Transcript Independent Samples T Test

Hypothesis Testing
• Hypothesis testing is an inferential process
• Using limited information to reach a
general conclusion
• Observable evidence from the sample
data
• Unobservable fact about the population
• Formulate a specific, testable research
hypothesis about the population
Null Hypothesis
• no effect, no difference, no change, no
relationship, no pattern, no …
• any pattern in the sample data is due to
random sampling error
Type I Error
• A researcher finds evidence for a
significant result when, in fact, there is no
effect (no relationship) in the population.
• The researcher has, by chance, selected
an extreme sample that appears to show
the existence of an effect when there is
none.
• The p-value identifies the probability of a
Type I error.
Significance in Comparisons
• The key to becoming confident in a
sample mean is determining the standard
error of the mean.
• But if research was simply about
becoming confident in and reporting our
confidence in a sample mean, research
would not be all that interesting.
Hillary C. Feeling Thermometer
Gender
Female
Sample Standard
Squared
Mean
Error of
Standard
the
Error (Mean
Sample
Variance)
Mean
55.7
.96
.92
1.10
1.20
(n=994, s=30.21)
Male
46.4
(n=784, s=30.67)
TOTAL
Mean difference
Variance of the mean difference
Standard error of the mean
difference
51.2
(1778)
9.3
Here is the difference between the
means in our sample, but how
possible is it that the real
difference in the population = 0?
H0: The difference between the
means in the population = 0.
2.12
1.46
Significance in Comparisons
• The key to figuring out whether the
difference between two means is
significant (different from 0 in the
population), we need to calculate the
standard error of the mean difference.
Std. Err. of the mean difference
• Hillary Clinton Feeling Thermometer
Ratings, by Gender
Gender
Sample
Mean
Standard Error of
the Sample Mean
Squared Standard
Error (Mean Variance)
Female
55.7
.96
.92
1.10
1.20
(n=994, s=30.21)
Male
46.4
(n=784, s=30.67)
TOTAL
Mean difference
Variance of the mean difference
Standard error of the mean
difference
51.2
(1778)
9.3
Step 1: Calculate these as
before:
s .
 (n-1) 2.12
1.46
Std. Err. of the mean difference
• Hillary Clinton Feeling Thermometer
Ratings, by Gender
Gender
Sample
Mean
Standard Error of
the Sample Mean
Squared Standard
Error (Mean Variance)
Female
55.7
.96
.92
1.10
1.20
(n=994, s=30.21)
Male
46.4
(n=784, s=30.67)
TOTAL
Mean difference
51.2
(1778)
Step 2: Square the standard
errors
9.3
Variance of the mean difference
2.12
Standard error of the mean
difference
1.46
Std. Err. of the mean difference
• Hillary Clinton Feeling Thermometer
Ratings, by Gender
Gender
Sample
Mean
Standard Error of
the Sample Mean
Squared Standard
Error (Mean Variance)
Female
55.7
.96
.92
1.10
1.20
(n=994, s=30.21)
Male
46.4
(n=784, s=30.67)
TOTAL
Mean difference
51.2
(1778)
Step 3: Sum the squared
standard errors
9.3
Variance of the mean difference
2.12
Standard error of the mean
difference
1.46
Std. Err. of the mean difference
• Hillary Clinton Feeling Thermometer
Ratings, by Gender
Gender
Sample
Mean
Standard Error of
the Sample Mean
Squared Standard
Error (Mean Variance)
Female
55.7
.96
.92
1.10
1.20
(n=994, s=30.21)
Male
46.4
(n=784, s=30.67)
TOTAL
Mean difference
Variance of the mean difference
Standard error of the mean
difference
51.2
(1778)
9.3
Step 4: Take the square root
of the variance of the mean
difference
2.12
1.46
95% confidence interval
• Armed with the standard error of the
mean difference, we can proceed with
the familiar process of setting up our 95%
confidence interval:
• 95% c.i. = mean difference + (standard
error of mean difference)(critical t-score)
• 95% c.i. = 9.3 + (1.46)(1.96) = 6.44 <
mean difference in population < 12.16
Significance in Comparisons
• H0: Difference in means in the population
= 0.
• H1: Difference in means in the population
> 0.
• We are 95% confident that the difference
in the means in the population falls
somewhere between 6.44 and 12.16.
• Therefore, we can reject H0.
But we should be more precise. There is a 2.5% chance that the difference
between the means in the population is < 6.44, and 2.5% chance that the
difference between the means in the population is > 12.16.
We don’t really care what chance there is that the difference between the
means is > 12.16. We only care that it is > 0.
We don’t care about
the results in this tail.
2.5% of
results in
this tail
1.96 s.d.
34%
34%
47.5%
47.5%
1 s.d.
m
1 s.d.
2.5% of
results in
this tail
1.96 s.d.
Two-tailed tests of significance: used to test alternative hypotheses that
state that something is significantly different from zero.
One-tailed tests of significance: used to test alternative hypotheses that
state that something is significantly greater than or less than zero.
H0: Difference in means in the population = 0.
H1: Difference in means in the population > 0.
We don’t really care what chance there is that the difference between the means
is > 12.16. We only care that it is > 0.
So we take all of the
critical area under the
curve from here. .
2.5% of
results in
this tail
1.96 s.d.
34%
34%
47.5%
47.5%
1 s.d.
m
1 s.d.
2.5% of
results in
this tail
1.96 s.d.
Two-tailed tests of significance: used to test alternative hypotheses that
state that something is significantly different from zero.
One-tailed tests of significance: used to test alternative hypotheses that
state that something is significantly greater than or less than zero.
H0: Difference in means in the population = 0.
H1: Difference in means in the population > 0.
We don’t really care what chance there is that the difference between the means
is > 12.16. We only care that it is > 0.
So we take all of the
critical area under the
curve from here. .
And add it here. .
2.5% of
results in
this tail
1.96 s.d.
5% of values under the
curve
34%
34%
47.5%
47.5%
1 s.d.
m
1 s.d.
1.96 s.d.
One-tailed tests of significance: used to test alternative hypotheses that
state that something is significantly greater than or less than zero.
H0: Difference in means in the population = 0.
H1: Difference in means in the population > 0.
“What is the probability that the population difference could be as low as 0? Is this probability
less than .05?”
mean difference - (standard error of mean difference)(critical t-score)
9.3 – (1.46)(1.645) = 6.9
The lowest plausible value
for the difference in the
population is 6.9, NOT 0.
Therefore, H0 is rejected.
Now find the critical t-score
by determining your degrees
of freedom, then looking at
the .05 column (rather than
the .025 column).
34%
34%
47.5%
47.5%
1.96 s.d.
5% of values under the
curve
1 s.d.
m
1 s.d.
1.96 s.d.
Maximum Precision: Test Statistics and P-values
In our example, instead of saying “there is a less than 5% chance that the mean
differences in the population = 0”, we can be even more precise. We might ask:
“What is the exact probability that the mean difference = 0?” For this, we use test
statistics.
From our example, to generate a test statistic on the difference of means, we need
3 components:
1. The difference associated with the alternative hypothesis, H1 (which we
observed in the sample) = 9.3
2. The difference claimed by the null hypothesis, H0 = 0
3. The standard error of the difference = 1.46
t = (H1 – H0) / standard error of the difference
t = (9.3 – 0) / (1.46)
t = 6.37
p = .00000000012088
This is a rule of thumb:
take any sample statistic
and divide it by its
standard error. If the
result > 2, the coefficient
is significant.
Every t-score has an associated probability value, or p-value. Most statistical
packages cut of the measure at .000 (one in ten-thousand).
To find the standard error of the difference of proportions, we proceed exactly
as before, with only one difference—calculating the initial standard error of the
sample proportions.
Every other step is exactly
s=
 (p)(q)
the same.
(n-1)
--Proportions Favoring Gay Adoptions, by Gender
Gender
Sample
Proportion
Standard Error of
the Sample
Proportion
Variance of Sample
Proportion
Female
.497
.016
.00027
.018
.00032
(n=922)
Male
.394
(n=734)
Difference in proportions
.103
Variance of the difference
.00059
Standard error of the difference
.0243