Download Course Review Chapter 9 Testing Hypotheses

Chapter 9
Testing Hypotheses
Course Review
1.
Levels of Measurement (nominal, ordinal,
interval/ratio)
2. Measures of central tendency (mean, median, mode)
3. Measures of variability (range, variance, standard
deviation)
4. Normal Distribution (normal curve)
5. Sampling (random, sampling distribution)
6. Estimation of population (confidence intervals,
confidence levels)
•
•
•
•
•
Overview
5 Steps
St
f
for ttesting
ti
h
hypotheses
th
One and two-tailed tests
Type 1 and Type 2 Errors
Z tests and t tests
7. Testing hypotheses (z test, t test)
Chapter 13 – 1
Chapter 13 – 2
Research Hypothesis
(and Null Hypothesis)
Testing Hypotheses
(and Null Hypotheses)
A research hypothesis is a statement
reflecting a substantive hypothesis (i.e., the
stated relationship between two population
parameters).
Testing Hypotheses is a procedure that
allows us to evaluate hypotheses about
population
p
p
parameters
p
based on sample
p
statistics.
A null hypothesis is a statement of
“no difference” that is in opposition to the
research hypothesis (for example: The GPA
of Sociology majors is no different than that
of the UNT student population.)
Example of an hypothesis:
Sociology majors at UNT have a
higher GPA than the UNT student
population.
Chapter 13 – 3
One-Tailed Tests
Chapter 13 – 4
It is one tailed since we are predicting
that the income of Native Americans
falls to the right of the income of the
U.S. population.
One-tailed hypothesis test – A
hypothesis test in which the population
parameter is likely to fall above or
below another.
Is this research hypothesis a one-tailed
or two-tailed test:
The family income of Native
Americans is less than that of the
U.S. population.
Native Am.
U.S. population
What would be the null hypothesis?
Chapter 13 – 5
Chapter 13 – 6
Two-Tailed Tests
One-Tailed Tests
• Right-tailed test – A one-tailed test
in which the sample outcome is
hypothesized to be at the right tail
of the sampling distribution. (e.g., Gas
prices
i
are hi
higher
h in
i CA than
th the
th U.S.)
US)
• Left-tailed test – A one-tailed test
in which the sample outcome is
hypothesized to be at the left tail of
the sampling distribution. (e.g., Gas
prices are lower in TX than the U.S.)
• Two-tailed hypothesis test – A hypothesis
test in which a parameter statistic might
fall within either tail of the sampling
distribution.
W are not sure which
We
hi h tail
il of
f the
h curve the
h
statistic is likely to fall.
For example: Gas prices in Michigan are
not the same as that of the U.S. (we
haven’t specified greater or lesser than)—
what would be the null hypothesis of this
example?)
Chapter 13 – 7
The Five Steps In Hypothesis Testing
1.
Make sure the data meet the assumptions
for hypothesis testing
--a random sample is being used
--either the variable is normally
distributed or the sample is over 50
cases, this will allow us to apply the
Central Limit Theorem
--knowing the level of measurement of the
data, in the examples that we will be using,
we will be assuming the dependent variable
is interval/ratio.
Chapter 13 – 8
The Five Steps In Hypotheses Testing
2.
State the research and null hypotheses and
selecting alpha.
What is a research hypothesis (H1) –
A statement reflecting the substantive
hypothesis.
For example, we might state: The average
salary of Native Americans is less than that
of the U.S. population.
Chapter 13 – 9
The Five Steps In Hypotheses Testing
The Five Steps In Hypotheses Testing
2.
Chapter 13 – 10
Stating the research and null hypotheses
and selecting alpha.
Alpha is the probability level at which the null
hypothesis is rejected. That is, it is the
probability level at which we are willing to
conclude that the null hypothesis is false.
What is a null hypothesis (Ho) –
Alphas are usually set at .05, .01, or .001.
– A statement of “no difference,” which
contradicts the research hypothesis.
An Example of Alpha: It is the level at which
we conclude that the average income for Native
Americans is so different from the general
population that we can safely say that the two
incomes are different (if our sample doesn’t
reach the “alpha level” we have set, then our
null hypothesis is not rejected).
For example, we might state: The average
salary of Native Americans is no different
than that of the U.S. population.
Chapter 13 – 11
Chapter 13 – 12
The Five Steps In Hypotheses Testing
The Five Steps In Hypotheses Testing
For example: We want to test the null hypothesis
that there is no difference in family income between
Native Americans and the general U.S. population.
If indeed there is no difference, then if we took
100 samples
p
of Native Americans,, we would expect
p
to find that the 100 sample means tend to fall
around the U.S. population mean.
Of course we only have a single sample. But, we can
calculate where that single sample would fall within
the normal distribution, when making the assumption
that it is no different from the population mean (null
hypothesis).
The question becomes:
What is the probability of obtaining a sample mean
that is this different from the population mean,
assuming there is really no difference between the
two groups?
If we find the probability to be extremely small, that
is, if we find that there is very little chance that we
would draw a sample that is this different from the
population if the sample and population are really not
different, then we will reject the null hypothesis and
accept the research hypothesis that the two groups
are different with regard to family income.
Chapter 13 – 13
The Five Steps In Hypotheses Testing
Chapter 13 – 14
The Five Steps In Hypotheses Testing
So, in other words, our question is:
3. Selecting the sampling distribution and
specifying the test statistic.
Does the mean for family income we have obtained
from our sample of Native Americans cast doubt on
the null hypothesis of no difference between Native
Americans and the general population with regard to
income?
Or: What are the chances that we would have
randomly selected a sample of Native Americans
with the income we found if there is no difference
between Native Americans and the U.S. population.
Our sampling distribution will be either the
Z distribution or t distribution. That is, we
will use either the Z or t distribution table
to assist us in testing our hypothesis.
To examine just how different our sample
statistic is from the population parameter
(e.g., how different our sample mean is from
the population mean), our test statistic will
be either the Z statistic or the t statistic.
Chapter 13 – 15
Chapter 13 – 16
The Five Steps In Hypotheses Testing
The Five Steps In Hypotheses Testing
4. Computing the test statistic, e.g., converting the
sample mean to a Z statistic or t statistic.
Let’s use an example to compute the test statistic:
If we assume that our sample mean family income for
Native Americans is $22,400 and that our mean
family income for the U.S. population is $28,985,
then we would ask:
What are the chances that we would have randomly
selected a sample of Native Americans such that
their average family income is lower than $28,985
(the average family income for the U.S. population).
Chapter 13 – 17
$22,400
$28,985
We can determine the chances or
probability because of what we know
about the sampling distribution.
Chapter 13 – 18
The Five Steps In Hypotheses Testing
So, lets suppose:
The mean family income for our sample of Native
Americans = $22,400 with a sample size of 100
The mean family income for U.S. population = $28,985
The standard deviation for U.S. population = $7,345
$22,400
$28,985
We can take our sample mean, convert it into a
Z score, and find the Z score in the Z table.
The number we find in the Z table will indicate
the probability (known as a “p value”) of having
a sample mean that is this different from the
U.S. population mean.
We want to test the null hypothesis that Native
American income is no different than the U.S.
population income.
Chapter 13 – 19
Z=
Y – uy
oy
Group Mean – Population Mean
or
Population SD
N
Oy
Z = 22,400 – 28,985
734.5
=
4,585.0
734.5
Y – uy
oy
or
Group Mean – Population Mean
Population SD
=
N
Z = 22,400 – 28,985
734.5
= $7,345/10 = $734.5
N
Z=
N
N
Standard Error =
Chapter 13 – 20
-8.97
=
4,585.0
734.5
-8.97
To determine the probability of observing a Z
value of -8.97, we look up the value in Appendix B
to find the area beyond a Z of -8.97 (Column C).
Chapter 13 – 21
Z = -8.97
8 97
=
0
Chapter 13 – 22
Z = -8.97
8 97
0
Our calculated Z value is not listed in Appendix B, so we will
need to rely on the Last Z value reported in the table, 4.0.
The “P value” for 4.0 (that is, the area beyond the Z) is
.0001.
Our calculated Z value is not listed in Appendix B, so we will
need to rely on the Last Z value reported in the table, 4.0.
The “P value” for 4.0 (that is, the area beyond the Z) is
.0001.
This value is the probability of getting the sample mean we
got for Native Americans if the null hypothesis is true.
Because it is so unlikely, we will reject the null hypothesis.
This value is the probability of getting the sample mean we
got for Native Americans if the null hypothesis is true.
Because it is so unlikely, we will reject the null hypothesis.
Chapter 13 – 23
Chapter 13 – 24
The Five Steps In Hypotheses Testing
We can then conclude that our
“Z statistic” (-8.97) is statistically
significant, that is, it is very unlikely
that the difference we found occurred
by chance or sampling error.
5. Making a decision and interpreting the
results
Thus, if our p value is less than the
alpha level we selected (e.g., .05 or
.01,
01 or .001),
001) then we can reject the
null hypothesis.
We can say that the difference
between the family income of Native
Americans and the U.S. population is
significant at the .0001 level.
Since the alpha is typically set at .05
in the social sciences and our P value
is much smaller than .05, we can
reject our null hypothesis.
Chapter 13 – 25
Errors that we try to avoid
Type I error: if the null hypothesis is true
but we reject it.
Type II error: if the null hypothesis is false
but we accept it.
it
Chapter 13 – 26
Type I and Type II Errors
and their relationship to alpha
• When selecting our alpha, we need to be
aware that if we set alpha too large (e.g.
.10)) we may
y create a Type
yp I error—that is,
we might reject the null hypothesis when it
is actually true.
• Or, if we set the alpha too small (e.g., .001)
we may create a Type II error by failing to
reject a false null hypothesis.
Chapter 13 – 27
Chapter 13 – 28
Let’s take another example:
--We will test the null hypothesis that the
salary for women is no different than that for
the U.S. population as a whole.
yp
is that the salary
y
--Our research hypothesis
for women is less than that for the U.S.
population
--The test statistic used is
either the Z statistic or
t statistic and since we have
the population standard
deviation we will use the Z
statistic.
--We sample at least 50 women so that our
theoretical sampling distribution will be
normally distributed (a required assumption).
Chapter 13 – 29
Chapter 13 – 30
Computing the test statistic. The formula
for the Z statistic is:
4. Computing the test statistic.
The formula for the Z statistic is:
Y – uy
Z = oy
N
Group Mean – Population Mean
or
Z=
Y – uy
oy
24,100 – 28,985
or
100
N
Population SD
= -2.09
23,335
,
Where the population mean is $28,985
N
the sample mean for women is $24,100
the population standard deviation of 23,335
and the sample size is 100
Chapter 13 – 31
Chapter 13 – 32
Making a Decision and Interpreting the
results, in our example:
Probability Values
--we confirm that the Z is on the left tail
of the distribution (-2.09)
--the P value found in the Z table (where Z =
2.09) is .0183, which is less than a .05 alpha.
--thus, we can reject the null hypothesis
of no difference and can conclude that
the average income for women is less than
that of the general population.
24,100
Women
28,985
U.S. Population
Chapter 13 – 33
When to use the t statistic and when to
use the Z statistic
1.
The Z statistic can only be used if the
population standard deviation is known.
Typically, this is not the case.
2
2.
When the sample standard deviation must
be used in lieu of the population SD then
the t statistic should be used.
3.
Chapter 13 – 34
The Five Steps In Hypotheses Testing
4. Computing the test statistic. The
formula for the t statistic is:
Z=
Y – uy
Sy
N
The formula for the t statistic is identical
to the formula for the Z statistic except
that the sample SD is used in place of
the population SD
Chapter 13 – 35
G
Group
Mean
M
– Population
P
l
Mean
M
or
Sample SD
N
Chapter 13 – 36
t distribution
Steps for interpreting the t statistic
1.
The t statistic has its own t distribution table
(Appendix C in text) that is used rather than the Z
distribution table (Appendix B). This is because the
Z distribution uses the population SD while the
t distribution uses the sample SD. Using the sample
SD results in some degree of error that is accounted
for, in part, by the different shaped curves.
Chapter 13 – 37
Steps for interpreting the t statistic
2.
Reading the t distribution table
requires knowing the degrees of
freedom (a concept used in calculating
a number of statistics including the
t statistic).
--Reading the t distribution table also
requires knowing the alpha (which you
select) and the number of cases.
A smaller degree of freedom produces a flatter
curve. When there is a large sample of 120+ cases,
then the shape of the t distribution mirrors that
of the Z distribution.
Chapter 13 – 38
Steps for interpreting the t statistic
3.
The degrees of freedom represent the number
of scores that are free to vary in calculating
each statistic (this is a statistical concept that
is too advanced for this class).
4
4.
Typically the degrees of freedom are
N – 1 when comparing a group to a whole
population and so, in some sense, it is simply
reflecting the size of the sample.
5.
Once we have selected our alpha and know the
number of cases and subsequently the degrees
of freedom, we can interpret the t statistic.
Chapter 13 – 39
Chapter 13 – 40
6. calculating the t statistic
For Example:
• In our previous example we knew that:
The population mean is $28,985 and the
sample mean for women is $24,100 with a
population standard deviation of 23,335
and sample size of 100.
100
The population mean is $28,985 and the
sample mean for women is $24,100 with a
sample standard deviation of 24,897 and
sample
l size
i of
f 100
100.
• We subsequently calculated the Z score.
• If we did not know the population SD, we
would need to use the sample SD (which is
$24,897) to calculate the SE and, in turn,
to calculate the “t” score (or t statistic).
Chapter 13 – 41
24,100-28,985
24,897
=
-4885
2489.7
= -1.96
100
Chapter 13 – 42
t distribution table
7. Finding the t statistic in the t distribution
table
• Our degrees of freedom for this example is N – 1 or
99 and our t statistic is -1.96 (the larger the
t statistic the more likely it will be significant).
• On page 520
520-21
21 of your book we can find the
t distribution table. It displays the degrees of
freedom for 60 and for 120 (or see next slide). Since
ours is 99 it falls between these.
• We can assume a one-tailed test since existing
knowledge indicates that women make less than the
population as a whole and certainly not more (the mean
will fall on the left side of the curve).
Chapter 13 – 43
Determining Statistical Significance
From the t distribution table we can conclude that our
sample mean is statistically significant from the
general population mean because:
1. On page 520 of your book we can see that, for 60
degrees of freedom, a t statistic
stat st c of 1.671
.67 has a
p value of .05 when using a one-tailed test. This is
large enough to be statistically significant at alpha .05.
2. the degrees of freedom in our analysis is 99,
therefore if our t statistic is 1.671 or larger we can
conclude that our sample mean is statistically
significant with a confidence level of 95%.
Chapter 13 – 44
3.Since our actual t statistic is -1.96 we can
conclude statistical significance at the .05
level.
4.We have assumed a one-tailed test since
existing knowledge indicates that women
make less than the population as a whole
and certainly not more. If we did not
know whether women make more or less
than men we would need to use a two
tailed test.
Chapter 13 – 45
Chapter 13 – 46
t test in Sum
Summary: Steps in Testing an
Hypothesis
• t statistic – The test statistic computed to
test the null hypothesis about a population
mean when the population standard deviation is
unknown and is estimated using the sample
standard deviation.
1.
• t distribution
d str ut on – A family
fam ly of cur
curves,
es, each
determined by its degrees of freedom (df). It
is used when the population standard deviation
is unknown and the standard error is estimated
from the sample standard deviation.
3. Select sampling distribution and test
statistic to be used (Z or t statistic)
• Degrees of freedom (df) – The number of
scores that are free to vary in calculating a
statistic (typically N – 1).
Chapter 13 – 47
Verify/make assumptions
2. State research and null hypotheses
4. Compute test statistic
5. Make a decision and interpret results
Chapter 13 – 48
Comparing Two Sample Means
(Rather than a Sample and a
Population as just learned)
Comparing Two Sample Means
(Rather than a Sample and a
Population as just learned)
Example for comparing two means:
– Comparing the mean salary for Native
American men to the mean salary for
White men (instead of the mean salary of
Native American men to the mean salary
of the whole U.S. population)
Steps for comparing two
sample means are the same
as we have
h
learned
l
d for
f
comparing a sample mean to
the population
except for:
Chapter 13 – 49
Chapter 13 – 50
except for:
(1) the formula for calculating the
t statistic,
Calculating the t statistic:
(2) calculating the degrees of freedom
(since each group has its on number
of cases we have to account for
this) and
this),
Mean of
1st Group
-
Mean of
2nd Group
Standard Error of the
Differences Between
the Means
(3) determining whether the two
groups have equal or unequal
variances. If the Levene’s test is
significant then their variances are
unequal (one is at least twice as
large as the other).
Chapter 13 – 51
Calculation of the t Statistic is done by computer
(so you won’t be responsible for calculating it by hand)
Mean of
1st Group
-
(N 1) S1
(N-1)
S1y11 - (N2-1)
(N2 1) S2y2
S2 2
(N1 + N2) – 2
Example: Comparing Two Sample Means
(male and female job burnout)
In SPSS:
Mean of
2nd Group
1. Analyze
Standard Error of the
Differences Between
the Means
SE =
Chapter 13 – 52
2. compare means
3. independent sample t test
N1+N2
N1 N2
N1N2
Where:
1st
N1 = Number of cases for
group
N2 = Number of cases for 2nd group
(N1 + N2) -2 = Degrees of freedom
S1y1 = Sample variance for 1st group
S2y2 = Sample variance for 2nd group
move V
V101 (s
(sex)) to “group
group variable”
ar a
box
o
4.. mo
5. click “define group”
6. in group 1 put “1” (female) and
in group 2 put “2” (male)
7. click continue
Chapter 13 – 53
8. move “burnout”, V102 (age), V100 (education),
V114 (# residents assigned) to “test
variables” box and then click Okay
Chapter 13 – 54