Title: Comparing Two Groups
1Comparing Two Groups Means or Proportions
- Independent Samples t-tests
2Review
- Confidence Interval for a Mean
- Slap a sampling distribution over a sample mean
to determine a range in which the population mean
has a particular probability of beingsuch as 95
CI. - If our sample is one of the middle 95, we know
that the mean of the population is within the CI.
Significance Test for a Mean Slap a sampling
distribution over a guess of the population mean
to determine if the sample has a very low
probability of having come from a population
where the guess is truesuch as a-level .05. If
our sample mean is in the outer 5, we know to
reject the guess, our sample has a low chance of
having come from a population with the mean we
guessed.
Y-bar?
Y-bar?
µ?
2.5
2.5
2.5
2.5
-1.96z
1.96z
Y-bar 95 CI Y-bar /- 1.96 (s.e.)
-1.96z
1.96z
µoguess z or t (Y-bar - µo)/ s.e.
sampling distribution the way statistics for
samples of a certain size would stack up or be
distributed after all possible samples are
collected
3Review
- Lets collect some data on educational
aspirations and produce a 95 confidence interval
to tell us where the population parameter likely
falls and then lets do a test of significance
where we guess that average aspiration will be 16
years. - I collected a sample of 625 kids who reported
their educational aspirations where 12 high
school, 16 equals 4 years of college and so
forth. The average for the sample was 15 years
with a standard deviation of 2 years. - 95 confidence interval 95 CI Sample Mean /-
z s.e. - Find the standard error of the sampling
distribution - s / ?n 2/v625 2/25 0.08
- Build the width of the Interval. 95 corresponds
with a z of /- 1.96. - /- z s.e 1.96 0.08 0.157
- Insert the mean to build the interval
- 95 CI Sample Mean /- z s.e 15 /-
0.157 - The interval 14.84 to 15.16
- We are 95 confident that the population mean
falls between these values. (What does this say
about my guess???)
4Review
- Lets collect some data on educational
aspirations and produce a 95 confidence interval
to tell us where the population parameter likely
falls and then lets do a test of significance
where we guess that average aspiration will be 16
years. - I collected a sample of 625 kids who reported
their educational aspirations where 12 high
school, 16 equals 4 years of college and so
forth. The average for the sample was 15 years
with a standard deviation of 2 years. - Significance Test z or t (Y-bar - µo)/ s.e.
- Decide ?-level (? .05) and nature of test
(two-tailed) - Set critical z or t (/- 1.96)
- Make guess or null hypothesis,
- Ho ? 16
- Ha ? ? 16
- Collect and analyze data
- Calculate Z or t z/t Y-bar - ?o (s.e. s/vn
2/v625 2/25 .08) -
s.e. - z/t (15 16)/.08 -1/.08 -12.5
- Make a decision about the null hypothesis (reject
the null -12.5 lt -1.96) - Find the P-value (look up 12.5 in z or t table).
P lt .0001 - It is extremely unlikely that our sample came
from a population where the mean is 16.
5Comparing Two Groups
- Were going to move forward to more sophisticated
statistics, building on what we have learned
about confidence intervals and significance
tests. - Sociologists look for relationships between
concepts in the social world. - For example
- Does ones sex affect income?
- Focus on the relationship between the concepts
Sex and Income - Does ones race affect educational attainment?
- Focus on the relationship between the concepts
Race and Educational Attainment
I love sophisticated statistics!
6Comparing Two Groups
- In this section of the course, you will learn
ways to infer from a sample whether two concepts
are related in a population. - Independent variable (X) That which causes
another variable to change when it changes. - Dependent variable (Y) That which changes in
response to change in another variable. - X ? Y
- (X Sex or Race) (Y Income or
Education) - The statistical technique you use will depend of
the level of measurement of your independent and
dependent variablesthe statistical test must
match the variables! - Levels of Measurement Nominal, Ordinal,
Interval-Ratio
7Comparing Two Groups
- The test you choose depends on level of
measurement - Independent Dependent Statistical Test
- Dichotomous Interval-ratio Independent Samples
t-test - Dichotomous
- Nominal Nominal Cross Tabs
- Ordinal Ordinal
- Dichotomous Dichotomous
- Nominal Interval-ratio ANOVA
- Ordinal Dichotomous
- Dichotomous
- Interval-ratio Interval-ratio Correlation and
OLS Regression - Dichotomous
8Comparing Two Groups
- Independent Dependent Statistical Test
- Dichotomous Interval-ratio Independent Samples
t-test - Dichotomous
- An independent samples t-test is concerned with
whether a mean or proportion is equal between two
groups. For example, does sex affect income?
? Income
? Income
µ
µ
Womens mean
Mens Mean ???
9Comparing Two Groups
- Independent Samples t-tests
- Earlier, our focus was on the mean. We used the
mean of the sample (statistic) to infer a range
for what our population mean (parameter) might be
(confidence interval) or whether it was like some
guess or not (significance test). - Now, our focus is on the difference in the mean
for two groups. We will use the difference of
the sample (statistic) to infer a range for what
our population difference (parameter) might be
(confidence interval) or whether it is like some
guess (significance test).
10Comparing Two Groups
- The difference will be calculated as such
- D-bar Y-bar2 Y-bar1
- For example
- Average Difference in Income by Sex
- Male Average Income Female Average Income
- (What would it mean if mens income minus womens
income equaled zero?)
11Comparing Two Groups
- Like the mean, if one were to take random sample
after random sample from two groups and calculate
and record the difference between groups each
time, one would see the formation of a Sampling
Distribution for D-bar that was normal and
centered on the two populations difference.
average difference between two groups samples
Sampling
Distribution of D-bar
Z -3 -2 -1 0 1 2 3
95 Range
12Comparing Two Groups
- Its Power Time Again!
- Using just a sample, our statistics will allow us
to pinpoint the difference between two groups in
the population (confidence interval) or to
determine whether our sample could have come from
a population with a difference between two groups
that we guessed (significance test).
13Comparing Two Groups
- So the rules and techniques we learned for means
and proportions apply to the differences in
groups means and proportions. - One creates sampling distributions to create
confidence intervals and do significance tests in
the same ways. - However, the standard error of D-bar has to be
calculated slightly differently. - For Means (s1)2
(s2)2 - s.e. (s.d. of the sampling distribution)
n1 n2 - For Proportions
- s.e. ?1 (1 - ?1) ?2
(1 - ?2) - n1 n2
14Comparing Two Groups
- Calculating a Confidence Interval for the
Difference between Two Groups Means - By slapping the sampling distribution for the
difference over our samples difference between
groups, D-bar, we can find the values between
which the population difference is likely to be. - 95 C.I. D-bar /- 1.96 (s.e.)
- (Y-bar2 Y-bar1) /- 1.96
(s.e.) - Or (?2 ?1) /- 1.96 (s.e.)
- 99 C.I. D-bar /- 2.58 (s.e.)
- (Y-bar2 Y-bar1) /- 2.58
(s.e.) - Or (?2 ?1) /- 2.58 (s.e.)
15Comparing Two Groups
- EXAMPLE
- We want to know what the likely difference is
between male and female GPAs in a population of
college students with 95 confidence. - Sample 50 men, average gpa 2.9, s.d. 0.5
- 50 women, average gpa 3.1, s.d. 0.4
- 95 C.I. Y-bar2 Y-bar1 /- 1.96 s.e.
- Find the standard error of the sampling
distribution - s.e. ?(.5)2/ 50 (.4)2/50 ? .005
.003 ? .008 0.089 - Build the width of the Interval. 95 corresponds
with a z or t of /- 1.96. - /- z s.e /- 1.96 0.089 /- 0.174
- Insert the mean difference to build the interval
- 95 C.I. (Y-bar2 Y-bar1) /- 1.96 s.e.
3.1 - 2.9 /- 0.174 0.2 /- 0.174 -
- The interval 0.026 to 0.374
16Comparing Two Groups
- Conducting a Test of Significance for the
Difference between Two Groups Means - By slapping the sampling distribution for the
difference over a guess of the difference between
groups, Ho, we can find out whether our sample
could have been drawn from a population where the
difference is equal to our guess. - Two-tailed significance test for ?-level .05
- Critical z or t /- 1.96
- To find if there is a difference in the
population, - Ho ?2 - ?1 0
- Ha ?2 - ?1 ? 0
- Collect Data
- Calculate z or t z or t (Y-bar2 Y-bar1)
(?2 - µ1) -
s.e. - Make decision about the null hypothesis (reject
or fail to reject) - Report P-value
17Comparing Two Groups
- EXAMPLE
- We want to know whether there is a difference in
male and female GPAs in a population of college
students. - Two-tailed significance test for ?-level .05
- Critical z or t /- 1.96
- To find if there is a difference in the
population, - Ho ?2 - ?1 0
- Ha ?2 - ?1 ? 0
- Collect Data
- Sample 50 men, average gpa 2.9, s.d. 0.5
- 50 women, average gpa 3.1, s.d. 0.4
- s.e. ?(.5)2/ 50 (.4)2/50 ? .005 .003
? .008 0.089 - Calculate z or t z or t 3.1 2.9
0 0.2 2.25 -
0.089 0.089 - Make decision about the null hypothesis Reject
the null. There is enough difference between
groups in our sample to say that there is a
difference in the population. 2.25 gt1.96 - Find P-value p or (sig.) .0122
- We have a 1.2 chance that the difference in our
sample could have come from a population where
there is no difference between men and women.
That chance is low enough to reject the null, for
sure!
18Comparing Two Groups
- The steps outlined above for
- Confidence intervals
- And
- Significance tests
- for differences in means are the same you would
use for differences in proportions. - Just note the difference in calculation of the
standard error for the difference.
19Comparing Two Groups
- Now lets do an example with SPSS, using the
General Social Survey.