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Quantitative Methods

QMS 202

Clare Chua

Winter

Description

Company
i.e. they are not related)
Company A has recently conducted aggressive advertising
Ha: The two categorica
variables are dependent
campaigns to maintain and possibly increase its share of obse
corected
the market for fabric softener. Their main competitor
e. they are related
Company B, has 40% of the market and a number of
The test statistics:
other competitors account for the remaining 15%. To
determine whether the market shares changed after the
Calculator steps
advertising campaign, the marketing manager f
input fo val
in List 1 and f values
List 2
Company A solicited the preferences of a random
TEST CHI GOF (F
allce
sample of 200 customers of fabric softener. Of the 200
GOF Test
customers, 102 indicated a preference for Company A's
Observed: List 1
(rowtotai
Column total
Expected: List 22
allCe
product, 82 preferred Company B's fabric softener, and
For each ce
re
(2-1) (3-1) 2
the remaining 16 preferred the products of one of the
Differences between X2 test of independence
CNTRB
List 3
competitors. Can the analyst infer at the 5% significance
Save Res: None
and test for equality of proportions
t customer preferences have changed from their Execute
Press EXE
levels before the advertising campaigns were launched?
ontingency table with 2
Contingency table with r
Calculator steps
ng the formula
rows and C columns
rows and C columns
Company Sample
ne factor of interest
Two factors of interest
Company
Datac f
The categorical responses
Summary:22 Test for the differenc
n c>2
are classified into two
16
categories success and proportions
mple data)
102
102-90-12
144 90-1.6
ure
Expected Hypothesized
Note: Test Statistics
90
proportions
he objective is to make
Critical Value
omparisons and evaluate
Existing Market Sh
ifferences between the
Calculator Results
Reject Ho
proportions of success
Do Not Reject Ho
344 618
22 GOF Test
among the various levels
x2 Test:
Note for
depends on
the rejection region is always to the right.
nd df c-1
0.05,df 2) 5.9915.) Since
8.1833 is greater than 5.9915 therefore
2. the table size is always 2 x c
reject Ho.
p 0.01671154
Decision if 0.05, p value is less than 0.05, reject Ho, there
the degrees of freedom is always (c-1)
allCe
is a significant change in the
shares since the advertising campaigns).
The hypotheses are always:
df 2
CNTRB: List
-value 0.0167
Ha Not all my, areequal (where j J1,2
The critical value will have
Summary
Test of indepen
2 and
nC
c 2
Y potprxa Axatpxat B.x.+ Psxste :The Population regression
li
2-1)(c-1
c-1) c-1 degrees of freedom
Note: T
er to X
Table (on your flowchart package)
2 Required Conditions
Criti
ical Values of y2 (Note: For
y2 est, the
SSE
Sample size considerations:
cal value is
Do Not Reject Ho
est forc proportions requires that:
The
MASE
All f
2 0.5 for all cells
1. The
always
the right.
2. The hypothe
f the fo
H The tw
depend
At least 80% of the cells have all fe 20.5 for all cells and
mber of p
nd k
(i.e.. there
nship betw
them).
no fe 1
NOTE: IN SPSS printout, s is called the MSE (Mean Square for
H. The
ategorical variables are dependent
We will
MSE t
relati
hip betw
1. Check the utility of the model
All fe z 1 for all cells (suggested by the text authors
Do Not Reject Ho
2. To provide a measure of the reliability of predictions and
mates wher
Analyzing a Multiple Regression Mode
the model
All fez 5 for all cells
Step 1: Collect
X, Step 5: Statisti
te the utility
ple data. Th
The sample size must be large enough so that the expected value
of th
del
for each cell must be 5 or more
Step 2: Hypothe
the fo
odel. Th
clud
When necessary, cells should be combined to satisfy this
tisfied and A rejection of null hypothesis
choosing which independent variables to include in the
Step 6: Check th
pti
condition.
make
odel modifications, if necessary.
Ak-0 in the global F-test leads
Rationale: The actual sampling dist of the test statistic is discrete
Step 7: Finally, ifthe model is deemed adequate
but it can be approximated by chi-squared dist provided the
to the conclusion that the model is statistically
Step 3: Use th
thod
least sq
to estimate th
sample size (n
s large
use the fitted model to estimate the mean
usefu
value of y or to predict a particular value of y
Step 4: Specify the probability distribution
for a given values of the independent
nt and estimates its variance a2
variables, and to make other inferences.
G2 variance of the random error E
Company i.e. they are not related) Company A has recently conducted aggressive advertising Ha: The two categorica variables are dependent campaigns to maintain and possibly increase its share of obse corected the market for fabric softener. Their main competitor e. they are related Company B, has 40% of the market and a number of The test statistics: other competitors account for the remaining 15%. To determine whether the market shares changed after the Calculator steps advertising campaign, the marketing manager f input fo val in List 1 and f values List 2 Company A solicited the preferences of a random TEST CHI GOF (F allce sample of 200 customers of fabric softener. Of the 200 GOF Test customers, 102 indicated a preference for Company A's Observed: List 1 (rowtotai Column total Expected: List 22 allCe product, 82 preferred Company B's fabric softener, and For each ce re (2-1) (3-1) 2 the remaining 16 preferred the products of one of the Differences between X2 test of independence CNTRB List 3 competitors. Can the analyst infer at the 5% significance Save Res: None and test for equality of proportions t customer preferences have changed from their Execute Press EXE levels before the advertising campaigns were launched? ontingency table with 2 Contingency table with r Calculator steps ng the formula rows and C columns rows and C columns Company Sample ne factor of interest Two factors of interest Company Datac f The categorical responses Summary:22 Test for the differenc n c>2 are classified into two 16 categories success and proportions mple data) 102 102-90-12 144 90-1.6 ure Expected Hypothesized Note: Test Statistics 90 proportions he objective is to make Critical Value omparisons and evaluate Existing Market Sh ifferences between the Calculator Results Reject Ho proportions of success Do Not Reject Ho 344 618 22 GOF Test among the various levels x2 Test: Note for depends on the rejection region is always to the right. nd df c-1 0.05,df 2) 5.9915.) Since 8.1833 is greater than 5.9915 therefore 2. the table size is always 2 x c reject Ho. p 0.01671154 Decision if 0.05, p value is less than 0.05, reject Ho, there the degrees of freedom is always (c-1) allCe is a significant change in the shares since the advertising campaigns). The hypotheses are always: df 2 CNTRB: List -value 0.0167 Ha Not all my, areequal (where j J1,2 The critical value will have Summary Test of indepen 2 and nC c 2 Y potprxa Axatpxat B.x.+ Psxste :The Population regression li 2-1)(c-1 c-1) c-1 degrees of freedom Note: T er to X Table (on your flowchart package) 2 Required Conditions Criti ical Values of y2 (Note: For y2 est, the SSE Sample size considerations: cal value is Do Not Reject Ho est forc proportions requires that: The MASE All f 2 0.5 for all cells 1. The always the right. 2. The hypothe f the fo H The tw depend At least 80% of the cells have all fe 20.5 for all cells and mber of p nd k (i.e.. there nship betw them). no fe 1 NOTE: IN SPSS printout, s is called the MSE (Mean Square for H. The ategorical variables are dependent We will MSE t relati hip betw 1. Check the utility of the model All fe z 1 for all cells (suggested by the text authors Do Not Reject Ho 2. To provide a measure of the reliability of predictions and mates wher Analyzing a Multiple Regression Mode the model All fez 5 for all cells Step 1: Collect X, Step 5: Statisti te the utility ple data. Th The sample size must be large enough so that the expected value of th del for each cell must be 5 or more Step 2: Hypothe the fo odel. Th clud When necessary, cells should be combined to satisfy this tisfied and A rejection of null hypothesis choosing which independent variables to include in the Step 6: Check th pti condition. make odel modifications, if necessary. Ak-0 in the global F-test leads Rationale: The actual sampling dist of the test statistic is discrete Step 7: Finally, ifthe model is deemed adequate but it can be approximated by chi-squared dist provided the to the conclusion that the model is statistically Step 3: Use th thod least sq to estimate th sample size (n s large use the fitted model to estimate the mean usefu value of y or to predict a particular value of y Step 4: Specify the probability distribution for a given values of the independent nt and estimates its variance a2 variables, and to make other inferences. G2 variance of the random error Eonset of a heart attack. A total of 40,845 patients were studied. Each was given one of the four drug regimens. Ca
31 14 52
A B C D
The outcome measure compared was the proportion of severe adverse events G.e, death or disabling strok
Expanded ranch 2 1 12 15
94 88 76 82
reported within 30 days of treatment. The data are presented below
Colonial 6 3 9 23
100 89 69 80
Drug Regimen
Ranch 16 20 24 60
90 88 76 82
Result B C D Total
Split-level 19 17 2 38
33 19 78
Tota 74 60 99 233
Not severe 9.630 9.543 9042 9557 37772
101 79 30 39
Ho:There is no relationship between the geographic location and architectural
Total 10.344 10.323 2796 10.37 40,845
90 32 72 SO
style houses in the region.
At the a 05 level of significance, determine whether there is evidence of a significant difference among
disabling stroke) within 30 davs following treatment for a heart attack
style houses in the region
Test
a 3.05
8.38265
At least one proportion is different
a 3.05
p-value .0000021073 (see calculator output)
Reject Ho if p-walue a .05
Epected Mat Ans
There is sufficient sample evidence to indicate a significant Decision:
There is sufficient sample evidence to indicate that
difference among the four drug regimens with respect t
relationship does exist between the geographic location and
Reject Hosince p-value .039 a .05
proportion of patients suffering
severe adverse events
e, dea
the architectural style of houses in the region.
or disabling stroke) within 30 days following treatrment for hea
attack.
Commercial
Savings
outside NY. City
Test
Program
4.81
3.00
4.70
40.974
4.55
3.75
4.80
p 3.0000021073
66 72 61 63
4.35
5.00
4.84
74 51 60 61
4.05
3.50
4.00
d Matt Ans
82 59 57 76
4.65
5.50
3.00
75 62 60 84
3.00
5.34
5.20
4.81
4.55
5.00
Ans
30 24 41214 97 64 55 65
4.20
3.75
4.60
4.20
4.00
4.65
7.3047 5.9227 9.7725
4.25
5.80
4.25
78 63 71 80
RE
4 19.055 15.45 254193
4.71
5.15
5.40
12.068 9.7854 16.145
4.10
5.25
4.28
4.25
4.78
5.45
3.85
H:At least one mean is different
2.52
4.50
4.30
a 0.05
5.00
Decision rule: Reiect Hnif p-value
a 0.05
4.28
Decision:
Reject Hn since p-value 0.014
a 0.05
There is sufficient sample evidence to indicate that at least
5.02
two of the employee program groups have different mean
4.70
4.87
exam test scores.
Since Hois rejected in (d) it is appropriate to use the Tukeytest.
Using the Tukey output (at the .05 level of significance Program.A produces
higher test/exam scores than B or C
Within pairs of groups CAD, BC, BD, CD) there does not appear to be any
ignificant difference in their means/average.
Sinc
is not reiected in (b) it is inappropriate to use the Tukev test. Note also no in the
onset of a heart attack. A total of 40,845 patients were studied. Each was given one of the four drug regimens. Ca 31 14 52 A B C D The outcome measure compared was the proportion of severe adverse events G.e, death or disabling strok Expanded ranch 2 1 12 15 94 88 76 82 reported within 30 days of treatment. The data are presented below Colonial 6 3 9 23 100 89 69 80 Drug Regimen Ranch 16 20 24 60 90 88 76 82 Result B C D Total Split-level 19 17 2 38 33 19 78 Tota 74 60 99 233 Not severe 9.630 9.543 9042 9557 37772 101 79 30 39 Ho:There is no relationship between the geographic location and architectural Total 10.344 10.323 2796 10.37 40,845 90 32 72 SO style houses in the region. At the a 05 level of significance, determine whether there is evidence of a significant difference among disabling stroke) within 30 davs following treatment for a heart attack style houses in the region Test a 3.05 8.38265 At least one proportion is different a 3.05 p-value .0000021073 (see calculator output) Reject Ho if p-walue a .05 Epected Mat Ans There is sufficient sample evidence to indicate a significant Decision: There is sufficient sample evidence to indicate that difference among the four drug regimens with respect t relationship does exist between the geographic location and Reject Hosince p-value .039 a .05 proportion of patients suffering severe adverse events e, dea the architectural style of houses in the region. or disabling stroke) within 30 days following treatrment for hea attack. Commercial Savings outside NY. City Test Program 4.81 3.00 4.70 40.974 4.55 3.75 4.80 p 3.0000021073 66 72 61 63 4.35 5.00 4.84 74 51 60 61 4.05 3.50 4.00 d Matt Ans 82 59 57 76 4.65 5.50 3.00 75 62 60 84 3.00 5.34 5.20 4.81 4.55 5.00 Ans 30 24 41214 97 64 55 65 4.20 3.75 4.60 4.20 4.00 4.65 7.3047 5.9227 9.7725 4.25 5.80 4.25 78 63 71 80 RE 4 19.055 15.45 254193 4.71 5.15 5.40 12.068 9.7854 16.145 4.10 5.25 4.28 4.25 4.78 5.45 3.85 H:At least one mean is different 2.52 4.50 4.30 a 0.05 5.00 Decision rule: Reiect Hnif p-value a 0.05 4.28 Decision: Reject Hn since p-value 0.014 a 0.05 There is sufficient sample evidence to indicate that at least 5.02 two of the employee program groups have different mean 4.70 4.87 exam test scores. Since Hois rejected in (d) it is appropriate to use the Tukeytest. Using the Tukey output (at the .05 level of significance Program.A produces higher test/exam scores than B or C Within pairs of groups CAD, BC, BD, CD) there does not appear to be any ignificant difference in their means/average. Sinc is not reiected in (b) it is inappropriate to use the Tukev test. Note also no in thents Ken new process. and standard deviation of assembly
produced Samples
or the process were minutes and The time
4 minutes with a standard deviation of 3.2 minutes for the components assembled
process. Assume that the times for both processes are normally distributed with the
does this evidence indicate that the new p
and a standard deviation of $0.32.
alue of all greeting eards in the s 3
estimate of the
Set up 95% confiden
rs inserted
provided bei
be marked)
s inventory fonly
the
stimating here?
What parameter
uicker than the old process?
ution. Let sample l be the existing process sample, and sample 2 the new process.
quired to construct the confiden
b) State the necessary
ption(s)
Main Menu select
2
Assume the valve
carts n
TAT F3 (test) F2(t) An auto
bile dealer
ts to estimate the proportion o
tomers who sti
purchased 5 years earlier. A random sample of200 customers selected fron
Before you construct
95% confidence interval estimate of the averag
-Sample Test
dealers records indicates that 82 still own cars that were purchased 5 years
ng cards in the st
inventory, state the critical value
F2IVar)
E
of the populatio
F3CH2 set up a est
t 2. o 730
n the cars 5 years after they were purchased
73.2
3.6
ue of
d) The point estimate
for the 95% confidence
erval estimate of the average
What parameter
g here?
50
greeting cards in the st
tory
5 E
b) statecheck the necessary assumption(s) required to construct the onfideace interv
f all greeting cards in
State the 95%
inte
stimate of th
40
F1 (On)
he store's inventory is
1. S 202.
f) State yo
terpretation of the 9
fid
interval
onstructed in part (d):
value
calculator w
Before you construct a 95% confidence interval estimate of the population proporti
ustomers who still own the cars years after they were purchased
state the cri
-Sample tTest
ali C
2.4749
e) The 90% confidence interval estimate of the average family dental expenses for a
7.6215
employees of this corporation
do The point estimate
for the 95% confidence
terval estimate of the population prop
fall customers who sti
73.2
the cars sy
E
go
e State your interpretation of the confidence interval constructed in part (d:
confidence interv
estimate of the population proportion of nil custo
e) State the 95%
R2on-
3.2
who still own the cars 5 years after they were purchased is
3.4284
E
0.34183
The personnel department of a large corporation wants to cstimate the family dental
expenses of its employees to determine the feasibility ofproviding a dental insurance
plan. A random sample of 10 employees reveals the following dental expenses (in
You learn to estimate the unknown population parameter (u or T) using Sample Size for Estimating the Proportion
dollars) for the preceding year:
sample data
bar or p) o construct the interval. This interval is most
110 362 246 85
510 208 173 425 316
likely to contain the population parameter (p or en a confidence leve
Solving for n, you have the sample size necessary to develop
(usually expressed in percentage)
a Cl for a proportion
Set up a 90%confidence interval estimate of the average family dental expenses for a
employees of this corporation
a) What parameter is being estimated here? (circle one only)
If you are ask to construct
95% confidence interval for the population
n
parameter, you use the CASIO calculator function "INTR
b) State the necessary assumption
required to construct the confidence interva
If you are ask to find the critical value you either use "INVN" or "INV1 To determine the sample size, you must know 3things
1.The desired confidence level (e.g. 95% or 99%), which determines
ssume the populaton
ly dental
you do not have the "INVt function in your calculator, you use the t-table
the value of
you have the "INVt" function, you may also use the t-table.
2. The acceptable sampling error, e
3. The population proportion Jl
The critical value is either a z
or a t-value based on the confidence
What is the value of the population
the critical yalue needed to construct the 90% confidence interval estimate of the
c) State
level. You use the "INVN function to obtain the z-vahue with mean equals
proportion T?
average family dental expenses for all employees of this corporation.
zero and standard deviation equals
one. Two ways to obtain the t-value, (1)
use Invit TNvt 833
You have two alternatives:
function
1. we do not have any information about
Find the sample size that meets your desired confidence level and margin of error.
d) State the point estimate and its value
used to construct the 90% confidence intervalestimate
e use the most conservative value of
Tt, VV
There are two formulae to compute the sample size. To pick the correct formula, you
T which is 0.5
of the average family dental expenses for all emplovees of th
coDOfatio
nts Ken new process. and standard deviation of assembly produced Samples or the process were minutes and The time 4 minutes with a standard deviation of 3.2 minutes for the components assembled process. Assume that the times for both processes are normally distributed with the does this evidence indicate that the new p and a standard deviation of $0.32. alue of all greeting eards in the s 3 estimate of the Set up 95% confiden rs inserted provided bei be marked) s inventory fonly the stimating here? What parameter uicker than the old process? ution. Let sample l be the existing process sample, and sample 2 the new process. quired to construct the confiden b) State the necessary ption(s) Main Menu select 2 Assume the valve carts n TAT F3 (test) F2(t) An auto bile dealer ts to estimate the proportion o tomers who sti purchased 5 years earlier. A random sample of200 customers selected fron Before you construct 95% confidence interval estimate of the averag -Sample Test dealers records indicates that 82 still own cars that were purchased 5 years ng cards in the st inventory, state the critical value F2IVar) E of the populatio F3CH2 set up a est t 2. o 730 n the cars 5 years after they were purchased 73.2 3.6 ue of d) The point estimate for the 95% confidence erval estimate of the average What parameter g here? 50 greeting cards in the st tory 5 E b) statecheck the necessary assumption(s) required to construct the onfideace interv f all greeting cards in State the 95% inte stimate of th 40 F1 (On) he store's inventory is 1. S 202. f) State yo terpretation of the 9 fid interval onstructed in part (d): value calculator w Before you construct a 95% confidence interval estimate of the population proporti ustomers who still own the cars years after they were purchased state the cri -Sample tTest ali C 2.4749 e) The 90% confidence interval estimate of the average family dental expenses for a 7.6215 employees of this corporation do The point estimate for the 95% confidence terval estimate of the population prop fall customers who sti 73.2 the cars sy E go e State your interpretation of the confidence interval constructed in part (d: confidence interv estimate of the population proportion of nil custo e) State the 95% R2on- 3.2 who still own the cars 5 years after they were purchased is 3.4284 E 0.34183 The personnel department of a large corporation wants to cstimate the family dental expenses of its employees to determine the feasibility ofproviding a dental insurance plan. A random sample of 10 employees reveals the following dental expenses (in You learn to estimate the unknown population parameter (u or T) using Sample Size for Estimating the Proportion dollars) for the preceding year: sample data bar or p) o construct the interval. This interval is most 110 362 246 85 510 208 173 425 316 likely to contain the population parameter (p or en a confidence leve Solving for n, you have the sample size necessary to develop (usually expressed in percentage) a Cl for a proportion Set up a 90%confidence interval estimate of the average family dental expenses for a employees of this corporation a) What parameter is being estimated here? (circle one only) If you are ask to construct 95% confidence interval for the population n parameter, you use the CASIO calculator function "INTR b) State the necessary assumption required to construct the confidence interva If you are ask to find the critical value you either use "INVN" or "INV1 To determine the sample size, you must know 3things 1.The desired confidence level (e.g. 95% or 99%), which determines ssume the populaton ly dental

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