CHAPT 7
Hypothesis Testing
Applied to Means
Part A
t-Static
1
t-Static
1. Single Sample or One Sample
t-Test AKA student t-test.
2. Two Independent sample
t-Test, AKA Between Subject Designs or
Matched subjects Experiment.
3. Related Samples t-test or Repeated
Measures Experiment AKA Within
Subject Designs or Paired Sample tTest .
2
Degrees of Freedom
df=n-1
3
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
4
1.The Values in the sample must consist of
independent observations.
5
2. The population sample must be normal.
6
FYI HYPOTHESIS for Z
7
FYI Steps in Hypothesis-Testing
Step 1: State The Hypotheses
H : µ ≤ 100
H : µ > 100
Statistics:
0
average
1
average
 Because the Population mean or µ is
known the statistic of choice is
 z-Score
8
FYI Hypothesis Testing
Step 2: Locate the Critical Region(s)
or Set the Criteria for a Decision
9
FYI Directional Hypothesis Test
10
11
FYI
Hypothesis Testing
Step 3: Computations/ Calculations or Collect
Data and Compute Sample Statistics
Z Score for Research

12
FYI Hypothesis Testing
Step 3: Computations/ Calculations or
Collect Data and Compute Sample
Statistics

13
Calculations for t-test
Step 3: Computations/ Calculations or
Collect Data and Compute Sample
Statistics

2
S /n
14
Hypothesis Testing
Step 4: Make a Decision
15
Inferential Statistics t-Static
Single Sample or One Sample t-Test
t-test is used to test hypothesis about an
unknown population mean (µ) when the
.
value of σ or σ² is unknown (S or S²) Ex. Is
this year class know more about STATS than the
last year? Mean for the last year class µ=80
Mean for this year class M=82
Note: We don’t know what the average/mean STATS
score should be for the population. We only
compare this year scores with the last year.
****Sample data can only be considered as
1
estimates of population values.****
6
17
FYI
Variability
SS, Standard Deviations and Variances
 X
1
2
4
5
σ² = ss/N
σ = √ss/N
s = √ss/df
s² = ss/n-1 or ss/df
Pop
Sample
SS=Σx²- (Σx)²/N
SS=Σ( x-μ)²
Sum of Squared Deviation from Mean
18
FYI d=Effect Size for Z
For t-test Use S instead of σ (next slide)
19
Cohn’s d=Effect Size for Single Sample t
Use S instead of σ for t-test
 d = (M - µ)
S
 S= (M - µ)
d
 M= (d . s) + µ
µ= (M – d) s
20
Percentage of Variance
Accounted for by the
TreaTmenT (similar To Cohen’s
d) Also known as ω² Omega
Squared (power of a test)
2
t
2
r  2
t  df
21
percentage of Variance
accounted for by the Treatment
 Percentage of Variance Explained
 r²=0.01-------- Small Effect
r2
r
2
 r²=0.09-------- Medium Effect
 r²=0.25-------- Large Effect
22
Problem 1
 A supervisor has prepared an
“Optimism Test” that is administered
yearly to factory employees. The test
measures how each employee feels
about its future. The higher the score,
the more optimistic the employee. Last
year’s employees had a mean score of
μ=15. A sample of n=9 employees from
this year was selected and tested..
23
Problem 1
 The scores for these employees 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
supervisor conclude that this year’s
employees has a different level of
optimism?
 Note that this hypothesis test will use a
t-statistic because the population
variance σ² is not known (S²). USE SPSS
24
Null Hypothesis
 t-Statistic:
 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
H : µ
= 15
H : µ
≠ 15
0
1
optimism
optimism
25
Step 2 Locate the Critical regions
 Step 2
26
Calculations for t-test
Step 3: Computations/ Calculations or
Collect Data and Compute Sample
Statistics
M-μ
 t=
s
Sm
Sm=
or
√n
2
S /n
df=n-1
s² = SS/df
Sm= estimated standard error of the mean
27
Step 4 Make a Decision
 Step 2
28
29
30
31
32
Bootstrap
 Bootstrapping is a method for
deriving robust estimates of
standard errors and confidence
intervals for estimates such as
the mean, median, proportion,
odds ratio, correlation coefficient
or regression coefficient.
33
Problem 2
 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.
34
Attractive
10
11
16
18
13
11
17
11
11
M=13
Problem 2
S=3
35
Problem 2
 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 S=3.
 Set the level of significance at α=0.01 and
then 0.05 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?
36
Null Hypothesis
 t-Statistic:
 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
H : µ
= 10 seconds
H : µ
≠ 10 seconds
0
1
attractive
attractive
37
STEP 2None-directional Hypothesis Test
Critical value of t=2.306
38
Calculations for t-test
Step 3: Computations/ Calculations or Collect
Data and Compute Sample Statistics

2
S /n
39
Hypothesis Testing
Step 4: Make a Decision
40
41
42
43
44
t-Static
2. Two Independent
sample t-Test, AKA
Between Subject
Designs or Matched
subjects Experiment
45
Two Independent
Sample
t-test
46
t-test
ANOVA
47
48
Two Independent Sample t-test
 An independent measures
study uses a separate group
of participants (samples) to
represent each of the
populations or treatment
conditions being compared.
49
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 Step 1
 H : µ -µ = 0
 H : µ -µ ≠ 0
0
1
1
1
2
2
50
None-directional
Hypothesis Test
 Step 2
51
STEP 3

52
Estimated Standard Error
S(M -M )
1
2
 The estimated standard error measures
how much difference is expected, on
average, between a sample mean
difference and the population mean
difference. In a hypothesis test, µ1 - µ2
is set to zero and the standard error
measures how much difference is
expected between the two sample
means.
53
Estimated Standard Error
S
(M1-M2)=
54
(1) Pooled Variance
s² P
55
(2) Pooled Variance
s² P
56
Step 4
57
Measuring d=Effect Size for the
independent measures

d
M1 M 2
2
S p
58
Estimated d
59
Estimated d
60
61
Percentage of Variance
Accounted for by the
TreaTmenT (similar To Cohen’s
d) Also known as ω² Omega
Squared
2
t
2
r  2
t  df
62
Problem 1
 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.
63
Problem 1
 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 n1=10
64
Problem 1
 students with a history of watching
“Sesame Street“ and a sample of
n2=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 and
Use non-directional or two-tailed
test
65
Problem 1
Average High School Grade
Watched Sesame St (1). Did not Watch Sesame St.(2)
86
87
91
97
98
99
97
94
89
92
n1=10
M1=93
SS1=200
90
89
82
83
85
79
83
86
81
92
n2=10
M2= 85
SS2=160
66
Two Independent Sample t-test
 Null Hypothesis:
 Two Independent Sample t-test,
Matched-Subject Experiment, or
Between Subject Design
 Step 1.
 H : µ -µ = 0
 H : µ -µ ≠ 0
0
1
1
1
2
2
67
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.
 H : µ Sesame St . ≤ µ No Sesame St.
0
H :
1
µ
Sesame
St.
>µ
No Sesame St.
68
Step 2
69
STEP 3

70
Estimated Standard Error
S
(M1-M2)=
71
(1) Pooled Variance
s² P
72
Estimated Standard Error
S
(M1-M2)=
73
Step 4
74
75
76
77
78
Measuring d=Effect Size for the
independent measures

d
M1 M 2
2
S p
79
FYI We use the Point-Biserial
Correlation (r) when one of our
variable is dichotomous, in this case
(1) waTChed sesame sT. and (2) didn’T
watch Sesame St.
2
t
2
r  2
t  df
80
Problem 2
 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
81
memorize the 40 noun pairs.
Problem 2
 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 tailed test.
82
Problem 2
Data (Number of words recalled)
Group 1 (Images)
Group 2 (No Images)
19
20
24
30
31
32
30
27
22
25
n1=10
M1=26
SS1=200
23
22
15
16
18
12
16
19
14
25
n2=10
M2= 18
SS2=160
83
Two Independent Sample t-test
 Null Hypothesis:
Step 1
H :
H :
0
1
µ -µ = 0
µ -µ ≠ 0
1
1
2
2
84
Step 2
85
STEP 3

86
(1) Pooled Variance
s² P
87
Estimated Standard Error
S
(M1-M2)=
88
Step 4
89
90