Case Notes 1
ANSWERS TO CASES Chapter 1 DiGiorno Pizza: Introducing a Frozen Pizza to Copete !ith Carr"#Out
In conducting research for the launching of a new product it is imperative that the target population be identified. In this case, who are the people most likely to be interested in purchasing and consuming frozen pizzas in lieu of carry-out pizzas !ow are these people to be identified for sampling "Chapter # refers to this group as the $frame%& $frame%& 'hould a test market city or area be used (hy or why not (hat mode of survey such as telephone, mail, or personal interview should be used (hen should these people be surveyed )oes time of day, day of the week, or season of the year make any difference (hat types of measurements should be taken 'ome possible measurements might might include dollar amounts spent per week of pizza per family, number of pizzas purchased per month, percentage of family pizza consumption that is frozen pizza, and total amount spent per month on take out food. 1. *ne population that that was identified was $pizza lovers%. +hese people may have been previously identified by market researchers based on number of pizzas purchased per month, use of coupons, or from previous surveys. nother population mentioned in the case was women ages to /. +he advertisements shown on national +0 +0 were likely aimed at the general population because raft was attempting to achieve broader goals such as brand name recognition and a dissemination of the $fresh-baked taste% message. In each of the research efforts, the market research company selected only a sample of the population. '2I-lcott sent out 1,333 surveys to pizza lovers, the 4oran 2arketing 5roup conducted focus groups "which usually have no more than 1 people per group& with women to /, and 6roduct )ynamics used focus groups to conduct blind taste tests. +he market research companies "'2I-lcott, the 4oran 2arketing 5roup, and 6roduct )ynamics&, took various measurements on sample members and from these measurements likely computed statistics. statistics. 'ome of these measurements may have included the ranking of various frozen pizza brands based on taste or status, numerical ratings of various types types of pizza in terms of taste "perhaps, for e7ample, on a scale from 1 to #&, amount of time sample members are willing to spend cooking a pizza, amount of money spent per month on pizzas, and percentage of sample who recognize the )i5iorno name. 8sing these measurements, sample sample statistics such such as average amount of money spent per
Case Notes
month on pizzas per family or proportion p roportion of the sample that recognized the )i5iorno name can be computed. 9rom these sample statistics, statistics, population parameters can be estimated such as the percentage percentage of all adults in the country who recognize the )i5iorno name: and the average amount a family family spends on take-out pizza per month +his is a good place to introduce the estimation estimation concepts of Chapter ; intuitively. intuitively. *ne can discuss point estimates "sample statistics& statistics& and the notion of sampling error. raft likely used known descriptive market statistics in their product decision making such as total annual amount of dollars spent in the 8.'. on frozen pizza: population demographics of the 8.'. including age, number and size of households, average household income: and number of competitors in the frozen pizza market. . a. number of pizzas pizzas per week b. age of purchaser c. zip code d. dollars spent per month e. time between purchases f. rating of taste g. ranking of four brands h. geographic location i.
ratio ratio level ratio level nominal level ratio level ratio level ordinal level "but some researchers treat as interval& ordinal level nominal level ordinal level nominal level nominal level
Case Notes
month on pizzas per family or proportion p roportion of the sample that recognized the )i5iorno name can be computed. 9rom these sample statistics, statistics, population parameters can be estimated such as the percentage percentage of all adults in the country who recognize the )i5iorno name: and the average amount a family family spends on take-out pizza per month +his is a good place to introduce the estimation estimation concepts of Chapter ; intuitively. intuitively. *ne can discuss point estimates "sample statistics& statistics& and the notion of sampling error. raft likely used known descriptive market statistics in their product decision making such as total annual amount of dollars spent in the 8.'. on frozen pizza: population demographics of the 8.'. including age, number and size of households, average household income: and number of competitors in the frozen pizza market. . a. number of pizzas pizzas per week b. age of purchaser c. zip code d. dollars spent per month e. time between purchases f. rating of taste g. ranking of four brands h. geographic location i.
ratio ratio level ratio level nominal level ratio level ratio level ordinal level "but some researchers treat as interval& ordinal level nominal level ordinal level nominal level nominal level
Case Notes >
Chapter $ Soap Copanie% Do &att'e
+he pie chart is useful in displaying the market shares in one device ad=acent to each other. 2any decision makers are used to to viewing pie charts in connection connection which budgets and therefore might feel more more at ease with a pie chart. *n the other hand, as mentioned mentioned in the te7t, when percentages are close such as with )ial and $*thers% in 1?;>, it can be difficult to discern the the difference using the pie chart slices. In this case, the bar chart shown above is more desirable.
1. 'hown below are pie charts for the 1?;>, the 1??1, and the 1??? market shares. shares.
Case Notes /
n e7amination of the pie charts ch arts from 1?;> through 1??? reveals that the slice sizes of 8nilever have grown and the sizes of the 6rocter @ 5amble slices have shrunk. 'hown below are the actual percentage figures for the three time periods so that you have the option of displaying the data in different waysA
Compan Company y 6rocter @ 5amble 8nilever )ial Colgate-6almolive *thers
1?;> 1?; > 'hare 'hare
1??1 'hare 'hare
1??? 'hare 'hare
>#.1 /.3 1.3 B. 1#./
>3. >1. 1?.3 ;.3 11.3
;./ >;. 1/.; ?.> ?.1
. 'hown below is a histogram histogram of the weekly sales of bars of soaps over the year. +he histogram was constructed constructed using 13 classes. In 2INI+, the student has the option of trying several different different values for the number of intervals. In D7cel, students can e7plore various bin options. +he shape of the histogram will will somewhat change according to the number of class intervals. Note the shape of this histogram is mound shaped with with some skewness to the right. right. +he center of the distribution distribution appears to be near to 3 million as would be e7pected since 6rocter @ 5amble sells about 3 million bars per week. Note, however that some weeks actually actually average as much as >? million million bars per week and others only 1 million million bars. (hat inventory, production, and human resource implications might this this have !ow does a company $cope% with such fluctuations fluctuations
Case Notes
+he stem and leaf plot for for these data is shown shown below. +he advantage of the stem and leaf over histograms, pie charts, bar charts, and others is that the stem and leaf retains the original data in case the researcher wants to calculate other statistics on the numbers. 6roduction people would likely find the histogram the most interesting because it displays to them where the bulk of production occurs and the magnitude of the unusual size runs. 'tem 1 1> 1/ 1 1B 1# 1; 1? 3 1 > / B # ; ? >3 >1 >
4eaf B # 3, /, /, ; 3, 1, 1, /, , >, /, , # 1, >, B, ? 3, >, >, /, /, B, #, ? >, /, / , ; 1, /, ;, ? 3, > , , , >, >, B, ?
B ;
Case Notes B >> >/ > >B ># >; >?
;
Case Notes #
Chapter ( Coca#Co'a Goe% Sa'' in Ru%%ia
'hown below is 2INI+ output describing the sample of 3 bottle fills. Note that the mean fill is 33.1 oz. with a standard deviation of 3./. +he minimum fill is 1??.3 oz. and the ma7imum is 31.13 oz. +he median fill is 33.1 oz. +he measure of skewness ".3>B#& demonstrates a very slight positive skewness. !owever the histogram and the high p-value associated with the normality test indicate that the data are appro7imately normally distributed. +he mean fill of 33.1 oz. indicates that, on average, the sample fills are very near to 1 oz. and are, if anything, giving a slight amount of free product away to the consumer. 8nder the empirical rule, using µ E 33.1 and σ E ./, ?F of the fill should be within 33.1 G ./ or between 1??.#3 and 33./ oz. Descriptive Statistics: Bottle Fills Variable Bottle F Variable Bottle F
N 50
Mean 200.12
Minimum 1"".20
Median 200.15
Maximum 201.10
TrMean 200.12 1 1"".#0
StDev 0.42 ! 200.40
)escriptive 'tatistics Variable: Bottle ill! /nder!on,arlin 'oralit3 e!t /)uared: -Value:
199.2
199.6
200.0
200.4
200.8
201.2
Mean )t,ev Variance )*e+ne!! (urto!i! ' Miniu 1!t &uartile Median #rd &uartile Ma$iu
95% Confidence Interval for Mu
0.21" 0.8## 200.118 0.419 0.1"5#84 #.6"02 1.401 50 199.200 199.800 200.150 200.400 201.100
95% Confidence Interval for Mu 199.999 199.95
200.05
200.15
200.25
200.2#"
95% Confidence Interval for )ia 0.#50
0.522
95% Confidence Interval for Median
95% Confidence Interval for Median
199.96"
200.2##
SE Mean 0.06
Case Notes ; . +he bottles have a label that claims there are 3 oz. of fluid therein. +his sample of 13 bottles has an average of 3.33> oz. with a median of 3.33 and a mode of 3.33/. ll three of these statistics indicate that, if anything, the bottles are slightly overfilled overall. +he standard deviation of fills is .3# oz. +he 2INI+ normality statistics and histogram overlaid with the normal curve indicate that the data are appro7imately normally distributed. (e can apply the empirical rule. ppro7imately B;F of the fills are within 3.33> G 1".3#&, from 1?.?#B to 3.3> oz.: and ?F within 3.33> G ".3#&, from 1?.?/? to 3.3#. +he measure of skewness "-.3;& indicates a slight negative skewness. +he bo7 plot indicates that there is an outlier on both the lower and the upper end "e7treme underfilled bottle and e7treme overfilled bottle&. n e7amination of the minimum value shows that there is a bottle with 1?.? oz. which has a H score of >.3# or over three standard deviations below the mean. +he ma7imum value is 3.3? which has a H score of >. or almost three-and-a- of an ounce.
Case Notes ?
Chapter ) Co'gate#Pa'o'i*e +a,e% a -Tota'. E//ort
1. +wo probabilities are given in this case. +he first is the marginal probability 8.'. household had purchased +otal for the first timeA
that a
6"+1& E .1 +he second is the conditional probability that a household purchased +otal "for a second time& given that it had purchased it beforeA 6"+ +1& E ./> +he percentage of 8.'. household that purchased +otal at least twice can be computed asA 6"+1 +& E 6"+1& J 6"++1& E ".1&"./>& E .3?3> little over nine percent "?.3>F& of 8.'. households purchased +otal at least twice during this time period.
. If age is independent of the willingness to try +otal, then the marginal probability of an age category should e
Case Notes 13
>. +he probability that a person is either in the /-B/ age category or purchased +otal is a union probability and is computed asA 6"/-B/ +& E 6"/-B/& G 6"+& - 6"/-B/ +& E .3 G .1 - .3?3> E .>1?# +he probability that a person purchased +otal given that they are in the /-B/ age category can be computed asA 6"+/-B/& E
P "T ∩ / − B/& P " / − B/&
=
.3?3> .-3
E ./1
/. 4et ' E saw the commercial, N E didnKt see the commercial, 6i E prior probability 6"'& E .>
6"N& E .B;
6"+'& E ./3
6"+ N'& E .13B
Dvent
6rior
6"+ 6i&
'
.>
./3
N
.B;
.13B
6"+ 6i&
6"6i +&
".>&"./3& E .1;
.B1
".B;&".13B& E .3;
.>?
6"+& E .13
Case Notes 11
Chapter 0 Fui Fi' Introduce% APS
1. n E >3, p E ./3. +he e7pected number isA µ E
nJ p E >3"./3& E 1
+he probability that si7 or fewer purchase an 6' camera isA 6rob" x L B n E >3 and p E ./3& E >3
C B "./3& B ".B3& -/
+>3
C "./3& ".B3& -
+ ⋅ ⋅ ⋅ +>3
C 3 "./3& 3 ".B3& >3 E
.311 G .33/1 G .331 G .333> G .3333 G .3333 G .3333 E .31#1 If the market share actually is /3F " p E ./3&, the e7pected number of purchasers from a sample of >3 is 1. 'i7 or fewer are considerably less than the e7pected number "1&. !ow often would one get si7 or fewer out of thirty when twelve is e7pect less than F ".31#1& of the time. +herefore, if si7 or fewer out of thirty actually purchase, it is likely that the market share is not /3F.
.
4et λ E ./ complaintsM133,333 rolls. 'hown below are some of the values for the 6oisson distribution with λ E ./ from 2INI+A x 3 1 > / B # ; ? 13 11 1
6robability .3?3# .1## .B1> .3?3 .1/ .3B3 .3/1 .33;> .33 .333# .333 .3333 .3333
Case Notes 1
(ith a 6oisson distributed average rate of ./ complaintsM133,333 rolls, the probability of getting # complaintsM133, 333 rolls by chance is .33;> or less than 1F. *ften researchers like to use the cumulative probability of x # on a problem like this. +he probability of randomly getting # or more complaintsM133,333 rolls when the average is only ./ is . 311# ".33;> G .33 G .333# G .333&. +his is still
>. +his is a hypergeometric problem withA N E , n E 13, A E 1; "customer satisfaction&, and x E #
+he probability is computed asA 1;
C # -
⋅>/
C 13
C >
E .313
+he probability that seven out of the ten successful products were created for customer service when only eighteen of the fifty-two original products were created for customer service is about 1F ".31&. 'ince this is unlikely to happen by chance, it is likely that there is something inherently more successful about creating a product for customer service reasons than for revenue growth.
Case Notes 1>
Chapter 2 +ercede% Goe% a/ter 3ounger &u"er%
1. 6rob" x /,333µ E />,1 and σ E ?;1&A z =
x − µ
=
σ
/-,333 − />,-1 -?;1
E -3./1
9rom +able ., the area for z E -3./1 is .1?1. 6rob" x /,333& E .333 G .1?1 E .B?1 E 204516 A'o%t 226 o/ +ercede% dea'er% !ou'd 7e priced out o/ copetition !ith thi% &+W ode'4
6rob " x />,1µ E >/,??3 and σ E >B#&A z =
x − µ
=
σ
/>,-1 − >/,??3 E >./# ->B#
9rom +able ., the area for z E >./# is ./??#. 6rob" x />,1& E .333 - ./??# E .333> E 848(6 9irtua''" none o/ the &+W dea'er% are pricing the ($ ci ore than the a*erage price o/ the +ercede% C;<($84
6rob" x L >/,??3µ E />,1 and σ E ?;1&A z =
x − µ σ
=
>/,??3 − />,-1 -?;1
E -.#B
9rom +able ., the area for z E -.#B is ./?#1. 6rob" x L >/,??3& E .333 - ./?#1 E .33? E 84$56 A7out 4(6 o/ the +ercede% dea'er% are pricing C;<($8 'e%% than the a*erage price o/ the &+W ($8 ci .
6rob" x L >#,3?µ E />,1 and σ E ?;1&A
Case Notes 1/
z =
x − µ
=
>#,3? − />,-1
σ
-?;1
E -.3#
9rom +able ., the area for z E -.3# is ./;3;. 6rob"O L >#,3?& E .333 - ./;3; E .31? E 145$6 ;e%% than $6 o/ the +ercede% dea'er% price the C;<($8 'e%% than =(>?8054 Conc'u%ion: There i% 'itt'e o*er'ap in the price% o/ the t!o car% and it cou'd 7e conc'uded that the" are not rea''" copeting !ith each other price!i%e4
. C4A 6rob. E
a E / >3 − -B >/ − -/
>;isA a E 6rob. E
>3 − -B > − -
b E >/
x1 E B
x E >3
E ./ E )86
b E >
x1 E B
x E >3
E ./ E )86
+he same proportion of >;is cars fall into this category "B->3 mpg&. !owever, an e7amination of the end points of the mileage distributions of each car reveals that the upper end for >;Ks is 1 mpg. higher than for the >;isKs and the lower end for C4Ks is 1 mpg. lower. oth cars have very close gas mileage figures.
Dach car more than >3 mpg.A 9or C4A 6rob. E
>/ − >3 >/ − -/
E ./ E )86
9or >;isA 6rob. E
> − >3 > − -
E . E 086
higher proportion of >;Ks are in the more than >3 mpg. category than C4Ks.
Case Notes 1
>.
λ E 1.># carsM> hours, µ E 1M1.># E .#> of > hours E .1? hours 9or 1 hourA 1 hour E .>>> of > hours. x3 E 3.>>>. +he cumulative probability of this time interval is .>BB>. +his means that there is a >B.B>F chance that there will be less than one hour between sales.
9or 1 hoursA 1 hours E / times > hours. x3 E /. +he cumulative probability for this time interval is . ??; meaning that there is a ??.;F chance that there will be less than 1 hours between sales. +he complement of this is that there is a 1 - .??; E .33/ E 3./F chance that there will be more than 1 hours between sales.
2anagers know that there is an almost >#F chance of a sale within every hour. +hey need to determine how much staffing it takes to sell a car every hour or less. 5iven that it takes several potential buyers and often multiple visits to the dea lership to sell one car and that it is relatively likely "probability almost #F& that they will close a sale every > hours " x3 E 1&, the dealership should never go without having salespeople around and may have to have several employees around all the time. y having good e7ponential and 6oisson distribution data, one can, to some e7tent, track the impact of advertising on sales by testing values of λ using random arrival data in time periods following advertising to determine if λ has increased. 9or e7ample, if λ E 1.># every > hours but following a advertising campaign, there is a randomly selected > hours period and cars are sold, then management might be able to statistically =ustify that the λ has increased and then conclude that the advertising campaign was the cause. In many businesses, the value of lambda changes with time of day, day of the week, and season of the year. In the car business, there may be an increase in sales on the weekend, in the evening, or perhaps in the fall when new models arrive. 'tudents should always be cautioned about using the same value of lambda for all time periods. 2any students know intuitively that lambda varies over time.
Case Notes 1B
Chapter > She'' Attept% to Return to Preier Statu%
1. +he answers to this
.
'hell contracted researchers appear to have stratified on age, ethnicity, household location, occupation, and previous employment with 'hell. (ith regard to opinions about 'hell as a $premier% company, some strata that might make sense are age, ethnicity, economic class, education, occupation, gender, and geographic location. It is important to 'hell that all segments of the 8.'. adult population be reached. In order to test to determine the effectiveness of marketing campaigns, 'hell ne eds to recognize that there will likely be differences in the perceptions of young people and those of old people of what a $premier% company should be because of their life e7periences and the types of messages that appeal to them. +he same thing is true for different ethic groups "different cultural values may appeal to different groups&, economic class "the economics of the household may determine what types of $premier% messages appeal to their needs&, education "at what level of education should the messages be targeted&, occupation "how does 'hell impact various occupations differently eg. environment, reliability of products, availability of products, pricing&, gender "often men and women seek different outcomes from firms&, and geographic location "state and regional cultures vary and thus consumer messages should be targeted so as to parallel geographic interests&.
Case Notes 1#
>. In 1?#?, p E .1. 6rob." z =
p Q
new survey of n E >3 resulted in
p Q
E ..
. n E >3 and p E .1&A
Q − p p p ⋅ q
=
.- − .1".-&".#& E .B >3
n
9rom +able ., the area for z E .B is .333. 6rob."
p Q
.& E .333 - .333 E .8888
It is virtually impossible to randomly select >3 people and have F declare that 'hell is a $premier% company if in the population only 1F believe that 'hell is a $premier% company. +his is strong statistical evidence that the 1F figure is no longer true and that the actual population figure is greater. +his is a nice segue into section ;.> of chapter ; in which the sample data can be used to estimate the actual proportion who now believe that 'hell is a $premier% company. It also lays the groundwork for the hypothesis testing to come in chapter ?.
/. 6rob." z =
x
.3 µ E 1.;, σ E .#, and n E >&A
x − µ σ
n
=
-.3 −1.; .# E 1.B? >
9rom +able ., the area for z E 1.B? is .// 6rob."
x
.3& E .333 - .// E 48)00
+here is only a /.F probability that the sample mean of .3 was obtained by chance. It is likely that the population mean is no longer 1.; and indeed, is now higher. +his is a good place for the instructor to mention .3 as a common standard for low probability and begin the groundwork for chapter ?.
Case Notes 1;
6rob." z =
x
. µ E 1.;, σ E .#, and n E >&A
x − µ σ
n
=
-. −1.; .# E .? >
9rom +able ., the area for z E .? is .333. 6rob."
x
.& E .333 - .333 E .8888
It is virtually impossible to randomly obtain a sample mean of . or more from a sample of > with this standard deviation if the population mean is only 1.;. +his is, of course, even more conclusive evidence than obtaining a sample mean of .3. discussion of the .3/ probability above and the .3333 value here may result in a preliminary understanding of the notion of p-value introduced in chapter ?.
Case Notes 1?
Chapter Theratri@
1. n E 11 8seA pQ ± z
1& ResA pQ
9or ?F confidence, z E 1.?B pQ ⋅ qQ n =
B> 11
E ./# "./#&"./-- &
./# G 1.?B
11
E ./#; G .3?13
.)02 p 42(
& ResA pQ
=
;B 11
.#/#; G 1.?B
E .#/#;
".#/#;&".---& 11
E .#/#; G .3#?/
422) p 4$>$
>& ResA pQ
=
131 11
.;#;> G 1.?B
E .;#;>
".;#;>&".1-1# &
410 p 45(1
11
E .;#;> G .3?;
Case Notes 3
/& ResA pQ
13
=
11
.?1>3 G 1.?B
E .?1>3 ".?1>3&".3;#3& 11
E .?1>3 G .31
.210 p 452)0
. n E ?
df E ; s
8seA
x ± t
1&
E >.#?
x
n
>.#? G .>3B (41(
&
x
.#/ G .>3B
>&
x
".;B& ?
E >.#? G .BB
s E 1.# "1.-# & ?
E .#/ G .?;
(4>$
E /.1;
/.1; G .>3B (4>8
s E .;B
)4)0
E .#/
14>2
9or ?F confidence, t .3,; E .>3B
s E .B> ".B>&
)422
?
E /.1; G ./;
Case Notes 1
/&
x
E >.>/
>.>/ G .>3B $4>$
&
x
".;1& ?
E >.>/ G .B
(452
E >.?
>.? G .>3B (4>5
s E .;1
s E .1 ".-1&
)411
?
E >.? G .1B
Case Notes
Chapter 5 Frito#;a" Target% the Bi%panic +ar,et
1.
a& !3A p E .B> !aA p ≠ .B>
4et α E .3
9or a two-tailed test,
αM E .3
z .3 E G 1.?B "critical value& n E ;3 Q p
=
z =
x E #
# ;3
E .B#B
Q − p p
=
p ⋅ q
.B#B − .B> ".B>&".>#& E $41
n
;3
'ince the observed z E .;1 z .3 E 1.?B, the decision is to reect the nu'' h"pothe%i% . +he proportion of !ispanics that are 2e7ican mericans is not .B>. +he sample data indicate that the proportion is higher.
b& !3A p E .?> !aA p L .?>
4et α E .3 n E B;? Q p
=
z =
z .3 E - 1.B/ "critical value& x E B3B
B3B B;?
E .;#?
Q − p p
p ⋅ q n
=
.;#? − .?> ".?>&".3#& E #04$8 B;?
'ince the observed z E -.3 L z .3 E -1.B/, the decision is to reect the nu'' h"pothe%i% . +he proportion of !ispanic grocery shoppers that are women is less than .?>.
Case Notes >
c& !3A p E .;> !aA p ≠ .;>
4et α E .3
9or a two-tailed test,
αM E .3
n E />; p Q
E .#?>#
'ince the p-value of .3/ L α E .3 "the p-value was ad=usted by 2INI+ for the two-tailed test&, the decision is to reect the nu'' h"pothe%i% . +he proportion of !ispanics who listen to advertisements in 'panish is not .;>. +he sample data indicate that the proportion may now be less than .;>.
. a& !3A !aA x
µ µ
E >1 ≠ >1
α E .31
E ;.;1
+he observed t E -1. 9or a two-tailed test, t >,.33 E -.3# 'ince the observed t E -1. t >,.33 E -.3#, the decision is to /ai' to reect the nu'' h"pothe%i% . There i% not enough e*idence to %a" that the a*erage age i% di//erent /ro (1 "ear% . 2arketing decision makers must assume that the average age has not changed.
b& !3A !aA x
/ µ L /
α E .3
E >.B#
s E 1?.B
t =
µ E
n E 1;
df E 1#
t .3, 1# E -1.#/3
x − µ >.B# − / = 1?.-B s E #$482 n
1;
'ince the observed t E -.3B L t .3,1# E -1.#/3, the decision is to reect the nu'' h"pothe%i% . We conc'ude that the a*erage e@penditure o/ Bi%panic cu%toer% on chip% per "ear i% 'e%% than = )0 .
Case Notes /
Chapter 18 Seitz Corporation Producing ua'it" Gear#Dri*en and ;inear#+otion Product%
1. Comparing last yearKs mean transactions to this yearKsA n1 E 3 n E
E >33 x - E /3
s1 E 33 s E /3
x 1
!3A µ1 - µ E 3 !aA µ1 - µ ≠ 3
α
df E n1 G n - E 3 G - E />
E .3
αM E .3
t .3,/> E G .31
" x1 t E
s1
-
" n1
−
n1
− x -
& − " µ 1 -
1& + s - " n+
n-
−
−
−
µ - &
1&
-
1 n1
+
E
1 n-
"->33 − -/3& − "3& -
-
"33& "1?& + "/3& "-/&
1
-3 + - − -
-3
+
1
E #8452
-
ecause t E -3.?B t .3,/> E -.31, the decision is to /ai' to reect the nu'' h"pothe%i% . +here is not enough evidence here to say that there is any difference in the average dollar amount of sales between this year and last.
Case Notes
. Comparison of tractors at two plants using a confidence intervalA n1 E / Q1 p
=
Q1 " p
1; /
x1 E 1;
n E 1
E ./333
Q p
Q - & ± z − p
pQ 1qQ1 n1
+
"./3 - .>& S 1.?B #4815
p1 # p$
=
x E 1 11
E .>>
Q - qQ p
n"./3&".B3& /
+
".->>&".#B/#& 1
E .1B/# G .1;/
4()5$
+he point estimate of the difference in
Case Notes B
>. 333 vs. 331A t E -1.;> with a p-value of .3#. +his is not significant at α E .3. +here is no significant difference in the mean ratings between 333 and 331. +his is underscored by the confidence interval that includes zero. !owever, if α E .13 were used, there would be a significant difference. D7amining the means reveals that the mean score for 33 was higher. +he sample sizes were # for 333 and ?> for 331.
/. Comparison of variances for week 1 and week A
E σ
!3A !aA
σ 1
4et
α E .3
σ 1
≠ σ
F .3,,B E .??
αM E .3
df 1 E n1 - 1 E
df E n - 1 E B
F .?#,B, E 1M.?? E 3.1B#
(eek 1A n1 E B s1 E 1.1# (eek A n E # s E 1.B;
F
=
s1 s -
-
"1.1#& =
"1.B;&
-
E 84)0
'ince the observed value of F E 3./; is the left tail critical value of F E 3.1B#, the decision is to fail to re=ect the null hypothesis. +he variances of product being produced these two weeks are not significantly different. 2anagement would probably like this because this indicative of consistent production patterns. (ide swings in variance would be of concern because it would indicate that some weeks the variability is more out-of-control than others and a less consistent product is being produced.
Case Notes #
Chapter 11 4 R4 C'ar,%on Copan"
1. +he two by three factorial design is analyzed using a two-way N*0. +here are two independent variables, temperature and supplier. +emperature has three treatment levelsA #3o, 113o, and 13o. 'upplier has two classifications levelsA supplier and supplier . +he dependent variable is strength of the valve as measured in psi. 'hown below is D7cel output for this analysis.
/'V/ Source of Variation )ulier e%erature Interaction 7itin otal
SS 20.056 800.111 84.""8 21".###
df 1 2 2 12
1122.2"8
1"
MS 20.056 400.056 42.#89 18.111
F 1.11 22.09 2.#4
P-value 0.#1##85 0.000095 0.1#8598
F crit 4."5 #.89 #.89
9irst, we e7amine the observed F for interaction which is .>/ with a p-value of .1>;B. 'ince interaction is not significant at any commonly used alpha, we proceed to e7amine main effects. +here is no significant difference between the two suppliers " F E 1.11, p-value E .>1>>?&. +here is a significant difference in the strength of the valves by temperature at α E .3331. +he mean psi for #3o is 1?.;>, for 113o is 1;.>>, and for 13o is 1/ psi. It appears that at 13o the valves are not as strong. 8sing 2INI+, +ukeyKs multiple comparison tests were done to determine if there were any significant differences in valve strength by temperature. +he results areA Tu$e%&' (air)i'e *om(ari'on' Famil% error rate + 0.0500 ,ndividual error rate + 0.020! -riti*al value + !.6 ,nterval' /or *olumn level mean ro) level mean 1 2
5.444 #.444
!
.#"0 21.
2
6.!"0 20.2
+hese results show that there were significant differences between #3o and 13o and between #3o and 13o. +he confirms what we observed above with the D7cel output.
Case Notes ;
. +he data are analyzed using a one-way N*0. +he independent variable is country with four classificationsA Canada, Columbia, +aiwan, and the 8.'. +he dependent variable is the dollar cost to replace a seal. 'hown below is the 2INI+ output for this one-way analysis with multiple comparisonsA One-way ANOVA: Cost versus Country 3nal%'i' o/ Varian*e /or -o't Sour*e DF SS MS -ountr% ! 64!!1 21444 Error 24 1550 656 Total 2 #00#1
evel 1 2 ! 4
N
Mean 244.2" !2!.5 1"5.00 222.14
ooled StDev +
StDev !1.01 !0.65 20.#2 1.04
25.62
F !2.6#
0.000
,ndividual "5 -,' For Mean Ba'ed on ooled StDev 7777 8 8 8 8 7777 200 250 !00 !50
Tu$e%&' (air)i'e *om(ari'on' Famil% error rate + 0.0500 ,ndividual error rate + 0.0110 -riti*al value + !."0 ,nterval' /or *olumn level mean ro) level mean 1
2
2
11.0 41.5
!
11.5 #.0
"0.# 166.!
4
15.6 5"."
6!. 1!".2
!
64." 10.6
+he results show that there is a significant difference in the cost of seal replacement between countries " F E >.B;, p-value E .333&. n e7amination of the means shows that there is potential significant differences between countries "Canada - T//.?, Columbia - T>>.#, +aiwan - T1?.33, and 8.'. - T.1/&. +he results of the +ukey multiple comparison analysis shows that there are significant pairwise differences between Canada and Columbia, between Canada and +aiwan, between Columbia and +aiwan, and between Columbia and the 8.'. Clarkson might be very effective in marketing to Columbian companies because in Columbia the cost of replacing the seal is significantly higher than in other countries.
Case Notes ? >. +his is a randomized block design. +he main independent variable of interest is type of valve. +he blocking variable was day of the week. elow is D7cel output for a two-way N*0 without replication. /'V/ Source of Variation ,a3 of te 7ee* 3e of Valve rror otal
SS 0.""6" 6.010" 0.""9#
df 4 5 20
".566"
29
MS 0.1942 1.2021 0.0#90
F 4.98 #0.85
P-value 0.00595206 0.000000010
F crit 2.8" 2."1
highly significant observed 9 value was obtained for type of valve " F E >3.; with a p-value of .333333331&. +he various valve types and their associated mean lead times areA 'afety 1.B/ utterfly .1 Clack 1.> 'lide 1.BB 6oppet .1; Needle 3.;; 'ince there is a significant difference in valve type and since the means appear to be
2
!
4
2
0."6 0.016
!
0.16 0.#16
0.!024 1.2"6
4
0.516 0.46
0.0!6 0."56
0.#!6 0.156
5
1.0!6 0.0424
0.556 0.4!6
1.!56 0.!624
1.016 0.0224
6
0.2624 1.256
0.424 1.!6
0.056 0."!6
0.2#24 1.26
5
0.#024 1."6
T9e re'ult' o/ t9e'e multi(le *om(ari'on te't' i' t9at lead time' /or t9e /ollo)in: (air' o/ valve' are 'i:ni/i*antl% di//erent;
Case Notes >3 Sa/et% and o((et< Sa/et% and Needle< Butter/l% and -la*$< Butter/l% and Needle< -la*$ and o((et< Slide and o((et< Slide and Needle< and o((et and Needle.
+he study was attempting to control for day of the week as a blocking variable. +he blocking variable produced an F value that was significant at α E .31. 4ead times differ by type of valve. +he needle and clack valves have the shortest lead times and the butterfly and poppet valves have the longest lead times. cursory e7amination of the mean lead times by day of the week indicates that 2ondayKs and 9ridayKs produce the longest lead times.
Case Notes >1
Chapter 1$ Foot ;oc,er in the Shoe +i@
1. !as the distribution of shoe sales by 6rice Category changed from the year 333 to the year 331 chi-s
$881
11 >; ># >3 1 11 1#
1B /3 > # 3 3 11 1;
chi#%4
1.31# 3.13B 3.13;11 3.>3333 3.1;1; 3.3/#B 3.33333 3.3;; χ E 140(8
+he observed chi-s;3 8sing an α E .3, and df of k -1 E #, the critical value isA
χ#,.3 E 1/.3B#1
'ince the observed chi-s
Case Notes >
. chi-s
Female 4! !0.!
Total 2
=.S. Sout9
4# !#."#
20 2".02
6#
=.S. Ea't
52 64.
61 4#.2!
11!
=.S. Nort9
2# !0.!#
25 22.62
5!
Euro(e
# 6!.05
!2 46."5
110
3u'tralia
4 4!.56
2" !2.44
6
2#2
210
4"2
=.S. >e't
Total -9iS? +
!.64 7 4.#"# 7 2.0"0 7 0.1#6 7 0.250 7 !.545 7 DF + 5< Value + 0.000
2.#06 7 2.51 7 4.61 7 0.22 7
!.!#0 7 0.!65 + 28.716
n observed chi-s ccording to sources, 9oot 4ocker has a 1?./F share of the sneaker market. 9oot 4ocker believes that it holds a higher share of the market in the 8.'. (est. +o test this notion, 9oot 4ocker hires a market research company that randomly samples 1,333 people in the 8.'. (est who have =ust purchased new sneakers. *f these, >3 have purchased their sneakers at 9oot 4ocker. +he hypotheses being tested areA !3A p E .1?/ !aA p .1?/
Case Notes >>
+he sample information isA
n E 1,333
Q p
x E >3
=
->3 1333
E .>3
4et alpha be .31. Chapter ? techni3 as the observed proportion, a 7 table can be set up and analyzed using the chi-s
−
f e &
f e
f o
9oot 4ocker
1?/
>3
B.B;3/
*ther 'tore
;3B
##3
1.B3#?
-
f e
+he resulting observed chi-s/?. +he decision is to re=ect the null hypothesis. +his indicates that proportion of the market share in the west is significantly higher than in other locales. 6erhaps 9oot 4ocker ought to study the clientele, the stores, the sales people, and the sale methods to determine why they are doing better in the (est and attempt to implement things that they learn in other locales.
Case Notes >/
Chapter 1( De'ta Wire %e% Training a% a Weapon
1. 'hown below is 2INI+ output for correlation and regression analysis for the education and sick day data. +he correlation between hours of education and number of sick days is a relatively strong negative correlation. +his indicates that the more hours of education received, the fewer the sick days. +here may be several reasons for this. *ne e7planation may be that as workers participate in an educational process about their work, they become more interested in what they are doing because they understand more about it and can see more potential. If the training is interesting, they may look forward to the time in the classroom. In addition, they may feel more a part of the team by having been included in training and feel more self worth for being selected for the training. Correlation of !rsDduc and 'ick)ays E -3.##>
+he regression e
Coef #.#/ -3.3#?/B
'tdev 3.#?;3 3.31>B
P-s< E ?.;F
t-ratio ?.#1 -.1#
p 3.333 3.333
P-s<"ad=& E #.F
nalysis of 0ariance '*8PCD Pegression Drror +otal
)9 1 1; 1?
'' 1B1.B 13;./? B?.#
2' 1B1.B B.3>
9 B.#
p 3.333
+he 9 test for the overall model is significant at .331 as is the t value testing the slope. +he r of .?; indicates that almost B3F of the variation of the sick days is accounted for by the hours of education. +he standard error of the estimate is ./ days which is a modest error. +he regression e
'hown below is a regression plot of this line and the dataA
Case Notes >
.
+he F value for the regression model indicates significant overall regression. +he t -ratio, which here is the s
Case Notes >B
>. 'hown below is D7cel regression output for this problem. +his output shows that hours of training is highly predictive of productivity "r E .?#B&. +he overall F value for this regression is e7tremely large and significant " F E BB.;, p-value of .333333U&. In addition, the standard error "133.B//& is
/'V/ :ere!!ion :e!idual otal
Intercet =our!
df 1 16 1"
SS MS 669"812#1.5 669"812#1.5 16181129.59 1011#20.599 685962#61.1
Coefficients Standard Error "0880.252 #94.546 5.09# 0.198
t Stat 1"9.65 25."#
F 662.28
P-value ".1#828 1.90#14
Significance F 1.902"14
Case Notes ># Chapter 1) Star7uc,% Introduce% De7it Card
1. +his model uses four independent variables in an effort to predict the amount of money people spend on their debit card. *verall, the model has modest to good predictability with an R of .# and a standard error of T.1. +his standard error indicates that about ?F of the time, the model will be within G"T.1& or GT//.>3 of the actual figurewhich is not particularly good.. (hile the overall test of the model is significant " F E 1.>;, p-value E .33333#&, an e7amination of the t tests and their associated p-values shows that only one of the predictors, income "t E B.B?, p-value E .33333& is significant. None of the other variables are even close. !ad a simple regression model been developed using =ust income to predict the amount of the prepaid card, the R would be . #>, the t value for income would increase to #.#/, the standard error of the estimate would reduce to T1.?B, and the overall F test would increase to ?.?3.
)MM/:; - Regression Statistics Multile : 0.869 : )uare 0."55 /d
/'V/ :ere!!ion :e!idual otal
Intercet /e ,a3! Cu! Inco%e
df 4 20 24 Coefficients 8#.826 0.2#" 1.190 1.422 2.40"
SS #01"5.042# 9810.95"" #9986 Standard Error 22.494 0.5"6 1.4"4 2.6#1 0.#60
MS "54#."606 490.54"9
t Stat #."# 0.41 0.81 0.54 6.69
F 15.#8
P-value 0.001# 0.6852 0.4291 0.5949 0.000002
Significance F 0.00000"
Case Notes >;
. +his model attempts to predict the number of days per month that a customer fre.; days indicates that the model would predict within G ">.;& or G B.B days about ?F of the time. perusal of the data shows that the range of number of days is 1B days. +he relatively large size of the standard error to this range is further evidence of the modelKs weakness. study of the t statistics reveals that the predictor variable, cups, is the only significant predictor "t E >./3, p-value .33#&. +he number of cups of coffee that a person drinks per day seems to be a good predictor of the number of times per month the person fre/ and the standard error is >.>.
Case Notes >?
)MM/:; - Regression Statistics Multile : 0.645 : )uare 0.416 /d
/'V/ :ere!!ion :e!idual otal
Intercet Cu! Inco%e /e
df # 21 24
SS 160."59# 225.800" #86.56
Coefficients Standard Error 5.9684 #.0651 1.0644 0.#12" 0.0"16 0.0509 0.0"85 0.08#5
MS 5#.5864 10."524
F 4.98
t Stat 1.95 #.40 1.41 0.94
P-value 0.0650 0.002" 0.1"42 0.#5"8
Significance F 0.0091
Case Notes /3
>. 'hown below is the output from an D7cel multiple regression analysis to predict sales revenue by number of stores, number of drinks, and average weekly earnings. +he predictability is e7tremely high with an R of .???;. In predicting sales revenues that range from /33 to B33, the standard error of the estimate is only 1B.B?. +he overall F of />?.1 is significant at alpha E .33331. (hile number of stores is not a significant predictor "t E -3.?, p-value E ./11/&, both number of drinks "t E -#./#, p-value E .33/?#& and average weekly earnings "t E 1>.#3, p-value .333;/& are significant at α E .31. Notice that for the predictor, number of drinks, both the t value and the coefficient are negative. +his indicates that, at least in this model with other variables in the model, there is a negative relationship between number of drinks and sales revenue. !owever, a cursory e7amination of the raw data shows that as sales revenues increase so do the number of drinks. +his points out one of the dangers in over interpreting the regression coefficients "discussed in Chapter 1 in section on multicollinearity&. (hen a simple regression model is run using number of drinks as the sole predictor of sales revenue, the r is .??, and more importantly the regression coefficient is positive as is the t statistic. +his might serve as an informalMintuitive introduction to the notion of collinearity. +he correlation between the two significant predictors in the multiple regression model, number of drinks and weekly earnings, is .?;/.
)MM/:; - Regression Statistics Multile : 0.9999 : )uare 0.9998 /d
/'V/ :ere!!ion :e!idual otal
Intercet )tore! ,rin*! arnin!
df # # 6
SS MS #"92"#5.8"9 1264245.29# 8#5.55 2"8.52 #"9#5"1.429
Coefficients Standard Error 1#500.2#" 946.164 0.0264 0.02"8 "5.20448212 10.0"005905 #8.989 2.8466"56#8
t Stat 14.2" 0.95 ".4" 1#."0
F 45#9.21
P-value 0.000"5 0.41145 0.0049" 0.00084
Significance F 0.000006
Case Notes /1
Chapter 10 9irginia Seiconductor
1. 'hown below is a multiple regression analysis which contains a model to predict the size of a companyKs purchase by four other predictors. elow that is a stepwise regression analysis of the same data. +he full multiple regression model has an R of ##.>F. !owever, the ad=usted R is only B?F indicating that there are some nonsignificant predictors in the model. +he stepwise regression analysis confirms this by showing that an R of ##.3F can be obtained with only two predictors, size of company and whether or not the company has a central purchasing agent. +he overall F value is significant at alpha E .31. +he standard error of the estimate is ?/./>. n e7amination of the p-values of the t ratios to test the slopes confirms the stepwise regression output. In the full multiple regression model, only size of the company has a significant t value with a p-value of .31. +he variable, central purchasing agent, has a p-value of .3;1 which is significant if alpha α E .13. In the stepwise regression analysis, size of company was the strongest single predictor yielding an R of B?.//F. +he inclusion of central purchasing agent as step two brings the R up to ##F. +hese are the only two significant predictors included in the stepwise analysis. oth analyses result in positive regression coefficients for these two significant predictors. +he larger the company, the larger the size of purchase tends to be which makes sense. lso, having a central purchaser tends to result in a larger size of purchase. +he distance that a customer company is away from 0irginia 'emiconductor does not seem to impact size of purchase nor does the percent of a customer companyKs imports. Pegression nalysis +he regression e# 'izeCo - 3.> CoFImp G 3.111 2ileswy G 113 Cent6ur 6redictor Constant 'izeCo CoFImp 2ileswy Cent6ur s E ?/./>
Coef -1.#; 1.>#> -3.>1 3.1113 113./> P-s< E ##.>F
'tdev B?. 3.//1 .3B 3.>#;? #./
t-ratio -3.3> >.11 -3.1B 3.? 1.?
p 3.?;3 3.313 3.;#? 3.## 3.3;1
P-s<"ad=& E B?.3F
Case Notes /
nalysis of 0ariance '*8PCD Pegression Drror +otal
)9 / 11 1
'' >>>B3 ?;3?# />1#/;
2' ;>/1> ;?1;
9 ?.>
p 3.33
'tepwise Pegression 9-to-DnterA
/.33
9-to-PemoveA
/.33
Pesponse is Cust6ur on / predictors, with N E 1B 'tep 1 Constant /.>1B 'izeCo +-Patio
1.; .B/
Cent6ur +-Patio ' P-'<
.
-#.##1 1./ /.B 13? .3#
?#.1 B?.//
;#./ ##.33
+he regression analysis attempts to develop a model which can be used to predict sales figures with the predictors of average hours worked per week and number of customers. 'hown below are scatter plots of each of these predictors with sales. Note the number of customers has a slight linear shape but is more like the upper left
Case Notes />
Case Notes //
Pegression nalysis +he regression e1B !rs'< - 3.31?B Cust'< G 3.33?> Int!rCus 6redictor Constant !rsM(k No.Custs !rs'< Cust'< Int!rCus
Coef /1. -.1 3.BB; 3.3>1B> -3.31?B 3.33?>>
'tdev 11#.? .3?? >.>#> 3.3B// 3.3?;1 3.3B??
s E 1.3?#
P-s< E ?3.1F
t-ratio 3.> -3.// 3.3 3.B -3.BB 3.1>
p 3.#>? 3.B;1 3.;1 3.?? 3.>? 3.;??
P-s<"ad=& E ;3.>F
nalysis of 0ariance '*8PCD )9 '' Pegression .3B Drror B.31B +otal 13 B1.3/
2' 11.33 1.3>
9 ?.1
p 3.31
'tepwise Pegression 9-to-DnterA
/.33
9-to-PemoveA
/.33
Pesponse is 'ales on predictors, with N E 11 'tep Constant
1 #.3;?
>.1>?
Int!rCus +-Patio
3.311 /./1
3.3/# B.#
Cust'< +-Patio ' P-'<
-3.31> ->.#1 1./B B;./1
3.?/ ;;.>;
Case Notes /
>. 'hown below is a 2INI+ scatter plot of sales and number of employees. Notice how the graph rises and then levels out. +his fits fairly closely with the upper left
/.33
9-to-PemoveA
/.33
Pesponse is 'ales on predictors, with N E 13 'tep Constant
1 -?/.B;
-;;>.13
logDmpl +-Patio
./3 >.;
3./3 /.#?
No.Dmpl. +-Patio ' P-'<
-1./ -/.B >.;? B1.#
.1? ;?.?
Notice that log of number employees enters the analysis first and accounts for almost BF of the variation of sales. t the second step, number employees enters the process adding another ;F. It was likely worthwhile to recode the data using logs. ccording to this, sales increases are associated with the log of number employees and not as much with the straight linear increase. +he company should e7ercise caution in merely hiring more people as sales increase.
Case Notes /B
Chapter 12 De&ourgh +anu/acturing Copan"
1. +he decomposition analysis shows several things. study of the original data shows that there is a general upward trend. +his is underscored by the graph showing the trend line fit through the data. +here appear to be important seasonal effects in these data. Note that for Vanuary, 9ebruary, and 2arch, the seasonal indices are less than 133. 9rom pril through ugust, seasonal indices are more than 133 peaking in the summer months of Vune, Vuly, and ugust. +his would indicate that these are the strongest months of locker sales. +his might make sense because schools might be purchasing lockers and installing them in the summertime in preparation for the opening of school. +he months of *ctober, November, and )ecember have low seasonal inde7es. +he upward trend marks good news for )eourgh demonstrating a consistent growth. +he seasonal indices can be used to help )eourgh plan for both production and shipping. y e7amining lead time and production times, )eourgh can more closely schedule raw materials from suppliers and do human resource planning.
. 8sing 2INI+, several forecasting techni years, and years producing 2)s of .1;, ./B1, and /.3#1#. +he -year moving average produced the least 2) and seemed to make the best fit. +he 2INI+ graphical output for the -year moving average model is shown below. Ne7t, single e7ponential smoothing models were e7amined with various values of α. 9or α E .>, the 2) was #.>313. 9or α E .B, 2) was #.3//#. 9or α E .?, 2) was >.B/. +hus, the higher the value of alpha, the better the forecast with a big improvement as alpha moved from .B to .?. 'ince alpha weights the actual previous value, the e7ponential smoothing works better here when the model actually shadows the previous value. 'ince there seems to be a downward overall trend in the data, tracking the previous value may be a good strategy. n optimal search for alpha resulted in an alpha of .?;#. +he value of 2) for this model was .?B?. +he 2INI+ graphical output for this optimal value of alpha is shown below. +rend analysis was preformed on these data by fitting a line through the data. 2INI+ trend analysis resulted in 2) of /.#1.
Case Notes /#
*verall, the best fitting model to these data was the t . 8sing t E 1 to represent the year 33, the predicted per-unit labor cost isA ?1.BB?1 - B.1?1B"1& G 3.BB11> "1& E ;.B. 'hown below is the graphical analysis for the
Movin /verae
/ctual
85
-redicted /ctual -redicted
"5 t ! o C
Movin /verae
65
55 0
5
10
ie
15
>ent:
2
M/-:
6.4#62
M/,:
4.0"1"
M),:
51.6"62
Case Notes /;
)inle $onential )ootin
/ctual
85
-redicted /ctual -redicted
t ! "5 o C r o b a > 65
)ootin Con!tant
55 0
5
10
/la:
0.98"
M/-:
4.61"6
M/,:
2.9569
M),:
#".2189
15
ie
rend /nal3!i! for >abor Co!t &uadratic rend Model ;t ? 91.6691 6.21916@t A 0.26611#@t@@2 /ctual
85
it! /ctual it!
t "5 ! o C r o b a > 65
M/-: M/,: M),:
55 0
5
10
ie
15
#.9"10 2.8426 16.""6"
Case Notes /?
Chapter 1> Sch!inn
1. 'ince two independent samples are being compared and the shape of the population distribution is unknown, the 2ann-(hitney U test is used to analyze the data rather than the t test for two independent samples. +he null hypothesis is that there is no difference between the age of 'chwinn customers in Colorado 'prings "near the mountain& and the age of 'chwinn customers in 't. 4ouis. +he 2INI+ computer output for the 2ann-(hitney test is shown below. +he analysis reveals that the median age for a customer in Colorado 'prings was >1 years of age and the median age for a customer in 't. 4ouis is 1/ years. It appears that the target market in Colorado 'prings is young adult versus 't. 4ouis where it is children and early teens. 'ince one of the sample sizes is greater than 13, a large sample 2ann-(hitney U test is the appropriate test. s noted in the te7t, the 2INI+ 2ann-(hitney test does not yield a z statistic but rather gives the value of W and a p-value. 'ince the p-value is .33>3, the null hypothesis is re=ected at α E .31. +here is a significant difference between the age of 'chwinn customers in Colorado 'prings and those in 't. 4ouis. +he median ages support the theory that a much older group of customers purchase 'chwinn bikes in Colorado 'prings perhaps to be used in mountain biking. +he 't. 4ouis population appears to be a youth market. Mann-"hitney Test an# C$: C%S% St% &ouis -.S. N + 11 Median + !1.00 St. oui N + " Median + 14.00 oint e'timate /or ET31ET32 i' 1.00 "5.2 er*ent -, /or ET31ET32 i' ".00<2!.00 > + 155.0 Te't o/ ET31 + ET32 v' ET31 not + ET32 i' 'i:ni/i*ant at 0.00!0 T9e te't i' 'i:ni/i*ant at 0.00!0 ad@u'ted /or tie'
. 'ince three independent groups are being compared, it would appear that this is a completely randomized design and that a one-way N*0 could be used to analyze the data. !owever, it is uncertain whether the data are normally distributed or not, so a ruskal-(allis test is used to analyze the data. +he independent variable is supplier with three classificationsA supplier 1, supplier , and supplier >. +he dependent variable is the weight of the handle bar. +he null hypothesis is that there is no difference in the weight of handle bars supplied by the three suppliers. +he alternative hypothesis is that there is a difference in the weights of handle bars by supplier. 2INI+ was used to analyze these data. +he results, given below, show that there is no significant difference in the weights of handle bars according to supplier. +he H value ">.3?& is 2INI+Ks e& denotes that there is no significant difference even at α E .13. +he differences in the medians, shown in the 2INI+ output, are merely due to chance. +o 'chwinn this might mean that, at least on handle bar weight, these suppliers are interchangeable.
Case Notes 3
'rus(al-"allis Test: "ei)ht versus Supplier Aru'$al>alli' Te't on )ei:9t -5 Su((lier 1 Su((lier 2 Su((lier ! verall + !.0"
N # 6 5 1" DF + 2
Median 201." 1"#.2 1"5.2
3ve an$ 12.4 ".5 6.# 10.0
C 1.5 0.26 1.4#
+ 0.21!
>. +he 2INI+ output for the Puns +est contains a p-value of .?3;> that is not significant "underscored by 2INI+Ks statement Cannot re=ect at alpha 3.3&. +his indicates that the paint flaws are occurring in a random manner. In this sample, there are 1? observations above K and B below "out of # bicycles&. 'ince the data were coded with a 1 to indicate at least one flaw and a 3 to indicate no flaws, there are 1? bicycles with at least one flaw out of the # bicycles. +he proportion of flawed bicycles is
1? #
E .>> "shown as K in the output&. (hile 'chwinn management
may by happy to know that there appears to be no systematic pattern of flaws, they are likely to be unhappy about having at least one flaw in .>>F of the bikes. 6erhaps, an intensive, manufacturer-wide
Case Notes 1
Chapter 1 Ro7otron
1.
6roduce → → → → → 'hipping illing
Item
↑
↑ ↑ ↑
eginning → → → → → → → →of ssembly 4ine ↑ ↑ ↑ ↑"no& ↑ eing +ag ↑ ↑ 2anufactured"yes& → Item → ↑
↑ ↑ ↑ 2ail Poom
*rder → 6rocessing
→
↑
6lant Clerk
→ ↓
'top
↑
↑
↑ ↑ ↑
↑ "no& ↑ ↑ 'tandard ↑"no& ↑ Item "yes& → (arehouse → vailable "yes& → → → →→
. 6art 1#>A +he x bar chart looks pretty good with the e7ception of sample number 1> which has a mean that is above the upper control limit. 2ost of the rest of the means seem to fluctuate randomly around the centerline mostly in the inner 1M> of the limit area. +he P chart indicates that the ranges are in control. None of the ranges are outside the control limits and most lie near the centerline with no obvious pattern occurring. 6art /;A +he x bar chart indicates a process that is in control. None of the points are outside of the limits. In fact, only two of the spikes are in the outer M> of the limits. +here seems to be little or no pattern with points occurring randomly. !owever, the range chart indicates some potential problems. +he ranges for samples ? and 1# are above the upper control
Case Notes limits. +hese samples and their origin should be investigated. !owever, the rest of the range values seem to be in order and well under control. >. p chartA +he centerline of the p chart is .3>B/3. *n average, >.B/F of the items in each sample are in nonconformance. Ret, there are / of the 133 samples that produced sample proportions above the upper control limit which is established at .11? or 11.?F. 2anagement has to be concerned about samples that have almost 1F nonconforming items when the overall average is only >.B/F. +here are many samples that are on the lower control limit. !owever, since the lower values of p for nonconformance indicate few items in nonconformance, these values are good. In fact, 1? of the 133 samples had no items nonconforming. *verall, the patterns seem to be random. 2anagers should investigate to determine why, seemingly out-of-the-blue, samples contain unacceptable proportions of nonconforming items.
Case Notes >
Chapter 15 F'etcher#Terr": On the Cutting Edge
1. +here are several decisions that management had to make during this time including 1& whether or not to invest in technology, & e7pand its line of offerings through imports, >& conduct a significant planning process, /& attempt to increase market share, & develop new products, B&create greater employee involvement, #& invest in employee education, ;& invest in plant improvements, and ;& implement a participatory management system. 'everal states of nature occurred which could have effected 9letcher-+erryKs outcomes. 'ome of these includeA 1& its largest customers decided to introduce their own privatelabel cutters made overseas, & the technology that 9letcher-+erry invested in would not work, >& dollar weakened, /& slow-down in demand for cutters, & employees do not respond to company efforts.
.
)ecision Import lternative Not Import
T 8p ".& T>3,33 3 -T,#33
T 'ame ".>& T#,33 3 -T,#33
T )own "./3& -T,333 -T,333
+he e7pected monetary value "D20& for each of these alternatives isA ImportA ".&"T>3,333& G ".>&"T>3,333& G "./3&"-T,333& E -TB>,333 Not ImportA ".&"-T,#33& G ".>&"-T,#33& G "./3&"-T,#33& E -T,#33 +he D20Ker would choose the highest of these alternatives which is to not import and take a T,#33 loss. +he risk avoider would also choose to not import. !owever, a risk taker might decide to import gambling that the dollar does not go down.
