1. in Quantitative Research II E.Shehniyilagh Ph.D
The Role of Statistics
in Quantitative Research II E.Shehniyilagh Ph.D

2. t-Static
t-test is used to test hypothesis about an unknown population mean, µ, when the value of σ or σ² is unknown.

3. Degrees of Freedom df=n-1

4. Assumption of the t-test (Parametric Tests)
1.The Values in the sample must consist of independent observations
2. The population sample must be normal
3. use a large sample n ≥ 30

5. Inferential Statistics
t-Statistics:
There are different types of t- Statistic
1. Single (one) Sample t-statistic (test)
2. Two independent sample t-test, Matched-Subject Experiment, or Between Subject Design
3.Repeated Measure Experiment, or Related/Paired Sample t-test

6. Hypothesis Testing Step 3: Computations/ Calculations or Collect Data and Compute Sample Statistics FYI Z Score for Research
ZZZ 

7. Hypothesis Testing Step 3: Computations/ Calculations or Collect Data and Compute Sample Statistics

8. Calculations for t-test
M-μ
t= s
Sm Sm= or
Sm= M-μ √n t
M=t.Sm+μ
μ=M- Sm.t

9. FYI Variability SS, Standard Deviations and Variances
X σ² = ss/N Pop
σ = √ss/N 2
s = √ss/df
s² = ss/n-1 or ss/df Sample
SS=Σx²-(Σx)²/N
SS=Σ( x-μ)²
Sum of Squared Deviation from Mean

10. d=Effect Size Use S instead of σ for t-test

11. Cohn’s d=Effect Size Use S instead of σ for t-test
d = (M - µ)/s
s= (M - µ)/d
M= (d . s) + µ
µ= (M – d) s

12. Percentage of Variance Accounted for by the Treatment (similar to Cohen’s d) Also known as ω² Omega Squared

13. percentage of Variance accounted for by the Treatment
Percentage of Variance Explained
r²=  Small Effect
r²=  Medium Effect
r²=  Large Effect

14. Problems
Infants, even newborns prefer to look at attractive faces (Slater, et al., 1998). In the study, infants from 1 to 6 days old were shown two photographs of women’s face. Previously, a group of adults had rated one of the faces as significantly more attractive than the other. The babies were positioned in front of a screen on which the photographs were presented. The pair of faces remained on the screen until the baby accumulated a total of 20 seconds of looking at one or the other. The number of seconds looking at the attractive face was recorded for each infant.

15. Problems
Suppose that the study used a sample of n=9 infants and the data produced an average of M=13 seconds for attractive face with SS=72.
Set the level of significance at α=.05
for two tails
Note that all the available information comes from the sample. Specifically, we do not know the population mean μ or the population standard deviation σ.
On the basis of this sample, can we conclude that infants prefer to look at attractive faces?

16. t-Statistic: Null Hypothesis
If the Population mean or µ and the sigma are unknown the statistic of choice will be t-Static
1. Single (one) Sample t-statistic (test)
Step 1
H0 : µ attractive = 10 seconds
H1 : µ attractive ≠ 10 seconds

17. Problems
A psychologist has prepared an “Optimism Test” that is administered yearly to graduating college seniors. The test measures how each graduating class feels about its future. The higher the score , the more optimistic the class. Last year’s class had a mean score of μ=15. A sample of n=9 seniors from this year’s class was selected and tested..

18. Problems
The scores for these seniors are 7, 12, 11, 15, 7, 8, 15, 9, and 6, which produced a sample mean of M=10 with SS=94.
On the basis of this sample, can the psychologist conclude that this year’s class has a different level of optimism?
Note that this hypothesis test will use a t-statistic because the population variance σ² is not known. USE SPSS
Set the level of significance at α=.01 for two tails

19. t-Statistic: Null Hypothesis
If the Population mean or µ and the sigma are unknown the statistic of choice will be t-Static
1. Single (one) Sample t-statistic (test)
Step 1
H0 : µ optimism = 15
H1 : µ optimism ≠ 15

20. Chapter 10 Two Independent Sample t-test Matched-Subject Experiment, or Between Subject Design
An independent-measures study uses a separate sample to represent each of the populations or treatment conditions being compared.

21. Two Independent Sample t-test
Null Hypothesis:
If the Population mean or µ is unknown the statistic of choice will be t-Static
Two independent sample t-test, Matched-Subject Experiment, or Between Subject Design
H0 : µ1 -µ2 = 0
H1 : µ1 -µ2 ≠ 0

22. Problems Research results suggest a relationship
Between the TV viewing habits of 5-year-old children and their future performance in high school. For example, Anderson, Huston, Wright & Collins (1998) report that high school students who regularly watched Sesame Street as children had better grades in high school than their peers who did not watch Sesame Street.

23. Problems
The researcher intends to examine this phenomenon using a sample of 20 high school students. She first surveys the students’ s parents to obtain information on the family’s TV viewing habits during the time that the students were 5 years old. Based on the survey results, the researcher selects a sample of n=10

24. Problems
students with a history of watching “Sesame Street“ and a sample of n=10 students who did not watch the program. The average high school grade is recorded for each student and the data are as follows: Set the level of significance at α=.05
for two tails

25. Average High School Grade
Problems
Average High School Grade
Watched Sesame St (1). Did not Watch Sesame St.(2)
n1= n2=10
M1= M2= 85
SS1= SS2=160

26. Two Independent Sample t-test
Null Hypothesis:
Two independent sample t-test, Matched-Subject Experiment, or Between Subject Design- non-directional or two-tailed test
Step 1.
H0 : µ1 -µ2 = 0
H1 : µ1 -µ2 ≠ 0

27. Two Independent Sample t-test
Null Hypothesis:
Two independent sample t-test, Matched-Subject Experiment, or Between Subject Design - directional or one-tailed test
Step 1.
H0 : µ Sesame St . ≤ µ No Sesame St.
H1 : µ Sesame St. > µ No Sesame St.

28. Measuring d=Effect Size for the independent measures

29. Percentage of Variance Accounted for by the Treatment (similar to Cohen’s d) Also known as ω² Omega Squared and Coefficient of Determination

30. We use the Point-Biserial Correlation when one of our variable is dichotomous, in this case (1) watched Sesame St. (2) and didn’t watch Sesame St.

31. Problems
In recent years, psychologists have demonstrated repeatedly that using mental images can greatly improve memory. Here we present a hypothetical experiment designed to examine this phenomenon.
The psychologist first prepares a list of 40 pairs of nouns (for example, dog/bicycle, grass/door, lamp/piano). Next, two groups of participants are obtained (two separate samples). Participants in one group are given the list for 5 minutes and instructed to memorize the 40 noun pairs.

32. Problems
Participants in another group receive the same list of words, but in addition to the regular instruction, they are told to form a mental image for each pair of nouns (imagine a dog riding a bicycle, for example). Later each group is given a memory test in which they are given the first word from each pair and asked to recall the second word. The psychologist records the number of words correctly recalled for each individual. The data from this experiment are as follows: Set the level of significance at α=.01
for two tails

33. Data (Number of words recalled)
Problems
Data (Number of words recalled)
Group 1 (Images) Group 2 (No Images)
n1= n2=10
M1= M2= 18
SS1= SS2=160

34. 3.Repeated Measure Experiment, or Related/Paired Sample t-test Within Subject Experiment Design
A single sample of individuals is measured more than once on the same dependent variable. The same subjects are used in all of the treatment conditions.

35. t-Statistics: Null Hypothesis
If the Population mean or µ is unknown the statistic of choice will be t-Statistic
3.Repeated Measure Experiment, or Related/Paired Sample t-test
For Non-directional or two tailed test
Step. 1
H0 : µD = 0
H1 : µD ≠ 0

36. t-Statistics: Null Hypothesis
If the Population mean or µ is unknown the statistic of choice will be t-Statistic
3.Repeated Measure Experiment, or Related/Paired Sample t-test
For Directional or one tailed tests
Step. 1
H0 : µD ≤ 0
H1 : µD > 0

37. Problems
Research indicates that the color red increases men’s attraction to women (Elliot & Niesta, 2008). In the original study, men were shown women’s photographs presented on either white or red background. Photographs presented on red were rated significantly more attractive than the same photographs mounted on whit.

38. Problems
In a similar study, a researcher prepares a set of 30 women’s photographs, with 15 mounted on a white background and 15 mounted on red. One picture is identified as the test photograph, and appears twice in the set, once on white and once on red.

39. Problems
Each male participant looks through the entire set of photographs and rated the attractiveness of each woman on a 12-point scale. The data in the next slide summarizes the responses for a sample of n=9 men.
Set the level of significance at α=.01
for two tails
Do the data indicate that the color red increases men’s attraction to women ??

40. Problems
Participants White background X1 Red Background X2 D=X2-X D² A B C D E F G H I
ΣD =27 ΣD²=99
MD =

41. Null Hypothesis Step. 1 H0 : µD = 0 H1 : µD ≠ 0
For Non-Directional or two tailed tests
Step. 1
H0 : µD = 0
H1 : µD ≠ 0

42. Problems
One technique to help people deal with phobia is to have them counteract the feared objects by using imagination to move themselves to a place of safety. In an experiment test of this technique, patients sit in front of a screen and are instructed to relax. Then they are shown a slide of the feared object for example, a picture of a spider, (arachnophobia). The patient signals the researcher as soon as feelings of anxiety begin to arise, and the researcher records the amount of time that the patient was able to endure looking at the slide.

43. Problems
The patient then spends two minutes imagining a “safe scene” such as a tropical beach (next slide) before the slide is presented again. If patients can tolerate the feared object longer after the imagination exercise, it is viewed as a reduction in the phobia. The data in next slide summarize the items recorded from a sample of n=7 patients. Do the data indicate that the imagination technique effectively alters phobia? . Set the level of significance at α=.05 for one tailed test.

45. Problems
Participant Before imagination X1 After Imagination X2 D=X2-X D² A B C D E F G
ΣD =56 ΣD²=502
MD =

46. Null Hypothesis For Directional or one tailed tests Step. 1
H0 : µD ≤ 0 (The amount of time is not increased.)
H1 : µD > 0 (The amount of time is increased.)

47. Relationship of Statistical Tests
Does this Diagram Make Sense to You?

48. Estimation
The inferential process of using sample statistics to estimate population parameter is called estimation.
We use estimation 1. After hypotheses test when H0 is rejected. 2. when we know there is an effect and simply want to find out how much. 3. When we want some basic information about an unknown population.
See the logic behind hypothesis tests ans estimation.

49. Estimation Point Estimate: Interval Estimate: µ= M ± tsM
µ1- µ 2 = M1-M2 ± ts(M1-M2)
µD = MD ± tsMD

51. ANALYSIS OF VARIANCE ANOVA
TESTS FOR DIFFERENCES AMONG TWO OR MORE POPULATION MEANS
σ²=S²=MS
MS=Mean Squared Deviation
Ex of ANOVA Research: The effect of temperature on recall.

52. Statistics Standard Deviations and Variances
X σ² = ss/N Pop
σ = √ss/N 2
s = √ss/df Sample
s² = ss/n-1 or ss/df
MS = SS/df

53. Effects of Temperature (IV) on Recall (DV)

56. MS bet = SS bet / df bet
MS with = SS with /df with

58. ANOVA SS bet =Σ(T²/n-G²/n) SS with =Σss SS total =SS bet + SS with
df bet = K-1
df with =N-K
df total= df bet + df with

59. Post Hoc Tests (Post Tests)
Post Hoc Tests are additional hypothesis tests that are done after an ANOVA to determine exactly which mean difference are significant and which are not.

60. Post Hoc Tests (Post Tests)
Tukey’s Honestly Significant Difference(HSD) Test
HSD= q

61. Problems
The data in next slide were obtained from an independent-measures experiment designed to examine people’s performances for viewing distance of a 60-inch high definition television. Four viewing distances were evaluated, 9 feet, 12 feet, 15 feet, and 18 feet, with a separate group of participants tested at each distance. Each individual watched a 30-minute television program from a specific distance and then completed a brief questionnaire measuring their satisfaction with the experience.

62. Problems
One question asked them to rate the viewing distance on a scale from 1 (Very Bad definitely need to move closer or farther away) to 7 (Excellent-perfect viewing distance). The purpose of the ANOVA is to determine whether there are any significant differences among the four viewing distances that were tested. Before we begin the hypothesis test, note that we have already computed several summary statistics for the data in next slide. Specifically, the tretment totals (T) and SS values are shown for the entire set of data.

63. Problems
9 feet 12 feet 15 fet 18 feet N= G= ΣX² = K= T1=5 T2=10 T3=25 T4=20 SS1=8 SS2= 8 SS3=10 SS4=6 M1=1 M2=2 M3=5 M4=4 n1=5 n2=5 n3=5 n4=5

64. H0 : µ1= µ2=µ3=µ4 (There is no treatment effect.)
Problems
Having these summary values simplifies the computations in the hypothesis test, and we suggest that you always compute these summary statistics before you begin an ANOVA.
Step 1)
H0 : µ1= µ2=µ3=µ4 (There is no treatment effect.)
H1 : (At least one of the treatment means is different.)
We will set alpha at α =.05

65. Step 2

66. Problems
A human factor psychologist studied three computer keyboard designs. Three samples of individuals were given material to type on a particular keyboard, and the number of errors committed by each participant was recorded. The data are on next slide.

67. Problems
Keyboard A Keyboard B Keyboard C N= G= ΣX² = T1=5 T2=25 T3=30 SS1=12 SS2=20 SS3=14 M1=1 M2=5 M3=6 Is there a significant differences among the three computer keyboard designs ?

68. H0 : µ1= µ2=µ3 (No differences between the computer keyboard designs )
Problems
Step 1)
H0 : µ1= µ2=µ3 (No differences between the computer keyboard designs )
H1 : (At least one of the computer keyboard designs is different.)
We will set alpha at α =.01

69. Step 2

70. 2-WAY ANOVA

71. Correlation & Regression

72. Correlation
Correlation measures the strength and the direction of the relationship between two or more variables.
A correlation has three components:
The strength of the coefficient
The direction of the relationship
The form of the relationship
The strength of the coefficient is indicated by the absolute value of the coefficient.
The closer the value is to 1.0, either positive or negative, the stronger or more linear the relationship.
The closer the value is to 0, the weaker or nonlinear the relationship.

73. Correlation
The direction of coefficient is indicated by the sign of the correlation coefficient.
A positive coefficient indicates that as one variable (X) increases, so does the other (Y).
A negative coefficient indicates that as one variable (X) increases, the other variable (Y) decreases.
The form of the relationship
The form of the relationship is linear.
In correlation variables are not identified as independent or dependent because the researcher is measuring the one relationship that is mutually shared between the two variables
As a result, causality should not be implied with correlation.

74. Correlation
Remember, the correlation coefficient can only measure a linear relationship.
A zero correlation indicates no linear relationship. However, does not indicate no relationship.
a coefficient of zero rules out linear relationship,
but a curvilinear could still exist.
The scatterplots below illustrate this point:

75. The Correlation is based on a Statistic Called Covariance
Variance and Covariance are used to measure the quality of an item in a test.
Reliability and validity measure the quality of the entire test.
σ²=SS/N  used for one set of data
Variance is the degree of variability
of scores from mean.

76. The Correlational Method SS, Standard Deviations and Variances
X σ² = ss/N Pop
σ = √ss/N 2
s = √ss/df
s² = ss/n-1 or ss/df Sample
SS=Σx²-(Σx)²/N
SS=Σ( x-μ)²
Sum of Squared Deviation from Mean

77. Variance X σ² = ss/N Pop 1 s² = ss/n-1 or ss/df Sample 2 4 5
SS=Σx²-(Σx)²/N
SS=Σ( x-μ)²
Sum of Squared Deviation from Mean

78. Covariance
Correlation is based on a statistic called Covariance (Cov xy or S xy) ….. r=sp/√ssx.ssy
Covariance is a number that reflects the degree to which 2 variables vary together.
Original Data
X Y

79. 2 ways to calculate the SP SP= Σxy-(Σx.Σy)/N Computation
Covariance
COVxy=SP/N-1
2 ways to calculate the SP
SP= Σxy-(Σx.Σy)/N Computation
SP= (x-μx)(y-μy) Definition
SP requires 2 sets of data
SS requires only one set of data

80. The Correlational Method
Correlational data can be graphed and a “line of best fit” can be drawn
1- Pearson
Correlations
2-Spearman
3-Point-Biserial Correlation
4- Partial Correlation

81. Types of Correlation
In correlational research we use continues variables (interval or ratio scale) for Pearson Correlation (for linear relationship).
If it is difficult to measure a variable on an interval or ratio scale then we use Spearman Correlation
Spearman Correlation uses ordinal or rank ordered data
Spearman Correlation measures the consistency of a relationship (Monotonic Relationship). Ex. next

82. Monotonic Transformation
They are rank ordered numbers (DATA), and use ordinal scale(data) examples; 1, 2, 3, 4, or 2, 4, 6, 8, 10
Spearman Correlation can be used to measure the degree of Monotonic relationship between two variables.

83. Ex. of Monotonic data
X Y

84. Types of Correlation
Ex. A teacher may feel confident about rank ordering students’ leadership abilities but would find it difficult to measure leadership on some other scale.
The Point-Biserial Correlation.
However, we can use both continues and discrete variables(data) in The Point-Biserial Correlation. (can be a substitute for two independent t-test)
In special situations we can use Partial Correlations.

85. The Point-Biserial Correlation
The point-biserial correlation is used to measure the relationship between two variables in situations in which one variable consist of regular, numerical scores (non-dichotomies), but the second variable has only two values (dichotomies).
We can calculate the correlation from t-test
r² = Coefficient of Determination which measures the effect size=d
r² = t²/t²+df
r = √r²

86. The Correlational Method
Correlation is the degree to which events or characteristics vary from each other
Measures the strength of a relationship
Does not imply cause and effect
The people chosen for a study are its subjects or participants, collectively called a sample
The sample must be representative

87. A Partial Correlation
Measures the relationship between two variables while controlling the influence of a third variable by holding it constant.
Ex. The correlation between churches and crime.

88. The Correlational Method
Correlational data can be graphed and a “line of best fit” can be drawn

89. Positive correlation: variables change in the same direction

90. Positive Correlation

91. Negative correlation: variables change in the opposite direction

92. Negative Correlation

93. Unrelated: no consistent relationship
No Correlation
Unrelated: no consistent relationship

94. No Correlation

95. The Correlational Method
The magnitude (strength) of a correlation is also important
High magnitude = variables which vary closely together; fall close to the line of best fit
Low magnitude = variables which do not vary as closely together; loosely scattered around the line of best fit

96. The Correlational Method
Direction and magnitude of a correlation are often calculated statistically
Called the “correlation coefficient,” symbolized by the letter “r”
Sign (+ or -) indicates direction
Number (from 0.00 to 1.00) indicates magnitude
0.00 = no consistent relationship
+1.00 = perfect positive correlation
-1.00 = perfect negative correlation
Most correlations found in psychological research fall far short of “perfect”

97. The Correlational Method
Correlations can be trusted based on statistical probability
“Statistical significance” means that the finding is unlikely to have occurred by chance
By convention, if there is less than a 5% probability that findings are due to chance (p < 0.05), results are considered “significant” and thought to reflect the larger population
Generally, confidence increases with the size of the sample and the magnitude of the correlation

98. The Correlational Method
Advantages of correlational studies:
Have high external validity
Can generalize findings
Can repeat (replicate) studies on other samples
Difficulties with correlational studies:
Lack internal validity
Results describe but do not explain a relationship

99. External & Internal Validity
External Validity
External validity addresses the ability to generalize your study to other people and other situations.
Internal Validity
Internal validity addresses the "true" causes of the outcomes that you observed in your study. Strong internal validity means that you not only have reliable measures of your independent (predictors) and dependent variables (criterions) BUT a strong justification that causally links your independent variables (IV) to your dependent variables (DV).

100. The Correlational Method Pearson
r=sp/√ssx.ssy
Original Data
X Y
SP requires 2 sets of data
SS requires only one set of data
df=n-2

101. The Correlational Method Spearman
r=sp/√ssx.ssy
Original Data  Ranks
X Y X Y
SP requires 2 sets of data
SS requires only one set of data

102. Percentage of Variance Accounted for by the Treatment (similar to Cohen’s d) is known as ω² Omega Squared also is called Coefficient of Determination

103. Coefficient of Determination
If r = then, r ² = 0.64
This means 64% of the variability in the Y scores can be predicted from the relationship with X.

104. Problems
Test the hypothesis for the following n=4 pairs of scores for a correlation.
r=sp/√ssx.ssy
Original Data
X Y

105. Step 1) Problems H0 : ρ=0 (There is no population correlation.)
H1 : ρ≠0 (There is a real correlation.)
Ρ: probability or chances are…
We will set alpha at α =.01

106. STEP 2

108. Problems
Test the hypothesis for the following set of n=5 pairs of scores for a positive correlation.
Original Data
X Y

109. Problems
Step 1)
H0 : ρ≤0 ((The population correlation is not positive.)
H1 : ρ>0 (The population correlation is positive.)
Ρ: probability or chances are…
We will set alpha at α =.05

110. Bi-Variate Regression Analysis
Bi-variate regression analysis extends correlation and attempts to measure the extent to which a predictor variable (X) can be used to make a prediction about a criterion measure (Y). E
Bi-variate regression uses a linear model to predict the criterion measure.
The formula for the predicted score is:
Y' = a + bX

111. Bivariate Regression
The components of the line of best fit (Y' = a + bX) are:
the Y-intercept (a) Constant
the slope (b)
Variable (X)

112. Bivariate Regression
The Y-intercept is the average value of Y when X is zero.
The Y-intercept is also called constant.
Because, this is the amount of Y that is constant or present when the influence of X is null (0).
The slope is average value of a one unit change in Y for a corresponding one unit change in X.
Thus, the slope represents the direction and intensity of the line.

113. Regression and Prediction
Y=bX+a
Regression Line e

115. Bivariate Regression Line of Best Fit: Y' = 2.635 + .204X
With this equation a predicted score may be made for any value of X within the range of data.
a= and b=.204
Y-intercept
Slope = .204
2.635

116. Multiple Regression Analysis
Multiple regression analysis is an extension of bi-variate regression, in which several predictor variables are used to predict one criterion measure (Y).
In general, this method is considered to be advantageous; since seldom can an outcome measure be accurately explained by one predictor variable. E
Y' = a + b1X1 +b2X2 +b3X3

119. Relationship of Statistical Tests
Does this diagram make sense to you?

120. PARAMETRIC AND NONPARAMETRIC STATISTICAL TESTS
Parametric tests are more accurate and have 3 assumptions:
1. Random selection
2. Independent of observation
3. Sample is taken from a normal population with a normal distribution

121. NONPARAMETRIC STATISTICAL TESTS
CHI SQURE: It is like frequency distribution
Is is used for comparative studies.
Ex. Of the two leading brands of cola, which is preferred by most American?

122. CHI SQURE
= Σ(fo-fe)² /fe
df = C-1
fe = n/c

123. CHI SQURE

124. df=C-1

126. Problems
A psychologist examining art appreciation selected an abstract painting that had no obvious top or bottom. Hangers were placed on the painting so that it could be hung with any one of the four sides at the top. The painting was shown to a sample of n=50 participants, and each was asked to hung the painting in the orientation that looked correct. The following data indicate how many people choose each of the four sides to be placed at the top. fo
Top up (correct) 18
Bottom up 17
Left side up 7
Right side up 8

127. Problems
The question for the hypothesis test is whether there are any preferences among the four possible orientations. Are any of the orientations selected more (or less) often than would be expected simply by chance?
We will set alpha at α =.05

128. Step 1) Problems H0 : fo=fe no preference for any specific orientation
H1 : fo≠fe preference for specific orientation

129. Step df=C-1