Download Directional Hypothesis Test

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Document related concepts
no text concepts found
Transcript
CHAPTER 7
Probability and
Samples:
 Distribution of Sample Means
1
Probability and Samples:
Chap 7
Sampling Error
The amount of error between a sample
statistic (M) and population parameter (µ).
Distribution of Sample Means: is the
collection of sample means for all the
possible random samples of a particular size
(n) that can be obtained from a population.
2
3
Sampling Distribution
 Sampling Distribution is a
distribution of statistics obtained
by selecting all the possible
samples of a specific size from a
population. Ex. Every distribution
has a mean and standard deviation.
The mean of all sample means is
called Sampling Distribution. The
mean of all standard deviations is
called Standard Error of Mean (σM)
4
Expected Value of M
The mean of the distribution of (M) sample
means (statistics) is equal to the mean of the
Population of scores (µ) and is called the
Expected Value of M
M= µ
And, the average standard deviation (S) for all
of these means is called Standard Error of
Mean, σM. It provides a measure of how much
distance is expected on average between a
sample mean (M) and the population mean (µ)
5
The Law of Large Numbers
The Law of Large Numbers
states that the larger the
sample size (n), the more
probable it is that the sample
mean (M) will be close to the
population mean (µ)
n≈ N
6
Probability and Samples
 The Central Limit Theorem:
Describes the distribution of sample means
by identifying 3 basic characteristics that
describe any distribution:
1. The shape of the distribution of sample
mean has 2 conditions 1a. The population
from which the samples are selected is
normal distribution. 1b. The number of scores
(n) in each sample is relatively large
(30 or more) The larger the n the shape of the
distribution tends to be more normal.
7
The Central Limit Theorem:
 2. Central Tendency: Stats that the mean of
the distribution of sample means M is equal
to the population mean µ and is called the
expected value of M. M= µ
 3. Variability: or the standard error of mean
σM .
 The standard deviation of the distribution of
sample means is called the standard error of
mean σM.
 It measures the standard amount of
difference one should expect between M
and µ simply due to chance.
8
Computations/ Calculations or Collect
Data and Compute Sample Statistics
Z Score for Research

9
Computations/ Calculations or Collect
Data and Compute Sample Statistics
Z Score for Research

10
Computations/ Calculations or Collect
Data and Compute Sample Statistics
Z Score for Research

11
Computations/ Calculations or Collect Data
and Compute Sample Statistics
d=Effect Size/Cohn d
 Is the difference between the
means in a treatment condition.
 It means that the result from a
research study is not just by
chance alone
12
13
Computations/ Calculations or Collect
Data and Compute Sample Statistics
d=Effect Size

14
15
Problem 1
 The population of scores on the
SAT forms a normal distribution
with µ=500 and σ=100. If you take
a random sample of n=25
students, what is the probability
(%) that the sample mean will be
greater than 540. M=540?
 First calculate the Z Score then, look for proportion and convert into
percentage.
16
Problem 2
 Once again, the distribution of SAT forms a
normal distribution with a mean of µ=500
and σ=100. For this example we are going
to determine what kind of sample mean (M)
is likely to be obtained as the average SAT
score for a random sample of n=25
students. Specifically, we will determine the
exact range of values that is expected for
the sample mean 80% of the time.
17
CHAPTER 8
Hypothesis
Testing
18
Chap 8Hypothesis Testing
 Hypothesis : Statement such as “The
relationship between IQ and GPA. Topic of
a research.
 Hypothesis Test: Is a statistical method
that uses sample data to evaluate a
hypothesis about a population.
 The statistics used to Test a hypothesis is
called “Test Statistic”
i.e., Z, t, r, F, etc.
19
Hypothesis Testing
 The Logic of Hypothesis: If the
sample mean is consistent with
the prediction we conclude that
the hypothesis is reasonable
but, if there is a big discrepancy
we decide that hypothesis is not
reasonable.
 Ex. Registered Voters are Smarter than Average People.
20
Role of Statistics in Research
21
Steps in Hypothesis-Testing
Step 1: State The Hypotheses
H =Null Hypothesis
0
H :Alternative or
Researcher Hypothesis
1
22
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
23
Hypothesis Testing
Step 2: Locate the Critical Region(s) or
Set the Criteria for a Decision
24
Directional Hypothesis Test
25
None-directional
Hypothesis Test
26

Hypothesis Testing
Step 3: Computations/ Calculations or
Collect Data and Compute Sample
Statistics
27
Hypothesis Testing
Step 4: Make a Decision
28
Hypothesis Testing
Step 4: Make a Decision
29
Uncertainty and Errors in
Hypothesis Testing
 Type I Error
 Type II Error
see next slide
True H0
False H0
α
Reject
Type I Error
Retain
Correct Decision
Correct Decision
Power=1-β
Type II error β
30
True State of the World
True H0
Reject Type I
Error
Retain
α
Correct
Decision
False H0
Correct Decision
Power=1-β
Type II error
β
31
Power
 Power:
 The power of a statistical test is the
probability that the test will correctly
reject a false null hypothesis.
 That is, power is the probability that
the test will identify a treatment effect
if one really exists.
32
The α level or the level of
significance:
 The α level for a hypothesis test is
the probability that the test will
lead to a Type I error.
 That is, the alpha level determines
the probability of obtaining sample
data in the critical region even
though the null hypothesis is true.
33
The α level or the level of
significance:
It is a probability value
which is used to define
the concept of “highly
unlikely” in a hypothesis
test.
34
The Critical Region
 Is composed of the extreme sample
values that are highly unlikely (as
defined by the α level or the level of
significance) to be obtained if the
null hypothesis is true.
 If sample data fall in the critical
region, the null hypothesis is
rejected.
35
Computations/ Calculations or Collect Data
and Compute Sample Statistics
d=Effect Size/Cohn d
 It is the difference between the
means in a treatment condition.
 It means that the result from a
research study is not just by
chance alone
36
Effect Size=Cohn’s d
 Effect Size=Cohn’s d=
Result from the research
study is bigger than what
we expected to be just by
chance alone.
37
38
Cohn’s d=Effect Size

39
Evaluation of Cohn’s d Effect Size
with Cohn’s d
Magnitude of d
Evaluation of Effect Size
d≈0.2
Small Effect Size
d≈0.5
Medium Effect Size
d≈0.8
Large Effect Size
40
Problems
 Researchers have noted a decline in cognitive
functioning as people age (Bartus, 1990)
However, the results from other research
suggest that the antioxidants in foods such as
blueberries can reduce and even reverse
these age-related declines, at least in
laboratory rats (Joseph, Shukitt-Hale, Denisova,
et al., 1999). Based on these results one might
theorize that the same antioxidants might also
benefit elderly humans. Suppose a
researcher is interested in testing this theory. n
Next slide
Problems
 Standardized neuropsychological tests such as
the Wisconsin Card Sorting Test WCST can be use
to measure conceptual thinking ability and mental
flexibility (Heaton, Chelune, Talley, Kay, & Kurtiss,
1993). Performance on this type of test declines
gradually with age. Suppose our researcher selects a
test for which adults older than 65 have an average
score of μ=80 with a standard deviation of σ=20. The
distribution of test score is approximately normal. The
researcher plan is to obtain a sample of n=25 adults
who are older than 65, and give each participants a
daily dose of blueberry supplement that is very high in
antioxidants. After taking the supplement for 6 months
42
Problems
 The participants were given the
neuropsychological tests to measure their level
of cognitive function. M=92, 2 tailed,
α = 0.05
The hypothesis is that the blubbery
supplement does appear to have an effect
on cognitive functioning.
Step 1
H0 :
H1
μ with supplement = 80
:μ
with supplement
≠ 80
43
None-directional
Hypothesis Test
44
Problems
 M=92, one tailed,
α = 0.05
If the hypothesis stated that the consumption
of blubbery supplement will increase the
cognitive functioning (test scores) then,
Step 1
H0 :
H1
μ ≤ 80
: μ > 80
45
Directional Hypothesis Test
46
Problems
 M=92, one tailed,
α = 0.05
If the hypothesis stated that the consumption
of blubbery supplement will decrease the
cognitive functioning (test scores) then,
Step 1
H0 :
H1
μ ≥ 80
: μ < 80
47
Directional Hypothesis Test
48
Problems
 Alcohol appears to be involve in a variety
of birth defects, including low birth weight
and retarded growth. A researcher would
like to investigate the effect of prenatal
alcohol on birth weight . A random
sample of n=16 pregnant rats is obtained.
The mother rats are given daily dose of
alcohol. At birth, one pop is selected from
each litter to produce a sample of n=16
newborn rats. The average weight for the
sample is M=16 grams.
49
Problems
 The researcher would like to compare the
sample with the general population of rats. It
is known that regular new born rats have an
average weight of μ=18 grams. The
distribution of weight is normal with σ=4,
set α=0.01, and we use a 2 tailed test
consequently, 0.005 on each tail and the critical
value for Z=2.58
Step 1
H0 :
H1 :
μ alcohol exposure = 18
μ
≠ 18
alcohol exposure
grams
grams
50
Degrees of Freedom
df=n-1
51
Standard Deviation of Sample
52
Related documents