Transcript Education 793 Class Notes

Education 793 Class Notes
Inference and Hypothesis Testing
Using the Normal Distribution
Presentation 6
Inferential statistics
Statistics that allow one to draw conclusions from data
(as opposed to simply describing data)
Characteristics of samples are described by statistics;
characteristics of populations are described by
parameters
Using probability and information about a sample to
draw conclusions (make inferences) about a
population or how likely it is that a result could have
been obtained simply as a function of chance
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Chain of reasoning
Population with
parameters
Random
selection
Inference
Probability
Sample with
statistics
Underlying
distributions
of the statistic
Random selection can take several forms (such as simple,
systematic, cluster, or stratified random sampling), and is
intended to generate a sample that represents the
population.
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Three distributions of note
• Distribution in the population (parameters)
• Distribution in the sample (statistics)
• Distribution of sample statistics from all
possible samples of a given size drawn from
the population, or the sampling distribution of
a statistic (mean, correlation coefficient,
regression coefficient, etc.)
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Comparing the three
distributions
Population
Sample
Sample means
Mean = 
Mean =
Mean = 
Stddev = 
Stddev = s
Stddev = 
Population
distribution
Sample
distribution
Sampling
distribution of the
mean 5
Hypothesis testing
Used to assess the statistical significance of findings, and
involves the comparison of empirically observed findings (drawn
from a sample) with those which were expected.
Null hypothesis (H0):
The hypothesis about the population, suggesting no difference
(hence null), such as:
1) a distribution has a specified mean
2) two or more statistics (such as means) are not the same
Research hypothesis (HA):
Any hypothesis that is an alternative to the one being tested;
usually the opposite of the null hypothesis. By rejecting the null
hypothesis one shows that there is evidence to suggest that the
research hypothesis may be true.
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Null and Alternative Hypotheses
• Examples, One population
– Null: =some specific value
– Alternative:   some specific value (nondirectional - 2-tailed)
OR  < some specific value (directional – 1-tailed)
OR  > some specific value (directional – 1-tailed)
• Examples, Two populations
– Null: 1=2
– Alternative 1  2 (nondirectional)
OR 1 > 2 (directional)
OR 1 < 2 (directional)
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Inferential error
• Type I: Alpha
– Rejecting the null hypothesis with the null
hypothesis is really true
• Type II: Beta
– Failing to reject the null hypothesis when the null
hypothesis is in fact false
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Possible outcomes
Decision
Reject H0
(H0 false)
Fail to
reject H0
(H0 true)
In the population, In the population,
H0 is true
H0 is false
Type I error
a
Correct decision
Correct decision
Type II error
b
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Basic Concepts of
Hypothesis Testing
• The null hypothesis is assumed to be
true
• The greater the difference between the
sample mean and hypothesized mean,
the lower the probability that the
difference is due to chance
• Need to understand sampling
distribution of the mean
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Dancing distributions
Population
Mu
Sample 1
X1
freq
freq
X
Mu
Sample 2
X
X2
freq
X
X
Sample n
Xn
freq
X
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Sampling distribution
of the mean
• Shape of this distribution is defined by the Central
Limit Theorem: As sample size increases the
sampling distribution of the mean approximates a
normal distribution.
• Central tendency: The mean of the distribution is an
unbiased estimator of the population parameter value
().
• Variability: The variance of the distribution is a
function of the variance of the population () and the
size of the sample.
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Central Limit Theorem
• For a given a distribution with a mean  and
variance 2, the sampling distribution of the mean
approaches a normal distribution with a mean ()
and a variance 2/N as N, the sample size,
increases.
• The sampling distribution of the mean approaches a
normal distribution regardless of the shape of the
original distribution. Really, it does! Hard to believe,
but true!!
simulation
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Which sample size?
•
According to the Central Limit Theorem, as the sample size
increases, the sampling distribution:
A. becomes more normal in shape
B. decreases in variability (less spread)
•
In this case, sample size refers to the size of each individual
sample, not the number of samples drawn (which, in practice,
is typically one).
•
To generate a complete sampling distribution of the mean
requires drawing an infinite number of samples of a particular
sample size, and no one – literally – has time for that!
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X
Three population distributions
Each distribution
ranges from 0 to
100, with a  X
of 50
freq
freq
freq
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Central limit: Normal population
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Central limit: Skewed population
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Central limit: Rectangular population
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How can we use this
information?
• We can draw a sample from a population, and generate
statistics that describe that particular sample.
• If the sample is of sufficient size, based on the central limit
theorem, we can make inferences about unmeasured
characteristics of the population (parameters, such as  and 2)
based on characteristics of the sample (statistics, such a mean
and variance).
• Since the sampling distribution of the mean approaches a
standard normal distribution as the sample size increases
(regardless of the shape of the original distribution), with a
sufficient sample we can assign probability statements to the
inferences described above.
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Point Estimates
• Most statistics are point estimates -- the single best
guess ( X   X ) of the population parameter:
z
X  0
X
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Standard deviation or
standard error?
• A standard deviation is a measure of the variability in a
distribution of scores.
• A standard error is a measure of sampling error, and is
the standard deviation of the sampling distribution of a
statistic (such as a mean).
• We seldom have a direct measure of the standard error
of a statistic, so it must be estimated. For means,
standard error can be estimated in two ways.
– If the population standard deviation is known:
X 

n
– If not, standard error can be estimated from the sample’s
standard deviation, s:
s
sX 
n
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Hypothesis Testing Process
• Compute test statistic from sample data
• Identify sampling distribution of statistic (assuming nothing
unusual is happening). This becomes the null hypothesis –
necessary for testing purposes, but not typically the research
hypothesis in which we are interested.
• Compare computed statistic to sampling distribution based upon
null and research hypotheses (e.g., one or two tailed test? Need
to reduce erroneous conclusions?)
• Make inference – If the value of the test statistic is unusual (lies
in the critical region) , reject the null hypothesis and infer that
something besides expected variation is going on.
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A sample problem
• A random sample of 36 teachers scores an average of
1100 on a national teacher assessment test. The test
is designed to produce a standard normal distribution,
with a mean of 950 and variance of 250,000.
– A. Using alpha =.05, test the hypothesis that the
mean teachers’ score was significantly higher than
the national average.
Reject Null
CV
1100  950
z
.05
250,000 =1.8
36
1.64
 =950
1.8
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Problem to Try
• 49 seniors scored a mean of 54 on a
national chemistry exam. Using
alpha=.05, test the hypothesis that the
seniors were below the national norm of
58 with an SD of 14.
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Next week
• Chapter 10 p. 272-302
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