Transcript 10-w11-stats250-bgunderson-chapter-11-ci-for

Author(s): Brenda Gunderson, Ph.D., 2011
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Pooled CI for Difference in Population Means
Pooled Two Independent-Samples
t Confidence Interval for m1 - m2
x1  x 2   t * pooled s.e.( x1  x 2 ) 
where pooled s.e. x1  x 2   s p
1
1

n1 n 2
(n1  1) s12  (n 2  1) s 22
and s p 
n1  n 2  2
and t* is the value from a t (n1 + n2 – 2) distribution.
This interval requires we have independent random samples from
normal populations with equal population variances.
If the sample sizes are large (both > 30), the assumption of normality
is not so crucial and the result is approximate.
Learning about the Difference in Two Popul Means
Computer Package Notes:


Most provide results for both unpooled and pooled two indep
samples t procedures.
Some provide Levene’s test for assessing if popul variances
can be assumed equal.



Null hypothesis = population variances are equal.
Small p-value  reject null hypothesis and conclude pooled
procedure should not be used. (10% level used, lab module 8.)
Graphical tool for assessing equal population variance
is side-by-side boxplots and comparing the IQRs.
If lengths of boxes and ranges very different, pooled method
may not be reasonable.
Learning about the Difference in Two Popul Means
Pooled or Unpooled? Guidelines:



If sample std devs similar, assumption of common
population variance reasonable and pooled can be used.
Note: pooled and unpooled standard errors equal anyway.
Advantage for pooled version is df is simpler.
But sample std devs almost never exactly equal.
So how similar is similar enough?
If larger std dev from group with larger sample size,
pooled acceptable, conservative (produces wider interval).
If smaller std dev from group with larger sample size,
pooled produces misleading narrower interval.
Bottom-line: Pool if reasonable;
but if sample std devs not similar, we have unpooled.
Try It! Comparing Reading scores
Study compared reading scores for students taught by
certified teachers vs those taught by uncertified teachers
Group Statistics
READING
SCORE
GROUP
1
2
N
10
10
Mean
34.21
28.20
Std. Deviation
9.118
7.036
Std. Error
Mean
2.883
2.225
Independent Sam ple s Te st
Levene's Test for
Equality of Varianc es
F
READING
SCORE
Equal variances
as sumed
Equal variances
not as sumed
.104
Sig.
.751
t-t est for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
St d. Error
Difference
1.651
18
.116
6.01
3.642
1.651
16.913
.117
6.01
3.642
m average reading score for popul of students taught by certified teachers
m2 = average reading score for popul of students taught by uncertified teachers
 1=
Try It! Comparing Reading scores (continued)
a.Assumption for pooled two indep samples CI is 2 populations have
same standard deviation. Look at two sample std devs and Levene's
test result. Does assumption seem to hold (at 10% level)? Explain.
Group Statistics
READING
SCORE
GROUP
1
2
N
10
10
Mean
34.21
28.20
Std. Deviation
9.118
7.036
Std. Error
Mean
2.883
2.225
Independent Sam ple s Te st
Levene's Test for
Equality of Varianc es
F
READING
SCORE
Equal variances
as sumed
Equal variances
not as sumed
.104
Sig.
.751
t-t est for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
St d. Error
Difference
1.651
18
.116
6.01
3.642
1.651
16.913
.117
6.01
3.642
Try It! Comparing Reading scores (continued)
b.Give a 95% confidence interval for the difference in the population
Group Statistics
mean reading scores.
READING
SCORE
GROUP
1
2
N
10
10
Mean
34.21
28.20
Std. Deviation
9.118
7.036
Std. Error
Mean
2.883
2.225
Independent Sam ple s Te st
Levene's Test for
Equality of Varianc es
F
READING
SCORE
Equal variances
as sumed
Equal variances
not as sumed
.104
Sig.
.751
t-t est for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
St d. Error
Difference
1.651
18
.116
6.01
3.642
1.651
16.913
.117
6.01
3.642
Yes or No (please click in your answer)
c. Based on the interval (-1.64, 13.66),
does there appear to be a difference in the
mean reading scores for the two populations?
Try It! Stroop Word Color Test

Stroop Word Color Test:



100 color names shown in colors different from word.
Task = correctly identify display color of each word.
Experiment:


Time to complete test for n = 16 individuals after
consumed alcohol and for n = 16 other individuals after
consumed placebo drink.
Each group balanced with 8 men and 8 women.
Ready to try it?
GREEN
RED
BLACK
BLUE
Try It! Stroops’ Word Color Test
Group
1 = alcohol
2 = placebo


Sample
size
16
16
Sample
mean
113.75
99.87
Sample standard
deviation
22.64
12.04
Assume two samples are indep random samples,
Model for completion time is normal for each popul.
As we go through the questions,
have your clicker ready for parts (a) and (e).
a. What graph(s) would you make to check the
normality condition?
A)
B)
C)
D)
Make bar chart(s)
Make qq plot(s)
Make histogram(s)
Make pie chart(s)
b. How did the two groups compare descriptively?
Group
1 = alcohol
2 = placebo
Sample
size
16
16
Sample
mean
113.75
99.87
Sample standard
deviation
22.64
12.04
c. Which procedure?
Pooled or
Unpooled?
Group
1 = alcohol
2 = placebo
Sample
size
16
16
Sample
mean
113.75
99.87
Sample standard
deviation
22.64
12.04
d. Calculate a 95% CI for the difference in popul means
Group
1 = alcohol
2 = placebo
Sample
size
16
16
Sample
mean
113.75
99.87
Sample standard
deviation
22.64
12.04
x1  x 2   t * s.e.( x1  x 2 ) 
s.e. x1  x 2  
Think about part (e) with your clicker ready …
s12 s 22

n1 n 2
Yes or No (please click in your answer)
e. Based on the interval (0.30, 27.46), can we conclude the
population means for the two groups are different?