DIRECTIONAL HYPOTHESIS
• The 1-tailed test:
– Instead of dividing alpha
by 2, you are looking for
unlikely outcomes on
only 1 side of the
distribution
– No critical area on 1
side—the side depends
upon the direction of the
hypothesis
– In this case, anything
greater than the critical
region is considered
“non-significant”
1.2
1.0
.8
.6
.4
.2
0.0
-2.07
-1.21
-.36
-1.96
-1.65
.50
0
1.36
Normal Curve, Mean = .5, SD = .7
2.21
3.07
Non-Directional & Directional Hypotheses
• Nondirectional
– Ho: there is no effect:
(X = µ)
– H1: there IS an effect:
(X ≠ µ)
– APPLY 2-TAILED TEST
• 2.5% chance of error in
each tail
-1.96
1.96
• Directional
– H1: sample mean is larger
than population mean
(X > µ)
– Ho x ≤ µ
– APPLY 1-TAILED TEST
•
5% chance of error in
one tail
1.65
Why we typically use 2-tailed
tests
• Often times, theory or logic does allow us
to prediction direction – why not use 1tailed tests?
• Those with low self-control should be more likely to
engage in crime.
• Rehabilitation programs should reduce likelihood
of future arrest.
• What happens if we find the reverse?
– Theory is incorrect, or program has the unintended
consequence of making matters worse.
STUDENT’S t
DISTRIBUTION
– We can’t use Z distribution with smaller
samples (N<100) because of large standard
errors
– Instead, we use the t distribution:
– Approximately normal beginning when sample size > 30
– Probabilities under the t distribution are
different than from the Z distribution for small
samples
– They become more like Z as sample size (N) increases
THE 1-SAMPLE CASE
– 2 Applications
• Single sample means (large N’s) (Z statistic)
– May substitute sample s for population standard
deviation, but then subtract 1 from n
» s/√N-1 on bottom of z formula
• Smaller N distribution (t statistic), population SD
unknown
STUDENT’S t
DISTRIBUTION
– Find the t (critical) values in App. B of Healey
– “degrees of freedom”
• # of values in a distribution that are free to vary
• Here, df = N-1
– When finding t(critical) always use lower df associated
with your N
Practice:
ALPHA
.05
.01
.10
.05
TEST
2-tailed
1-tailed
2-tailed
1-tailed
N
57
25
32
15
t(Critical)
Example: Single sample means, smaller N
and/or unknown pop. S.D.
1.
A random sample of 26 sociology grads scored an
average of 458 on the GRE sociology test, with a
standard deviation of 20. Is this significantly higher
than the national average (µ = 440)?
2.
The same students studied an average of 19 hours a
week (s=6.5). Is this significantly different from the
overall average (µ = 15.5)?
•
USE ALPHA = .05 for both
1-Sample Hypothesis Testing
(Review of what has been covered so far)
1.
If the null hypothesis is correct, the estimated
sample statistic (i.e., sample mean) is going to be
close to the population mean
2.
When we “set the criteria for a decision”, we are
deciding how far the sample statistic has to fall from
the population mean for us to decide to reject H0
–
Deciding on probability of getting a given sample statistic if H0
is true
3 common probabilities (alpha levels) used are .10, .05 & .01
–
•
These correspond to Z score critical values of 1.65, 1.96 & 258
1-Sample Hypothesis Testing
(Review of what has been covered so far)
3. If test statistic we calculate is beyond the
critical value (in the critical region) then we
reject H0
–
Probability of getting test stat (if null is true) is small
enough for us to reject the null
–
In other words: “There is a statistically significant
difference between population & sample means.”
4. If test statistic we calculate does not fall in
critical region, we fail to reject the H0
–
“There is NOT a statistically significant difference…”
2-Sample Hypothesis Testing (intro)
– Apply when…
• You have a hypothesis that the means (or proportions) of a
variable differ between 2 populations
– Components
– 2 representative samples – Don’t get confused here (usually
both come from same “sample”)
– One interval/ratio dependent variable
– Examples
» Do male and female differ in their aggression (# aggressive
acts in past week)?
» Is there a difference between MN & WI in the proportion
who eat cheese every day?
– Null Hypothesis (Ho)
• The 2 pops. are not different in terms of the dependent
variable
2-SAMPLE HYPOTHESIS TESTING
• Assumptions:
– Random (probability) sampling
– Groups are independent
– Homogeneity of variance
» the amount of variability in the D.V. is about equal in
each of the 2 groups
– The sampling distribution of the difference between
means is normal in shape
2-SAMPLE HYPOTHESIS TESTING
• We rarely know population S.D.s
– Therefore, for 2-sample t-testing, we must use 2 sample S.D.s,
corrected for bias:
» “Pooled Estimate”
• Focus on the t statistic:
t (obtained) = (X – X)
σ x-x
• we’re finding the
difference between the two means…
…and standardizing this difference with the pooled estimate
2-SAMPLE HYPOTHESIS TESTING
2-Sample Sampling
Distribution
– difference between
sample means (closer
sample means will have
differences closer to 0)
• t-test for the
difference between 2
sample means:
• Addresses the question
of whether the
observed difference
between the sample
means reflects a real
difference in the
population means or is
due to sampling error
-2.042
0
2.042
ASSUMING THE NULL!
Applying the 2-Sample t Formula
– Example:
• Research Hypothesis (H1):
– Soc. majors at UMD drink more beers per month than
non-soc. majors
– Random sample of 205 students:
» Soc majors: N = 100, mean=16, s=1.0
» Non soc. majors: N = 105, mean=15, s=0.9
» Alpha = .01
» FORMULA:
t(obtained) = X1 – X2
pooled estimate
Answers
• Null hypothesis:
– “There is no difference in mean number of
fights between inmates with tattoos and
inmates without tattoos.”
• Use a 1 or 2-tailed test?
– One-tailed test because the theory predicts
that inmates with tattoos will get into MORE
fights.
Answers
• Calculations
– Obtained value
• Reject the null?
– Yes because the t(obtained) (19.09) is greater than the
t(critical, one-tail, df=398) (1.658)
• This t value indicates there are 19.09 standard error units that
separate the two mean values
– VERY unlikely we got this big a difference due to sampling error
• Research hypothesis restated as non-directional:
– “There is a difference in the mean number of fights
reported by inmates with tattoos and inmates without
tattoos.”
• Would you come to a different conclusion if you
used a 2-tailed test?
– No, because 19.09 is still well beyond the 2-tailed critical
value (1.980).
2-Sample Hypothesis Testing in SPSS
• Independent Samples t Test Output:
– Testing the Ho that there is no difference in number
of adult arrests between a sample of individuals who
were abused/neglected as children and a matched
control group.
Group Statistics
SUB_CNLX
SUBJECT / CONTROL
ADULT_S NUMBER
1 Subjects
OF ADULT OFFENSES 2 Controls
N
397
192
Std. Error
Mean
Std. Deviation
Mean
9.24
13.821
.694
4.43
7.002
.505
Interpreting SPSS Output
• Difference in mean # of
adult arrests between
those who were abused
as children & control
group
Independent Samples Test
Levene's Test for
Equality of Variances
F
ADULT_S NUMBER
Equal variances
OF ADULT OFFENSES assumed
Equal variances
not assumed
36.864
Sig.
.000
t-test for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
4.547
587
.000
4.81
1.058
2.732
6.887
5.604
585.783
.000
4.81
.858
3.124
6.495
Interpreting SPSS Output
• t statistic, with degrees of freedom
Independent Samples Test
Levene's Test for
Equality of Variances
F
ADULT_S NUMBER
Equal variances
OF ADULT OFFENSES assumed
Equal variances
not assumed
36.864
Sig.
.000
t-test for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
4.547
587
.000
4.81
1.058
2.732
6.887
5.604
585.783
.000
4.81
.858
3.124
6.495
Interpreting SPSS Output
• “Sig. (2 tailed)”
– gives the actual probability of making a Type I (alpha)
error
• a.k.a. the “p value” – p = probability
Independent Samples Test
Levene's Test for
Equality of Variances
F
ADULT_S NUMBER
Equal variances
OF ADULT OFFENSES assumed
Equal variances
not assumed
36.864
Sig.
.000
t-test for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
4.547
587
.000
4.81
1.058
2.732
6.887
5.604
585.783
.000
4.81
.858
3.124
6.495
“Sig.” & Probability
• Number under “Sig.” column is the exact
probability of obtaining that t-value (finding that
mean difference) if the null is true
– When probability > alpha, we do NOT reject H0
– When probability < alpha, we DO reject H0
• As the test statistics (here, “t”) increase, they
indicate larger differences between our obtained
finding and what is expected under null
– Therefore, as the test statistic increases, the
probability associated with it decreases
Example 2: Education & Age
at which First Child is Born
H0: There is no relationship between whether an individual has a
college degree and his or her age when their first child is born.
Group Statistics
AGEKDBRN R'S AGE
WHEN 1ST CHILD BORN
COLDGREE R has
4-year college degree
1.00 No -- less than a
Bachelor's degree
2.00 Yes -- a Bachelor's
or Graduate degree
N
Mean
Std. Deviation
Std. Error
Mean
812
22.74
4.826
.169
222
26.82
5.343
.359
Independent Samples Test
Levene's Test for
Equality of Variances
F
AGEKDBRN R'S AGE
Equal variances
WHEN 1ST CHILD BORN assumed
Equal variances
not assumed
4.547
Sig.
.033
t-test for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
-10.926
1032
.000
-4.09
.374
-4.824
-3.355
-10.310
326.163
.000
-4.09
.397
-4.869
-3.309
Education & Age at which First Child is Born
1. What is the mean difference in age?
2. What is the probability that this t statistic is due to
sampling error?
3. Do we reject H0 at the alpha = .05 level?
4. Do we reject H0 at the alpha = .01 level?
Independent Samples Test
Levene's Test for
Equality of Variances
F
AGEKDBRN R'S AGE
Equal variances
WHEN 1ST CHILD BORN assumed
Equal variances
not assumed
4.547
Sig.
.033
t-test for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
-10.926
1032
.000
-4.09
.374
-4.824
-3.355
-10.310
326.163
.000
-4.09
.397
-4.869
-3.309