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Lecture

Lecture 5

11 Pages
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Department
Quantitative Methods
Course Code
QMS 202
Professor
Clare Chua

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2/7/2011
1
Two-Sample Tests
Chapter 11
Page 452
Objectives
1. Comparing The Means of Two Independent
Populations
Z-test for the difference between two means.
Pooled-Variance t-test for the difference between
Two Means.
2. Comparing The Means of Two Related Populations
3. Comparing Two Population Proportions
Z-Test
4. F-Test For The Difference Between Two Variances
Hypothesis Test
One Population
Hypothesis Test
Two Populations
Sample
Independent Sample
dependent
2 MEANS2 Proportions
Check the 4
conditions
If yes
Normal?
Yes
No
STOP
Equal V2?unequal V2
V1and V2known?
Z-test Pooled Variance t-test
Z-test
Paired t-test
Yes
No
YesYe s
No
T-test
“not” Pooled variance
Comparing Dependent and Independent
Samples
Independent SamplesIndependent Samples
The samples chosen at random are
not relatednot relatedto each other.
We wish to study the mean incomes
of companies Xand Y
Dependent SampleDependent Sample
Dependent samples are
characterized by a measurement,
then some type of intervention,
followed b
y
another measurement.
oGGGGGGGkG
GGf
of
companies
X
and
Y
.
We select a random sample of 28
employees from the Company X and
a sample of 19 employees in
Company Y.
A person cannot be an employee in
both companies.
The samples are independentindependent,
that is, unrelated.
y
Paired samples are also
dependent because the same
individual or item is a member of
both samples.
Examples: 10 participants in a
marathon were weighed prior to
and after competing in the race.
We wish to study the mean
amount of weight loss.
There are two types of dependent samples:
Characterized by a Measurement,
followed by an intervention of
some type, and then another
measurement
Called a before” and after study.
Example:
Suppose we want to show that, by
placing speakers in the production
area and playing soothing music,
weareable toincrease
Characterized by matching or
pairing of the observations.
Example:
Suppose you are selling a house
and you asked for two
appraisals of your property.
Suppose an industrial
psychologist wishes to study
the intellectual similarities of
we
are
able
to
increase
production.
We begin by selecting a sample of
workers and measuring their
output under current conditions,
The speakers are then installed in
the production area, and we again
measure the output of the same
workers.
There are 2 measurements: (1)
before placing the speakers
(2) After placing the speakers.
the
intellectual
similarities
of
newly married couples. She
selects a sample of
newlyweds.
Next she administers a standard
intelligence test to both the
man and woman to determine
the difference in the scores
Notice the matching that
occurred: comparing the
scores of the man and the
woman.
Comparing Dependent and Independent
Samples
2
2
2
1
2
1
2121
calc
n
1
n
1
)()xx(
z
Independent SamplesIndependent Samples Dependent SampleDependent Sample
z-test: Test of two indep
pop, VKnown
n
S
D
t
D
D
calc
Paired t-test:
Test of two dep pop, Vequal but
unknown
Where D is “difference scores”
df
=
n
-
1
Section 11.1Section 11.2
)1n()1n(
s)1n(s)1n(
s
21
2
22
2
11
2
p
)
n
1
n
1
(s
)()xx(
t
21
2
p
2121
calc
t-test: Test of two indep pop,
Vequal but unknown
df=(n1-1)+(n2-1)=n1+n2-2
t-test: Test of two indep pop,
Vunequal and unknown
2
2
2
1
2
1
2121
calc
n
s
n
s
)()xx(
t
P
P
)1n(n
s
)1n(n
s
)
n
s
()
n
s
(
df
2
2
2
4
2
1
2
1
4
1
2
2
2
1
2
1
df n
1
The seven-step Method of Hypothesis Testing
Steps:
1. Define the parameters: P1and P2
2. State the hypothesis
Ho: P1= P2 (or Ho: P1-P2 = 0)
Ha: P1 P2 (or Ha: P1-P2 0)
3. Determine the appropriate test (refer to the flowchart)
4. State the conditions/assumptions: (1) Normal? (2) Equal variance?
5
rejection region
5
.
-
rejection
region
6. Calculate the test statistics value
7. Make the statistical decision and Managerial conclusion
To reject Ho or DO NOT reject Ho
Compare the test statistics with the critical value (critical value approach)
or
compare p-value with .(p-value approach)
www.notesolution.com
2/7/2011
2
Illustration:
Z Test for Difference Between Two
Means (Vknown)
A company that makes bolts that are used on an automotive
component uses two machines to make these bolts. It has been
determined by past studies that the standard deviation of the bolt
diameters made by machine 1 is 0.025 mm. and the standard
deviation of the bolt diameters of machine 2 is 0.022 mm. Both
machines have a dial to set for the desired diameter. Recently they
used both machines to fill a large order. The customer found that
many of the boltsfrom acertain package were too large and made a
many
of
the
bolts
from
a
certain
package
were
too
large
and
made
a
complaint. It was determined that the package in question was
made by machine 2. The manufacturer decided to take samples of
the bolts from both machines to test to see whether the mean
diameter of the bolts from machine 2 was significantly larger than
the mean diameter from machine 1 when the dial was set to the
same diameter on each machine. The sample of 100 bolts from
machine 1 had a mean diameter of 5.023 mm and a sample of 100
bolts from machine 2 had a mean diameter of 5.031 mm when the
dial on both machines was set at 5.00 mm. At the 5% level of
significance what is the conclusion?
Calculator steps
From the Main Menu select:
STAT F3(test)F1(Z) F2(2-S) then enter the following items:
2-Sample ZTest
Data: F2(Var) z
1: F2(<2) z
11: 0.025 EXE
12: 0.022 EXE
1: 5.023 EXE
n1 : 100 EXE
2: 5.031 EXE
n2 : 100 EXE
Now key EXE or F1(Calc)
CFX-9850GB Calculator
Z Test for Difference Between Two Means (Vknown)
A company that makes bolts that are used on an automotive
component uses two machines to make these bolts. It has been
determined by past studies that the standard deviation of the bolt
diameters made by machine 1 is 0.025 mm. and the standard
deviation of the bolt diameters of machine 2 is 0.022 mm. Both
machines have a dial to set for the desired diameter. Recently they
used both machines to fill a large order. The customer found that
many of the boltsfrom acertain package were too large and made a
many
of
the
bolts
from
a
certain
package
were
too
large
and
made
a
complaint. It was determined that the package in question was
made by machine 2. The manufacturer decided to take samples of
the bolts from both machines to test to see whether the mean
diameter of the bolts from machine 2 was significantly larger
than the mean diameter from machine 1 when the dial was set to
the same diameter on each machine. The sample of 100 bolts from
machine 1 had a mean diameter of 5.023 mm and a sample of 100
bolts from machine 2 had a mean diameter of 5.031 mm when the
dial on both machines was set at 5.00 mm. At the 5% level of
significance what is the conclusion?
The calculator will now show the
results:
2-sample ZTest
1< 2
z=-2.4022
p
=8 1465E
03 (=0 0081465)
p
=8
.
1465E
-
03
(=0
.
0081465)
1=5.023
2=5.031
n1 =100
n2 =100
Illustration
Z Test for Difference Between Two Means (Vknown)
The operations manager at a light bulb factory wants to
determine if there is any difference in the average life
expectancy of bulbs manufactured on two different
types of machines. The process standard deviation of
machine I is 110 hours and of machine II is 125 hours.
A random sample of 25 light bulbs obtained from
machine Iindicates asample mean of 375 hours and
machine
I
indicates
a
sample
mean
of
375
hours
,
and
a similar sample of 25 from machine II indicates a
sample mean of 362 hours.
a) Using the 0.05 level of significance, is there any
evidence of a difference in the average life of bulbs
produced by the two types of machines?
b) Compute p-value in (a) and interpret its meaning.
Solution will be shown in class
www.notesolution.com
2/7/2011
3
Comparing Dependent and Independent
Samples
2
2
2
1
2
1
2121
calc
n
1
n
1
)()xx(
z
Independent SamplesIndependent Samples Dependent SampleDependent Sample
z-test: Test of two indep
pop, VKnown
n
S
D
t
D
D
calc
Paired t-test:
Test of two dep pop, Vequal but
unknown
Where D is “difference scores”
df
=
n
-
1
)1n()1n(
s)1n(s)1n(
s
21
2
22
2
11
2
p
)
n
1
n
1
(s
)()xx(
t
21
2
p
2121
calc
t-test: Test of two indep pop,
Vequal but unknown
df=(n1-1)+(n2-1)=n1+n2-2
t-test: Test of two indep pop,
Vunequal and unknown
2
2
2
1
2
1
2121
calc
n
s
n
s
)()xx(
t
P
P
)1n(n
s
)1n(n
s
)
n
s
()
n
s
(
df
2
2
2
4
2
1
2
1
4
1
2
2
2
1
2
1
df n
1
The seven-step Method of Hypothesis Testing
Steps:
1. Define the parameters: P1and P2
2. State the hypothesis
Ho: P1= P2 (or Ho: P1-P2 = 0)
Ha: P1 P2 (or Ha: P1-P2 0)
3. Determine the appropriate test (refer to the flowchart)
4. State the conditions/assumptions: (1) Normal? (2) Equal variance?
5
rejection region
5
.
-
rejection
region
6. Calculate the test statistics value
7. Make the statistical decision and Managerial conclusion
To reject Ho or DO NOT reject Ho
Compare the test statistics with the critical value (critical value approach)
or
compare p-value with .(p-value approach)
Illustration
t-Test for Difference Between Two Means (Vunknown)
A work team has developed a new process to assemble
a certain component. They would like to know if this new
process has significantly reduced the time to assemble
the component. They have taken samples of 50
components produced by the existing process and 40
components produced by the new process. The mean
dtd d d iti f th bltifth
an
d
s
t
an
d
ar
d
d
ev
i
a
ti
on o
f
th
e assem
bl
y
ti
mes
f
or
th
e
existing process were 73.2 minutes and 3.6 minutes,
respectively. The mean time was 71.4 minutes with a
standard deviation of 3.2 minutes for the components
assembled by the new process. Assume that the times
for both processes are normally distributed with the
same variance. At the 1% level of significance, does
this evidence indicate that the new process is quicker
than the old process?
Calculator Procedure for
t-Test for Difference Between Two Means (Vunknown)
Solution: Let sample 1 be the existing process sample, and sample 2 the new
process.
From the Main Menu select:
STAT F3(test) F2(t) F2(2-S)
then enter the following items:
2-Sample tTest
Data: F2
(
Var
)
z
()
1: F3(>2) z
1: 73.2 EXE
x11n-1: 3.6 EXE
n1 : 50 EXE
2: 71.4 EXE
x21n-1: 3.2 EXE
n2 : 40 EXE
Pooled : F1(On) EXE
The calculator will now show the
results:
2-Sample tTest
1> 2
t=2.4749
p=7.6215E-03 (=0.0076215)
df =88
1=73.2
2=71.4
x11n-1=3.6
x21n-1=3.2
xP1n-1=3.4284
n1 =50
n2 =40
Comparing Dependent and Independent
Samples
2
2
2
1
2
1
2121
calc
n
1
n
1
)()xx(
z
Independent SamplesIndependent Samples Dependent SampleDependent Sample
z-test: Test of two indep
pop, VKnown
n
S
D
t
D
D
calc
Paired t-test:
Test of two dep pop, Vequal but
unknown
Where D is “difference scores”
df
=
n
-
1
)1n()1n(
s)1n(s)1n(
s
21
2
22
2
11
2
p
)
n
1
n
1
(s
)()xx(
t
21
2
p
2121
calc
t-test: Test of two indep pop,
Vequal but unknown
df=(n1-1)+(n2-1)=n1+n2-2
t-test: Test of two indep pop,
Vunequal and unknown
2
2
2
1
2
1
2121
calc
n
s
n
s
)()xx(
t
P
P
)1n(n
s
)1n(n
s
)
n
s
()
n
s
(
df
2
2
2
4
2
1
2
1
4
1
2
2
2
1
2
1
df n
1
www.notesolution.com

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Description
272011 Hypothesis Test Hypothesis Test Two Populations Two-Sample Tests One Population 2 MEANS 2 Proportions Chapter 11 Page 452 Sample Independent Sample Check the 4 dependent Objectives conditions Normal? No 1. Comparing The Means of Two Independent STOP If yes Yess Populations Paired t-test No Z-test Z-test for the difference between two means. No 2 2 81and 8 2nown? Equal 8 ? unequal 8 Pooled-Variance t-test for the difference between Two Means. Yes Yes 2. Comparing The Means of Two Related Populations T-test not Pooled variance 3. Comparing Two Population Proportions Z-test Pooled Variance t-test Z-Test 4. F-Test For The Difference Between Two Variances Comparing Dependent and Independent There are two types of dependent samples: Samples Characterized by a Measurement, Characterized by matching or followed by an intervention of pairing of the observations. oZ k some type, and then another measurement Example: ZB Suppose you are selling a house Called a before and after study. and you asked for two IndepeendenttSampples DepeendenttSammplee Example: The samples chosen at random are Dependent samples are Suppose we want to show that, by appraisals of your property. nott eated to each other. characterized by a measurement, placing speakers in the productionSuppose an industrial then some type of intervention, psychologist wishes to study We wish to study the mean incomes followed byy another measurement. area and playing soothing music, offommpannessX and Y. wee ae able o ncceasee the intelectuallsmiiarries off We select a random sample of 28 Paired samples are also production. newly married couples. She employees from the Company X and dependent because the same We begin by selecting a sample of selects a sample of a sample of 19 employees in individual or item is a member of workers and measuring their newlyweds. Company Y. both samples. output under current conditions, The speakers are then installed inNext she administers a standard A person cannot be an employee in Examples: 10 participants in a intelligence test to both the both companies. marathon were weighed prior to the production area, and we again man and woman to determine The samples are indeppenddentt, and after competing in the race. measure the output of the same that is, unrelated. We wish to study the mean workers. the difference in the scores amount of weight loss. There are 2 measurements: (1) Notice the matching that before placing the speakers occurred: comparing the (2) After placing the speakers. scores of the man and the woman. Comparing Dependent and Independent The seven-step Method of Hypothesis Testing Section 11.1 Samples Section 11.2 Steps: IndeeppenndeennttSaamp pless DeepenndenntSaampplee Paired t-test: 1. Define the parameters: 2 1nd 2 2 z-test: Test of two indep Test of two dep pop, 8 equal but 2. State the hypothesis pop, 8 Known Ho: 2 = 2 (or Ho: 2 - 2 = 0) unknown t D D 1 2 1 2 z (1 x2) (1 2) calc SD Ha: 21 2 2 (or Ha: 1 - 2 0) calc 2 2 n 3. Determine the appropriate test (refer to the flowchart) 1 2 Where D is difference scores 4. State the conditionsassumptions: (1) Normal? (2) Equal variance? n1 n2 dffn-1 5. Determiine hee crrcallvalue haatdivide the ejeccton anddnoon-rejecton region 6. Calculate the test statistics value t-test: Test of two indep pop, 7. Make the statistical decision and Managerial conclusion t-test: Test of twoep pop, 8 unequal and unknown 8 equal but unknown To reject Ho or DO NOT reject Ho t (x 1 x 2 2 1 2 2) Compare the test statistics with the critical value (critical value approach) (x 1 x 2 (1 2) calc 2 2 or tcalc s1 s2 2 1 1 n1 n2 compare p-value with . (p-value approach) s p n n ) s2 s2 1 2 ( 1 ) ( 2 ) df=(1 -1)+(2 -1)1n 2n -2 n 1 n 2 2 (n 1 1)s1 (n2 1)s2 df 4 4 sp s1 s2 (n1 1)(n 2 1) n 1n 1 1) n 2n 2 1) 1 www.notesolution.com 272011 Illustration: Calculator steps Z Test for Difference Between Two Means (8 known) A company that makes bolts that are used on an automotive From the Main Menu select: component uses two machines to make these bolts. It has been STAT F3(test) F1(Z) F2(2-S)then enter the following items: diameters made by machine 1 is 0.025 mm. and the standard the bolt 2-Sample ZTest deviation of the bolt diameters of machine 2 is 0.022 mm. Both Data : F2(Var)z machines have a dial to set for the desired diameter. Recently they 1 : F2(<2)z used both machines to fill a large order. The customer found that 1 : 0.025 EXE maanyoffhebols om a ceran pacage were oo age andmade a 2 : 0.022 EXE complaint. It was determined that the package in question was 1 : 5.023 EXE the bolts from both machines to test to see whether the meanof n1 : 100 EXE diameter of the bolts from machine 2 was significantly larger than 2 : 5.031 EXE the mean diameter from machine 1 when the dial was set to the same diameter on each machine. The sample of 100 bolts from n2 : 100 EXE machine 1 had a mean diameter of 5.023 mm and a sample of 100 dial on both machines was set at 5.00 mm. At the 5% level ofe Now key EXE or F1(Calc) significance what is the conclusion? CFX-9850GB Calculator The calculator will now show the Z Test for Difference Between Two Means (8 known) results: A company that makes bolts that are used on an automotive component uses two machines to make these bolts. It has been 2-sample ZTest determined by past studies that the standard deviation of the bolt diameters made by machine 1 is 0.025 mm. and the standard 1 < 2 deviation of the bolt diameters of machine 2 is 0.022 mm. Both machines have a dial to set for the desired diameter. Recently they z =-2.4022 maanyoffhebols om a ceran pacage were oo age andmade a found that complaint. It was determined that the package in question was p =88.114665EE-003 ((=00.00081144655) made by machine 2. The manufacturer decided to take samples of 1 =5.023 the bolts from both machines to test to see whether the mean diameter of the bolts from machine 2 was significantly larger 2 =5.031 than the mean diameter from machine 1 when the dial was set to machine 1 had a mean diameter of 5.023 mm and a sample of 100om n1 =100 bolts from machine 2 had a mean diameter of 5.031 mm when the dial on both machines was set at 5.00 mm. At the 5% level of n2 =100 significance what is the conclusion? Illustration Z Test for Difference Between 8wknown)s ( The operations manager at a light bulb factory wants to determine if there is any difference in the average life expectancy of bulbs manufactured on two different types of machines. The process standard deviation of machine I is 110 hours and of machine II is 125 hours. Solution will be shown in class A random sample of 25 light bulbs obtained from maachine IIndcates a samppe meean off375 hours,andd a similar sample of 25 from machine II indicates a sample mean of 362 hours. a) Using the 0.05 level of significance, is there any evidence of a difference in the average life of bulbs produced by the two types of machines? b) Compute p-value in (a) and interpret its meaning. 2 www.notesolution.com
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