DSC 211
CLASS NOTES 22
Page 1
COMPARING TWO or MORE POPULATIONS
EXAMPLE
Issue:
Variability/consistency in service times for two bank tel ers.
Teller1*
Teller2*
(There are 3 worksheets that are a part of class notes 22.)
FTest
Suppose they had different training; we think tel er 1's training might be better.
7.2
10.9
SO FAR, IN CLASS NOTES 21:
Bus Question:
From our sample data, can we conclude that teller 1 is more consistent in
5.4
6
(1) COMPARING MEANS (MU'S):
service time than teller 2?
3.7
6.7
Data:
We recorded the service time (min) for 100 random customers for ea of the 2 tel ers.
Sample Std Devs
6.4
8.3
(Required condition:
Populations are approximately Normal)
Calculations with Excel:
We believe that the two populations of times are approximately Normal.
minutes
11.2
14.3
Test Stat:
Analysis:
FTest
minutes
8.7
3.7
[ s1^2/n1 + s2^2/n2 ] ^.5
df =
…
ttest: 2 sample assuming unequal variances
1. Hypotheses:
Ho:
var1/var2 >= 1
var1 >= var2
9.3
7.6
Ha:
var1/var2 < 1
var1 < var2
6.9
8
(Required condition:
Differences are approximately Normal)
2. Test Statistic:
9.1
8.1
Test Stat:
=TINV and =TDIST, or
3. Sig Level:
alpha = .05
8.6
1.3
s(D) / n^.5
df = n 1
ttest: paired 2 sample for means
4. Reject Rule:
Reject Ho if F < value of F that only 5% are below
Why smal F  and not large F??
7.3
10.6
(2) COMPARING PROPORTIONS (P'S):
Reject Ho if F <
6.1
8
5. Calculations:
Data set in columns L and M:
Observations in minutes per service transaction
6.7
10.8
4.4
10.5
Test Stat:
=NORMINV and =NORMDIST
7.4
9.7
F =
7.6
3.6
pvalue =
8.9
7.3
NOW THREE MORE SITUATIONS:
6. Conclusion:
For turnin,
10.9
7.8
(3) COMPARING THE VARIATION or SPREAD or CONSISTENCY of two populations  ARE THEY DIFFERENT?
Bus Answer:
print this
7.2
4.4
box
4.2
4
10.5
7.9
BusQues:
Is the variation (are the variances or stdevs) different? 
9.3
12.5
1. Hyp:
s1^2 =
10.3
s2^2 =
39.2
8.2
6.2
NOTE Can use the excel tool below for steps 4 and 5 of analysis:
5.8
10.3
2. TestStat:
Req Condition:
Populations are approx Normal.
(The tool is set up for onesided tests only.)
7.2
8
FTest TwoSample for Variances
7
4.3
NOTE:
If Ho is true, the ratio should NOT BE TOO DIFFERENT than one.
6.1
10.4
Tel er1*
Tel er2*
9.1
7.6
Mean
7.91
7.84
8.1
11.3
Variance
3.35
10.95
9.4
6.5
Observations
100
100
8.4
13.4
3. Sig Level:
alpha = .05
df
99
99
10.1
10
4. Reject
Reject Ho if F < smal less than 1 value
or F > large greater than 1 value
F
0.306
8.1
3.5
Rule:
smal value seen .025 or less =
0.207
=FINV(.975,7,9)
Note:
first argument is prob above.
P(F<=f) onetail
0.000
5.6
6.5
large value seen .025 or less =
4.197
=FINV(.025,7,9)
F Critical onetail
0.717
6.2
9
Reject Ho if F < 0.207 or F > 4.197
7.3
10.6
5. Calc's:
Data Set:
samples from population (1) and population (2)
Ave(xbar)
StDev
9.9
5.9
(1)
7
4
9
12
8
6
9
14
8.6
3.2
5.9
14.5
(2)
10
7
13
18
4
8
21
20
5
8
11.4
6.3
8.1
4.7
s1^2 =
10.27
8.4
2.2
s2^2 =
39.16
7.3
7.1
F =
0.262
7.7
5.3
pvalue =
0.091
9.1
4.1
8.4
5.2
6. Concl's:
Do not reject Ho.
7.8
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 Spring '07
 Dunne
 Normal Distribution, Variance, Student's ttest

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