This document provides an introduction to quantitative research methods. It discusses key concepts like research methodology, variables, hypotheses, experimental design, and statistical analysis. Specifically, it covers: - The difference between research methodology and methods, and examples of methodology scopes. - Key terms like variables, hypotheses, and types of errors in hypothesis testing. - How to plan, conduct, and analyze experiments, including best-guess experiments and one-factor-at-a-time experiments. - Basic statistical concepts like mean, variance, normal distribution, and the t-distribution. - Types of experimental designs like factorial experiments and comparative experiments.

1. An Introduction to Quantitative
Research Methods
Dr Iman Ardekani

2. From Research Methodology to Hypothesis
From Hypothesis to Experiments
Basic Statistical Concepts
Experimental Design and Analysis
Factorial Experiments
Comparative Experiments
Content

3. Part I
From Research Methodology to
Hypothesis

4. From Research Methodology to Hypothesis
Research Methodology
Method 1 Method 2
Research Questions……..…. Research Questions……… Research Questions

5. An example for Research Methodology
Each step may involve several research methods
From Research Methodology to Hypothesis
Step 1: Planning
and defining RQ
Step 2: Literature
Review
Step 3: Survey
Development
Step 5: Data
Analysis
Step 4: Data
Collection
Step 6:
Documentation
Methodology

6. Methodology Scopes (included but not limited to)
1. Descriptive research (aka statistical research): to describes data
and characteristics about the variables of a phenomenon.
2. Correlational research: to explore the statistical relationship
between variables.
3. Experimental research: to explore the causal effective
relationships between the variables in controlled environments.
4. Ex post facto research: to explore the causal effective
relationships between the variables when environment is not
under control.
5. Survey research: to assess thoughts and opinions.
From Research Methodology to Hypothesis

7. What is a variable?
Something that changes, takes different values, and that we
can alter or measure. It has two types:
1. Independent Variables (e.g. the aspect of environment)
2. Dependent Variables (e.g. behaviours of systems)
Example: when studying the effect of distance on the
transmission delay in radio telecommunication, the distance is
an independent variable and the delay is a dependent variable.
From Research Methodology to Hypothesis

8. From Research Methodology to Hypothesis
Difference Between Research Methods and Research Methodology
Research Methodology Research Methods
explains the methods by which you
may proceed with your research.
the methods by which you conduct
research into a subject or a topic.
involves the learning of the various
techniques that can be used in
conducting research, tests,
experiments, surveys and etc.
involve conduct of experiments,
tests, surveys and etc.
aims at the employment of the
correct procedures to find out
solutions.
aim at finding solutions to research
problems.
paves the way for research methods
to be conducted properly.

9. Classifications of Research Methods
1. Qualitative Research Methods
2. Quantitative Research Methods
From Research Methodology to Hypothesis

10. Quantitative Research Methods
 Examples are survey methods, laboratory
experiments, formal methods (e.g. econometrics),
numerical methods and mathematical modeling.
 Qualitative methods produce information only on the
particular cases studied, and any more general
conclusions are only hypotheses. Quantitative
methods can be used to verify, which of such
hypotheses are true.
From Research Methodology to Hypothesis

11. A number of descriptive/relational studies show that
people have difficulty navigating websites when the
navigational bars are inconsistent in their locations
through a Website.
 Inductive Reasoning?
 Deductive Reasoning?
 Variables?
 Hypothesis?
From Research Methodology to Hypothesis

12.  Inductive Reasoning?
People need consistency in navigational mechanisms.
 Deductive Reasoning?
People will have more difficulties with websites if the navigation is
inconsistent.
 Independent variables?
Navigational Consistency: defined as characteristics of navigational bars
and their elements such as location, font, colour, etc.
 Dependent variables?
Difficulty: defined as the efficiency of navigation by user. For example, time
taken to complete tasks, errors made, usage ratings.
From Research Methodology to Hypothesis

13.  Hypothesis?
People will take longer to complete tasks, make more
errors, and give lower ratings of acceptability on a website
with a navigation bar that varies in its location from screen
to screen in comparison to one in which the navigation
bar appears in a consistent position on all screens.
 How to test this hypothesis?
By using experiments and based on hypothesis testing
approaches!
From Research Methodology to Hypothesis

14. Part II
From Hypothesis to Experiments

15. What is a Hypothesis:
 A statement that specifically explain the
relationship between the variables of a system or
process.
 It is a proposed explanation.
 It should be tested. How?
From Hypothesis to Experiment

16. Statistical Hypotheses – Definition
A statement either about the parameters of a probability
distribution or the variables of a system.
This may be stated formally as
H0: A = B
H1: A ≠ B
Where A and B are statistics of two experiments.
From Hypothesis to Experiment
Null Hypothesis
Alternative
Hypothesis

17. Statistical Hypotheses – Notes
Note 1: The alternative hypothesis specified here is called
a two-sided alternative hypothesis because it would be
true if A>B or if A<B.
***
Note 2: A and B are two statistics (random variable) so for
examining A = B or A ≠ B, statistical distribution of them
should be considered.
***
From Hypothesis to Experiment

18. Statistical Hypotheses Testing
Testing a hypothesis involves in
1. taking a random sample
2. computing an appropriate test statistic, and then
3. rejecting or failing to reject the null hypothesis H0.
Part of this procedure is specifying the set of values for
the test statistic that leads to rejection of H0. This set of
values is called the critical region or rejection region for
the test.
From Hypothesis to Experiment

19. Errors in Hypothesis Testing
Two kinds of errors may be committed when testing
hypotheses:
Type 1: the null hypothesis is rejected but it is true.
 = P(type 1 error) = P(reject H0 | H0 is true)
Type 2: the null hypothesis is not rejected but it is false.
 = P(type 2 error) = P(fail to reject H0 | H0 is false)
Power of the test is defined as
Power = 1 -  =P(reject H0 | H0 is false)
From Hypothesis to Experiment

20. Significance Level
 is called the significance level.
The objective of a statistical test is to achieve low
significance level while still maintaining high test
power.
From Hypothesis to Experiment

21. Statistically Significant Hypotheses
The hypothesis verified using the statistical hypothesis testing
method is called statistically significant since it is unlikely to be
wrong in a probability sense.
From Hypothesis to Experiment

22. Experiment – Definition
 An experiment is a test or a series of tests.
 The hypothesis can describe the relationship between x, z and y
variables and an experiment can verify this hypothesis.
From Hypothesis to Experiment

23. How to plan, conduct and analyze an experiment?
Step 1 - Recognition of and statement of the problem
Step 2 - Selection of the response variable
Step 3 - Choice of factors, levels, and range
Step 4 - Choice of experimental design
Step 5 - Performing the experiment
Step 6 - Statistical analysis of the data
Step 7 - Conclusions and recommendations:
From Hypothesis to Experiment

24. Lets continue with the following example:
I really like to play golf. Unfortunately, I do not enjoy practicing, so I am
always looking for a simpler solution to lowering my score. Some of the
factors that I think may be important, or that may influence my golf score,
are as follows:
1. The type of driver used (oversized or regular sized)
2. The type of ball used (balata or three piece)
3. Walking and carrying the golf clubs or riding in a golf cart
4. Drinking water or drinking beer while playing
From Hypothesis to Experiment

25. Best-guess Experiments
Change one or several factors for the next round, based on the
outcome of the current test, in order to improve the output.
Example:
 Round 1: oversized driver, balata ball, walk, and water:
Score 87: Noticed several wayward shots with the big driver
 Round 2: regular-sized driver, balata ball, walk, and water:
Score 80: Notice that people will easily get tired by walking
 Round 3: regular-sized driver, balata ball, golf cart and water
Score 78: Notice that …
From Hypothesis to Experiment

26. One-factor-at-a-time Experiments
Select a starting point (a default setting for each factor)
Example:
Starting point: oversized driver, balata ball, walking, and
drinking water and successively varying each factor over its
range with other factors held constant at the baseline level.
From Hypothesis to Experiment

27. Example for one-factor-at-a-time approach:
Conclusion:
regular-sized driver, balata ball, riding, and drinking water
is the optimal combination.
From Hypothesis to Experiment

28. Problem with one-factor-at-a-time approach
 Interactions between factors are very common. If they occur, the one-
factor-at-a-time approach will usually produce poor results.
 For solving this problem, factorial experiment design can be used.
From Hypothesis to Experiment

29. Part III
Basic Statistical Concepts

30.  Mean (μ): a measure of central tendency.
μ = E{y}
 Variance (σ2): a measure of how far a set of
numbers is spread out.
σ2 = V(y) = E{(y-μ)2}
Basic Statistical Concepts

31.  If c is a constant and y is a random variable with the
mean of μ and variance of σ2, then
1. E(c) = c
2. E(y) = μ
3. E(cy) = c E(y) = cμ
4. V(c) = 0
5. V(y) = σ2
6. V(cy) = c2 V(y) = c2σ2
Basic Statistical Concepts

32.  If y1 is a random variable with the mean of μ1 and
variance of σ1
2, and y2 is another random variable with
the mean of μ2 and variance of σ2
2, then
1. E(y1+y2) = E(y1) + E(y2) = μ1 + μ2
2. E(y1-y2) = E(y1) - E(y2) = μ1 - μ2
3. V(y1+y2) = V(y1) + V(y2) = σ1
2 + σ2
2 (for independent and 0 mean y1 and y2)
4. V(y1-y2) = V(y1) + V(y2) = σ1
2 + σ2
2 (for independent and 0 mean y1 and y2)
5. E(y1y2) = E(y1) E(y2) = μ1 μ2 (for independent y1 and y2)
Basic Statistical Concepts

33.  Statistic: Statistical inference makes considerable
use of quantities computed from the observations
in the sample. We define a statistic as any function
of the observations in a sample that does not
contain unknown parameters:
1. Sample mean
2. Sample Variance
3. and even the random variable (quantity) itself!
Basic Statistical Concepts

34.  Sample Mean (shown by y)
 Sample Variance (shown by S2)
Basic Statistical Concepts

35. Find sample mean and sample
variance for each data set.
y1 = ?
y2 = ?
S1
2 = ?
S2
2 = ?
Basic Statistical Concepts

36. Sampling Distribution
The probability distribution of a statistic is called a
sampling distribution. Important examples are:
1. Normal distribution
2. Chi Square Distribution (Χ2 Distribution)
3. t Distribution
Basic Statistical Concepts

37. Normal Distribution
 y ~ N (μ,σ2)
 In general case, μ is the mean of the
distribution and σ is the standard
deviation.
 An important special case is the
standard normal distribution, where
μ=0 and σ=1.
 z = (y-μ)/σ has always an standard
normal distribution.
Basic Statistical Concepts

38. The Central Limit Theorem
If y1, y2, … yn is a sequence of n independent and
identically distributed random variables with E(yi)=
and V(yi)=2 and x= y1+ y2+ …+ yn then the following
random variable has standard normal distribution
zn=
Basic Statistical Concepts
n 2
x-n

39. Chi-Square Distribution
 x ~ Xk
2
 If x can be obtained as the sum
of the squares of k independent
normally distributed random
variables, then x follows the chi-
square distribution with k
degrees of freedom.
Basic Statistical Concepts

40. As an example of a random variable that follows the chi-
square distribution, suppose that y1, y2, …, yn is a random
sample from an N(μ,σ2) distribution. Then (SS=Sum of
Squares)
That is SS/σ2 is distributed as chi-square with n-1 degrees of
freedom.
Basic Statistical Concepts

41. Since S2 = SS/(n-1), then the distribution of S2 is
σ2 Xn-1
2
Thus, the sampling distribution of the sample
variance is a constant times the chi-square
distribution if the population is normally distributed.
Basic Statistical Concepts
n-1
S2 ~

42. t Distribution
 If z~N(0,1) and Xk
2 is a ch-square
variable, then the random
variable tk
follows t distribution with k
degrees of freedom.
Basic Statistical Concepts

43. If y1,y2, …, yn is a random sample from the N(μ,σ2)
distribution, then the quantity
is distributed as t with n-1 degrees of freedom.
Basic Statistical Concepts

44. Part IV
Experimental Design

45. Factorial Experiments
 Factors are varied together, instead of one at a time.
 An special kind of statsitical experiment design.
 22 Factorial Design (2 factors, each at 2 levels). For example:
Factorial Experiments

46. Example for 22 factorial design
- 8 sets
- replicated twice for each driver-ball combination
- Driver Effect?
Driver Effect = - = 3.25
That is, on average, switching from the oversized driver to
the regular sized deriver increases the score by 3.25 strokes
per round.
Factorial Experiments
92+94+93+91
4
88+91+88+90
4

47. - Ball Effect?
Ball Effect = -
= 0.75
That is, on average, switching from the balata ball to the three piece ball
increases the score by 0.75 strokes per round.
Factorial Experiments
88+91+92+94
4
88+90+93+91
4

48. - Driver-Ball Interaction Effect?
Driver-Ball Interaction Effect =
- = 0.25
That is, on average, switching of both ball and driver increases the score by
0.25 strokes per round.
Finally, one can concludes that
Driver effect > Ball effect > Intercation
Factorial Experiments
92+94+88+90
4
88+91+93+91
4

49. 23 Factorial Design (3 factors, each at 2 levels):
How to calculate ball-effect, driver effect, beverage
effect and interaction effects?
Factorial Experiments

50. Comparative experiments compare two experimental conditions. For
example, comparative experiments can be used to determine whether two
different formulations of a product give equivalent results.
Comparative Experiments
Apple three 1 (AKL) Apple three 2 (ChCH)
Apples weights:
0.101 kg
0.111
0.103
0.102
0.121
0.102
0.101
same cultivation conditions
Apples weights:
0.102 kg
0.101
0.105
0.106
0.111
0.98
0.110
Hypothesis: same apples weights?

51. Data Model for Comparative Experiments
The following model for each data set is considered:
yi =  + i
 is the mean of data set
i is assumed to be distributed by NID(0,2)
Comparative Experiments
Noise

52. Comparative Experiments Formulation:
In general case, we have two data sets:
y11, y12, …, y1
and n1
y21, y22, …, y2
The statistical hypothesis is formulated by
H0: 1 = 2
H1: 1 ≠ 2
Comparative Experiments
Null Hypothesis
Alternative
Hypothesis
n1
n2

53. Two Sample t-test
1. Assume that the variance of the two data sets are
equal: 1
2 = 2
2
2. Form the following statistic
Comparative Experiments
data set 1 sample mean
data set 2 sample mean
estimate of the common variance

54. Two Sample t-test
3. Assume To determine whether to reject H0 we would compare t0 to the t
distribution with n1+n2-2 degrees of freedom.
4. We would reject H0 if
Comparative Experiments
-

55. page(1/3)
Are the bond strength of the two cement
mortars similar at the significance level
of  = 0.05?
Comparative Experiments

56. page(1/3)
Using provided table:
Comparative Experiments

57. page(1/3)
Thus we would reject H0 and we can conclude
that the strengths of the two mortar are different.
Comparative Experiments
t0=-2.2

58. Repeat the previous problem with  = 0.01?
Comparative Experiments

59. Exercise
Two machines are used for filling plastic bottles with a net volume of 16.0 ounces.
The filling process can be assumed to be normal. The quality engineering
department suspects that both machines fill to the same net volume. An experiment
is performed by taking a random sample from the output of each machine. Would
you reject or accept the quality engineering department hypothesis?
Comparative Experiments

60. P Value
The smallest level of significance that would lead to
the rejection.
Comparative Experiments
t0
v
Min 

61. P Value Calculation for Previous Example
Comparative Experiments
t0=--2.2
V=18
Min  = 0.0411

62. Confidence Interval
It is often preferable to provide an interval within
which the statistics in question would be expected to
lie. These interval statements are called confidence
intervals.
This interval estimates the difference between the
statistics and the accuracy of this estimate.
Comparative Experiments

63. Confidence Interval Calculation
The 100(1-) percentage confidence interval is
L  1-2  U
L =
U =
Comparative Experiments

64. Confidence Interval
L  1-2  U 
1-2 = 0.5(L+U)  0.5(U-L)
It means the mean difference is 0.5(L+U) and the
accuracy of this estimate is  0.5(U-L).
If 0 is not in the interval H0 would be rejected.
Comparative Experiments

65. For the previous example:
The 95% confidence interval is
L = -0.55
U = -0.01
-0.55  1-2  -0.01 (1-2 =0 is not in the interval)
1-2= -0.28  0.27
It means the difference between the two mortars strengths is -0.28
with the accuracy of  0.27.
Comparative Experiments

66. For the previous example estimates the difference between the
tow mortars strengths and the accuracy of your estimation by
calculating the confidence interval of t-test.
The difference between the two mortars strengths is -0.28 with
the accuracy of  0.27.
Comparative Experiments

67. Some experiments involve comparing only one
population mean to a specified value, say,
H0: 1 = 0
H1: 1 ≠ 0
This problem is a simplified version of the two-sample t-test
problem, called one-sample Z test.
Comparative Experiments

68. 0ne-Sample Z-test
1. Assume that the variance of the sets is 2
2. Form the following statistic
3. If H0 is true, then the distribution of Z0 is N(0, 1).
Therefore, we would reject H0 if |Z0|>Z0.5
4. Z0.5 should be obtained from a table.
Comparative Experiments

69. Z0.5
=0.05
1-0.5 =0.975
Z=1.96
Comparative Experiments

70. In the population, the average IQ is 100 with a
standard deviation of 15. A team of scientists wants to
test a new medication to see if it has either a positive
or negative effect on intelligence, or no effect at all. A
sample of 30 participants who have taken the
medication has a mean of 140. Did the medication
affect intelligence, using alpha = 0.05?
Comparative Experiments

71. Comparative Experiments
1.96
If Z is less than -1.96, or greater than 1.96,
reject the null hypothesis.
-1.96
Result: Reject the null hypothesis.
Conclusion: Medication significantly
affected intelligence, z = 14.60, p < 0.05.

72. The confidence interval of one-sample z test is
Comparative Experiments

73. Find the confidence interval for the previous example.
L = 140 - 1.96 x 15 / √30 = 134.64
U= 140 + 1.96 x 15 / √30 = 145.36
140 ± 5.36
Comparative Experiments

74. Violation of Assumptions in t-test
Two main assumptions are:
1. Normal distribution: In practice, the assumption of
normal distribution can be violated to some extent
without affecting the effectiveness of t-test.
2. Equal variance: If this assumption is violated,
other test techniques should be used.
Comparative Experiments