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Transcript
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
CHAP 9 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 Ex. Is this class
know more about STATS than the last year class?
Mean for the last year class µ=80
Mean for this year class M=82 Note: We
don’t know what the average STATS score
should be for the population. We compare
this year scores with the last year.
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
6
Inferential
Statistics
 t-Statistics:
 There are different types of t- Statistic
 1. Single (one) Sample t-statistic/Test
(chap 9)
 2. Two independent sample t-test,
Matched-Subject Experiment, or
Between Subject Design t-test (chap10)
 3.Repeated Measure Experiment, or
Related/Paired Sample t-test (chap11)
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
FYI None-directional
Hypothesis Test
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
Hypothesis Testing
Step 4: Make a Decision
14
Calculations for t-test
Step 3: Computations/ Calculations or Collect
Data and Compute Sample Statistics
M-μ
 t=
s
Sm
Sm=
Sm= M-μ
√n
t
M=t.Sm+μ
μ=M- Sm.t
Sm= estimated standard error of the mean
or
2
S /n
15
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
16
FYI d=Effect Size for Z
Use S instead of σ for t-test
17
Cohn’s d=Effect Size for 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
18
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
19
percentage of Variance
accounted for by the Treatment
 Percentage of Variance Explained r 2
r2
 r²=0.01-------- Small Effect
 r²=0.09-------- Medium Effect
 r²=0.25-------- Large Effect
20
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.
21
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?
22
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
23
None-directional
Hypothesis Test
24
None-directional
Hypothesis Test
25
Calculations for t-test
Step 3: Computations/ Calculations or Collect
Data and Compute Sample Statistics
M-μ
 t=
s
Sm
Sm=
Sm= M-μ
√n
t
M=t.Sm+μ
μ=M- Sm.t
Sm= estimated standard error of the mean
or
2
S /n
26
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..
27
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
28
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
29
None-directional
Hypothesis Test
 Step 2
30
Calculations for t-test
Step 3: Computations/ Calculations or Collect
Data and Compute Sample Statistics
M-μ
 t=
s
Sm
Sm=
Sm= M-μ
√n
t
M=t.Sm+μ
μ=M- Sm.t
Sm= estimated standard error of the mean
or
2
S /n
31
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 .
32
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.
33
Independent Sample t-test
 An independent measures
study uses a separate group
of participants to represent
each of the populations or
treatment conditions being
compared.
34
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
35
None-directional
Hypothesis Test
 Step 2
36
STEP 3

37
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.
38
Estimated Standard Error
S
(M1-M2)=
39
Pooled Variance
s² P
40
Pooled Variance
s² P
41
Step 4
42
Measuring d=Effect Size for the
independent measures

d
M1 M 2
2
S p
43
Estimated d
44
Estimated d
45
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
46
FYI in Chap 15 We use the Point-Biserial
Correlation (r) when one of our variable
is dichotomous, in this case (1) watched
Sesame St. (2) and didn’t watch Sesame
St.
2
t
2
r  2
t  df
47
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.
48
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 n1=10
49
Problems
 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
for two tails
50
Problems
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
51
Two Independent Sample t-test
Null Hypothesis:
 Two independent sample t-test,
Matched-Subject Experiment, or
Between Subject Design- nondirectional or two-tailed test
 Step 1.
 H : µ -µ = 0
 H : µ -µ ≠ 0
0
1
1
1
2
2
52
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
≤µ
Sesame St .
0
H :
1
µ
Sesame St.
No Sesame St.
>µ
No Sesame St.
53
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
54
memorize the 40 noun pairs.
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
55
Problems
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
56
57