test
It is used to compare the means of two different groups.
Assumptions:
1. Groups must be independent.
2. Dependent variables (height, weight, etc.) must be continuous
variables. In other words, variables must be obtained with interval
and proportional scales.
3. Dependent variable must have normal distribution for each group.
4. Group variances must be homogeneous. In other words, group
variances must be equal.
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Homogeneity test of variances
Varyansların homojenliğini test etmek için bir çok test ileri sürülmüştür. Bazıları
aşağıdadır.
1. F test
2. Pittman Test (1939)
3. Levene Testi (1960)
4. Bartlett Test
𝐻! : Variances are homogeneous
𝐻" : Variances are not homogeneous
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Research hypothesis:
𝐻( : There is no significant difference between the two group means
(𝐻( : 𝜇) = 𝜇* )
𝐻) : There is no significant difference between the two group means
(𝐻) : 𝜇) ≠ 𝜇* )
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Example 2:
■ A new drug development study is being conducted to be used in the treatment of a disease. In this
study, a group of 20 patients was randomly divided into two groups of 10 people each. The first group
was determined as the experimental group, and the second group was the control group. The newly
developed drug was given to the experimental group, and the other drug used to treat this disease
was given to the control group. The reaction times in both groups to the drug were measured.
Accordingly, test whether there is a difference between the average reaction times for the
experimental and control groups.
a. Express the hypothesis.
b. Which test should be used to test this hypothesis? Why? Show whether the assumptions are met.
c. Test the hypothesis and interpret the results.
Note: You can see this data in the R screenshot on slide 14 under the experimental and control group
variables
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a. 𝐻! : There is no difference between the mean reaction time of the experimental
group and the mean reaction time of the control group (𝐻! : 𝜇" = 𝜇# )
𝐻" : There is difference between the mean reaction time of the experimental group
and the mean reaction time of the control group (𝐻" : 𝜇" ≠ 𝜇# )
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Since 𝑝 = 0.349 > 𝛼 =0.05 H0 can
not rejected. There is no statistically
significant difference between the
mean response times for the new
(experimental group) and the old drug
(control group).
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Independent Sample t-test (Continue)
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Paired t test or two related samples t
test
Paired data occur if the measurements forming the two sets of observations are recorded on the
same individual or if they are related in some other important or obvious way. A classic example of
this is “before” and “after” observations, such as two measurements made on each person before
and after some kind of intervention treatment.
Assumptions:
1. Groups must be dependent.
2. Group elements must be independent of each other.
3. The difference between two measurements of the variable to be tested must be normally
distributed. If the differences do not show a normal distribution, they are tested with
nonparametric alternative tests.
4. Variables must be obtained with interval or ratio scales.
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For example;
■ Whether a new teaching method leads to more effective learning
■ Whether an improvement in a production line increases production speed
■ Whether a treatment applied to a patient has an effect on the patient's recovery
Hypotheses:
𝐻! : There is no significant difference between the two group means (𝐻! : 𝜇" = 𝜇# )
𝐻" :There is a significant difference between the two group means (𝐻" : 𝜇" ≠ 𝜇# )
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Example 3:
■ A new teaching method is applied in the computer programming course. It is claimed that the
new teaching method increases the success of the students. A randomly selected group of
students were given same exam at the beginning and end of this training and the results were
recorded. İD Pre-test Post-test
1 70 72
a) State the statistical hypothesis
2 63 69
b) Test the hypothesis and interpret the results 3 64 75
4 80 82
5 47 55
6 48 50
7 62 55
8 50 58
9 72 80
10 61 58
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. 𝐻! : The new method has no effect on student success (𝐻! : 𝜇" = 𝜇# )
𝐻" : The new method had an impact on student success (𝐻" : 𝜇" < 𝜇# )
Since p=0.0331<𝛼 = 0.05 , 𝐻! is rejected. In the other
words it can be said that the new method increases the
success.
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References
■ Fischetti, T. (2015). Data Analysis with R. Packt Publishing
■ Demir, İ. (Editör), (2017). R ile Uygulamalı İstatistik. Papatya Bilim
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