Download Chapter 12 Testing Hypotheses

Chapter 12:
Testing Hypotheses
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Overview
Research and null hypotheses
One and two-tailed tests
Errors
Testing the difference between two means
t tests
Chapter 13 – 1
You already know how
to deal with two
nominal variables
Overview
Independent Variables
Nominal
Interval
Considers the distribution
of one variable across the
categories of another
variable
Considers how a change in
a variable affects a
discrete outcome
Interval
Dependent
Variable
Nominal
Considers the difference
between the mean of one
group on a variable with
another group
Considers the degree to
which a change in one
variable results in a
change in another
Chapter 13 – 2
You already know how
to deal with two
nominal variables
Overview
Independent Variables
Nominal
Lambda
Interval
Dependent
Variable
Nominal
Considers the difference
between the mean of one
group on a variable with
another group
TODAY!
Testing the differences
between groups
Interval
Considers how a change in
a variable affects a
discrete outcome
Considers the degree to
which a change in one
variable results in a
change in another
Chapter 13 – 3
You already know how
to deal with two
nominal variables
Overview
Independent Variables
Nominal
Interval
Dependent
Variable
Nominal
Lambda
Confidence Intervals
t-test
TODAY!
Testing the differences
between groups
Interval
Considers how a change in
a variable affects a
discrete outcome
Considers the degree to
which a change in one
variable results in a
change in another
Chapter 13 – 4
General Examples
Independent Variables
Nominal
Interval
Interval
Dependent
Variable
Nominal
Is one group scoring significantly
higher on average than another group?
Is a group statistically different from
another on a particular dimension?
Is Group A’s mean higher than Group
B’s?
Chapter 13 – 5
Specific Examples
Independent Variables
Nominal
Interval
Interval
Dependent
Variable
Nominal
Do people living in rural communities live
longer than those in urban or suburban areas?
Do students from private high schools perform
better in college than those from public high
schools?
Is the average number of years with an employer
lower or higher for large firms (over 100
employees) compared to those with fewer than
100 employees?
Chapter 13 – 6
Testing Hypotheses
• Statistical hypothesis testing – A procedure that
allows us to evaluate hypotheses about
population parameters based on sample statistics.
• Research hypothesis (H1) – A statement
reflecting the substantive hypothesis. It is always
expressed in terms of population parameters, but
its specific form varies from test to test.
• Null hypothesis (H0) – A statement of “no
difference,” which contradicts the research
hypothesis and is always expressed in terms of
population parameters.
Chapter 13 – 7
Research and Null Hypotheses
One Tail — specifies the hypothesized direction
• Research Hypothesis:
H1: 2 1, or 2 1 > 0
• Null Hypothesis:
H0: 2 1, or 2 1 = 0
Two Tail — direction is not specified (more common)
• Research Hypothesis:
H1: 2 1, or 2 1 = 0
• Null Hypothesis:
H0: 2 1, or 2 1 = 0
Chapter 13 – 8
One-Tailed Tests
• One-tailed hypothesis test – A hypothesis test
in which the alternative is stated in such a way
that the probability of making a Type I error is
entirely in one tail of a sampling distribution.
• Right-tailed test – A one-tailed test in which
the sample outcome is hypothesized to be at
the right tail of the sampling distribution.
• Left-tailed test – A one-tailed test in which the
sample outcome is hypothesized to be at the
left tail of the sampling distribution.
Chapter 13 – 9
Two-Tailed Tests
• Two-tailed hypothesis test – A
hypothesis test in which the region of
rejection falls equally within both tails of
the sampling distribution.
Chapter 13 – 10
Probability Values
• Z statistic (obtained) – The test statistic
computed by converting a sample statistic
(such as the mean) to a Z score. The
formula for obtaining Z varies from test to
test.
• P value – The probability associated with
the obtained value of Z.
Chapter 13 – 11
Probability Values
Chapter 13 – 12
Probability Values
• Alpha ( ) – The level of probability at
which the null hypothesis is rejected. It is
customary to set alpha at the .05, .01, or
.001 level.
Chapter 13 – 13
Five Steps to Hypothesis Testing
(1) Making assumptions
(2) Stating the research and null hypotheses
and selecting alpha
(3) Selecting the sampling distribution and
specifying the test statistic
(4) Computing the test statistic
(5) Making a decision and interpreting the
results
Chapter 13 – 14
Type I and Type II Errors
• Type I error (false rejection error)the probability (equal
to ) associated with rejecting a true null hypothesis.
• Type II error (false acceptance error)the probability
associated with failing to reject a false null hypothesis.
Based on sample results, the decision made is to…
reject H0
do not reject H0
In the
population
H0 is ...
true
Type I
error ()
correct
decision
false
correct
decision
Type II error
Chapter 13 – 15
t Test
• t statistic (obtained) – The test statistic computed
to test the null hypothesis about a population mean
when the population standard deviation is unknow
and is estimated using the sample standard
deviation.
• t distribution – A family of curves, 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.
• Degrees of freedom (df) – The number of scores
that are free to vary in calculating a statistic.
Chapter 13 – 16
t distribution
Chapter 13 – 17
t distribution table
Chapter 13 – 18
t-test for difference between two means
Is the value of 2 1 significantly different from 0?
This test gives you the answer:
t( N1  N 2 2)
Y1 Y 2

SY1 Y2
The difference between the
two means
 the estimated standard
error of the difference
If the t value is greater than 1.96, the difference
between the means is significantly different from
zero at an alpha of .05 (or a 95% confidence level).
The critical value of t will be higher than 1.96 if the total N is less
than 122. See Appendix C for exact critical values when N < 122.
Chapter 13 – 19
Estimated Standard Error of the
difference between two means
assuming unequal variances
SY  Y 
1
2
2
s1
N1

2
s2
N2
Chapter 13 – 20
t-test and Confidence Intervals
t( N1  N 2 2)
Y1 Y 2

SY1 Y2
The t-test is essentially creating a confidence interval around
the difference score. Rearranging the above formula, we can
calculate the confidence interval around the difference
between two means:
Y  Y  t (S
1
2
)
Y 1 Y 2
If this confidence interval overlaps with zero, then we
cannot be certain that there is a difference between the
means for the two samples.
Chapter 13 – 21
Why a t score and not a Z score?
Y  Y  t (S
1
2
)
Y 1 Y 2
• Use of the Z distribution has assumes the population
standard error of the difference is known. In practice,
we have to estimate it and so we use a t score.
• When N gets larger than 50, the t distribution
converges with a Z distribution so the results would be
identical regardless of whether you used a t or Z.
• In most sociological studies, you will not need to
worry about the distinction between Z and t.
Chapter 13 – 22
t-Test Example 1
Mean pay according to gender:
N
Mean Pay S.D.
Women
46
$10.29
.8766
Men
54
$10.06
.9051
10.06  10.29
.23
T

 1.23
2
2
.1785
.9051
.8766

54
46
What can we conclude
Y2  Y1
T( N1  N 2  2 ) 
about the difference in
2
2
s1
s
 2
wages?
N1 N 2
Chapter 13 – 23
t-Test Example 2
Mean pay according to gender:
N
Mean Pay S.D.
Women
57
$9.68
1.0550
Men
51
$10.32
.9461
T
10.32  9.68

.94612 1.05502

51
57
Y2  Y1
T( N1  N 2  2 ) 
s12
s22

N1 N 2
.64
 3.32
.1925
What can we conclude
about the difference in
wages?
Chapter 13 – 24
In-Class Exercise
Using these GSS income data, calculate a t-test statistic to
determine if the difference between the two group means is
statistically significant.
Mean
Standard Deviation
N
Men
$22,052.51
$17,734.92
434
Women
$14,331.21
$12,165.89
448
T(N1  N 2  2) 
T(880) 
Y 2  Y1
s12
s 22

N1 N 2
22,052.51  14,221.21
7,831.30

 7.62
2
2
1,027.18
17,734.92
12,165.89

434
448
Chapter 13 – 25