Download 9.4 Day 2

Section 9.4
Day 2
To Pool or Not to Pool?
To Pool or Not to Pool?
Almost always select the unpooled option.
To Pool or Not to Pool?
Almost always select the unpooled option.
The only situation in which the pooled
procedure has definite advantage over
unpooled is when the population
standard deviations are equal but the
sample sizes are unequal.
To Pool or Not to Pool?
Almost always select the unpooled option.
The only situation in which the pooled
procedure has definite advantage over
unpooled is when the population standard
deviations are equal but the sample sizes
are unequal.
Population standard deviations usually
unknown
Significance test for the
difference of two means
Components of a significance test
for the difference of two means
1)
2)
3)
4)
Components of a significance test
for the difference of two means
1) Name test and check conditions
2)
3)
4)
Components of a significance test
for the difference of two means
1) Name test and check conditions
2) State hypotheses
3)
4)
Components of a significance test
for the difference of two means
1) Name test and check conditions
2) State hypotheses
3) Compute test statistic, find P-value, and
draw sketch
4)
Components of a significance test
for the difference of two means
1) Name test and check conditions
2) State hypotheses
3) Compute test statistic, find P-value, and
draw sketch
4) Write conclusion, linked to computations
and in context of problem
Name Test
Name:
One-sided significance test for the difference
of two means
or
Two-sided significance test for the difference
of two means
Check Conditions
1)
For survey,
Check Conditions
1)
For survey, two samples randomly and
independently selected from two
different populations.
Check Conditions
1)
For survey, two samples randomly and
independently selected from two
different populations.
For experiment,
Check Conditions
1)
For survey, two samples randomly and
independently selected from two
different populations.
For experiment, two treatments randomly
assigned to available experimental units.
Check Conditions
2) normality: two samples must look like
they came from normally distributed
populations
or
Check Conditions
2) normality: two samples must look like
they came from normally distributed
populations
or
Sample sizes are large enough that
sampling distributions of sample means
will be approximately normal
Check Conditions
15/40 guideline can be applied to each
sample or treatment group, although it is a
bit conservative.
What does this mean?
Check Conditions
15/40 guideline can be applied to each
sample or treatment group, although it is a
bit conservative.
What does this mean?
You can get by with smaller sample sizes
when taking a difference of two means
Check Conditions
You can get by with smaller sample sizes
when taking a difference.
Subtracting two sample means brings in the
tails.
With skewed populations, the sampling
distribution of the difference of two means
tends to be more symmetric than the two
separate sampling distributions of the
sample means.
Check Conditions
3) For survey, population sizes should be
at least ten times larger than sample sizes
for both samples.
Check Conditions
3) For survey, population sizes should be at
least ten times larger than sample sizes
for both samples.
Remember, this condition does not apply to
experiment.
Check Conditions
Conditions: same conditions as for
confidence interval for difference of two
means
State Hypotheses
Null hypothesis is usually that the two
population means are equal.
H o: μ 1 = μ 2
or
H o: μ 1 - μ 2 = 0
where μ1 is mean of first population and μ2 is
mean of second population
State Hypotheses
Three forms of alternative hypothesis:
State Hypotheses
Three forms of alternative hypothesis:
1) Ha: 1   2 or Ha: 1   2  0
State Hypotheses
Three forms of alternative hypothesis:
1) Ha: 1   2 or Ha: 1   2  0
or
2) Ha: 1   2 or Ha: 1   2  0
State Hypotheses
Three forms of alternative hypothesis:
1) Ha: 1   2 or Ha: 1   2  0
or
2) Ha: 1   2 or Ha: 1   2  0
or
3) Ha: 1   2 or Ha: 1   2  0
Compute test statistic, find P-value,
and draw sketch
Compute difference between sample means
(because hypothesized mean difference is
zero), measured in estimated standard
errors.
Compute test statistic, find P-value,
and draw sketch
Write conclusion, linked to computations and
in context of problem
a) when do you reject the null hypothesis?
b) when do you not reject the null
hypothesis?
c) when do you accept the null hypothesis?
Write conclusion, linked to computations and
in context of problem
If you are using fixed-level testing:
a) when do you reject the null hypothesis?
When P-value is less than significance
level
b) when do you not reject the null
hypothesis?
c) when do you accept the null hypothesis?
Write conclusion, linked to computations and
in context of problem
If you are using fixed-level testing:
a) when do you reject the null hypothesis?
When P-value is less than significance level
b) when do you not reject the null
hypothesis? When P-value is greater
than or equal to significance level
c) when do you accept the null hypothesis?
Write conclusion, linked to computations and
in context of problem
If you are using fixed-level testing:
a) when do you reject the null hypothesis?
When P-value is less than significance level
b) when do you not reject the null
hypothesis? When P-value is greater than
or equal to significance level
c) when do you accept the null hypothesis?
never
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Name: Two-sided significance test for the
difference in two means
Two-sided because we are testing to see if
there is evidence of a statistically
significant difference.
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Check conditions:
(1) We are told this class is a random
sample taken from all students in this
course.
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Check conditions:
(1) We are told this class is a random
sample taken from all students in this
course.
Samples are independent as one is females
and other is males and no pairing of
subjects is being done.
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Check conditions:
(2) Data appears fairly symmetric for each
sample so reasonable to assume both
samples came from normal populations.
However, each contains one unusually large
value, which may have great influence on
the results.
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Check conditions:
(2) Data appears symmetric for each sample so
reasonable to assume both samples came from
normal populations.
However, each contains one unusually large value,
which may have great influence on the results.
Two analyses will be done: once with all data
and once without outliers.
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Check conditions:
(3) It is reasonable to assume that there are
more than 460 females and 150 males in
the populations of students who take this
course.
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State hypotheses:
Ho: μf = μm, where μf is the mean study time
of all female students taking this course
and μm is the mean study time of all male
students taking this course.
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State hypotheses:
Ho: μf = μm, where μf is the mean study time
of all female students taking this course
and μm is the mean study time of all male
students taking this course.
H a : μf ≠ μ m
Computations
2-SampTTest
Inpt: Stats
x1: 10.93
sx1: 6.22
n1: 46
x2: 8.20
sx2: 5.94
n2: 15
μ 1: ≠ μ 2
Pooled: No Yes??
Calculate
Computations
2-SampTTest
Inpt: Stats
x1: 10.93
sx1: 6.22
n1: 46
x2: 8.20
sx2: 5.94
n2: 15
μ 1: ≠ μ 2
Pooled: No
Calculate
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Compute test statistic, find P-value, and
draw sketch
Use 2-SampTTest
If reverse males and females:
t  ±1.53
t  ± 1.53
P-value  0.1392
P-value  0.1392
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Write a conclusion in context, linked to your
computations
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Write a conclusion in context, linked to your
computations
I do not reject the null hypothesis because
the P-value of 0.1392 is greater than the
significance level of 0.05.
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Write a conclusion in context, linked to your
computations
I do not reject the null hypothesis because
the P-value of 0.1392 is greater than the
significance level of 0.05.
There is not sufficient evidence to support
the claim that there is a statistically
significant difference in mean weekly study
hours for females and males.
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Are we finished with the problem?
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Are we finished with the problem?
Need to conduct test again with outliers
removed.
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Removing outliers:
Name of test:
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Removing outliers:
Name of test: same
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Removing outliers:
Name of test: same
Conditions:
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Removing outliers:
Name of test: same
Conditions: still met
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Removing outliers:
Name of test: same
Conditions: still met
Hypotheses:
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Removing outliers:
Name of test: same
Conditions: still met
Hypotheses: same
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Removing outliers:
Name of test: same
Conditions: still met
Hypotheses: same
Computations:
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Removing outliers:
Name of test: same
Conditions: still met
Hypotheses: same
Computations: need to do without outliers
What do you get for t and P-value now?
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t ≈ ± 2.657
P-value ≈ 0.012
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Removing the outliers:
Reject the null hypothesis because the
P-value of 0.012 is less than 0.05.
There is sufficient evidence to support the
claim that there is a statistically significant
difference in mean weekly study hours for
females and males.
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Removing the outliers:
Reject the null hypothesis because the P-value of
0.012 is less than 0.05.
There is sufficient evidence to support the claim
that there is a statistically significant difference in
mean weekly study hours for females and
males.
So, now do we reject or not reject the null
hypothesis?
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So, now do we reject or not reject the null
hypothesis?
Since we have a split-decision, we
should:
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So, now do we reject or not reject the null
hypothesis?
Since we have a split-decision, we should:
(1) get more data
(2)
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So, now do we reject or not reject the null
hypothesis?
Since we have a split-decision, we should:
(1) get more data
(2) recheck the accuracy of the outliers (if
possible)
Power
Power of a significance test is the probability
of rejecting the null hypothesis.
Best way to get more power to reject a
false null hypothesis is to increase
sample sizes.
Power
If we have reason to believe the population
standard deviations are about equal,
make the sample sizes the same.
Power
If we have reason to believe the population
standard deviations are about equal, make
the sample sizes the same.
If we have reason to believe that one
population’s standard deviation is
larger than the other’s, allocate our
resources so you take a larger sample
from the population with the larger
standard deviation.
Questions?