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UNISEMINAR


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Introduction  

    Quantitative  Methods  I   Academic  Year  2012/2013,  Block  1  

 

 


Uniseminar  –  Quantitative  Methods  I  

Welcome  to  Uniseminar!  

 

Introduction  

I n t r o d u c t i o n  

Uniseminar   offers   E x a m   P r e p a r a t i o n   S e m i n a r s ,   S u m m a r y   S c r i p t s   a n d  

L e a r n i n g   C a r d s   for   students   of   the   Maastricht   University.   It   is   our   goal   to   optimally  prepare  you  for  your  exams  and  to  make  your  own  exam  preparation  as  

efficient  as  possible.  In  order  to  achieve  this  goal,  we  have  developed  a  system  of  

seminars  in  combination  with  an  extensive  summary  script,  which  is  proven  for   several  years  by  now.  

In   university   it   is   often   the   case   that   there   is   a   lot   of   material   available   for   a   course  and  that  the  importance  of  this  material  is  hard  to  evaluate.  Since  we,  as  

students,  have  made  this  experience  as  well,  you  are  provided  with  a  Uniseminar   Summary   Script   of   the   corresponding   course.   This   folder   contains   all   exam-­‐

relevant   material   and   it   gives   you   a   good   summary   of   all   course   topics.   The  

content  of  the  folder  is  created  by  experienced  Master  or  PhD  students,  who  have   taught  this  course  already  several  times.  As  a  consequence,  it  is  possible  for  you  

to   concentrate   on   the   actual   exam   preparation,   rather   than   spending   hours   searching  and  printing  the  right  material.  

At   the   end   of   week   6   of   your   block,   normally   during   the   weekend,   our   E x a m  

P r e p a r a t i o n   S e m i n a r s   take   place.   These   seminars   are   taught   by   above-­‐

average   students,   who   have   already   mastered   their   studies   at   the   Maastricht  

University  and  have  a  great  deal  of  experience  in  tutoring.  Since  they  have  studied   and   taught   at   the   Maastricht   University   they   know   exactly   where   potential   problems  may  lie  and  are  therefore  able  to  optimally  teach  you  the  whole  theory   of  the  course  and  practice  perfectly  tailored  examples  with  you.  Furthermore  you  

can   bring   in   your   own   questions   during   the   seminar   and   discuss   individual   problems  during  the  breaks.  

You   are   able   to   pick   up   your   S u m m a r y   S c r i p t   a n d   L e a r n i n g   C a r d s   in  

advance   of   the   Seminar   in   order   to   already   start   preparing   so   that   you   can   discover   your   own   difficulties   early   enough.   Later   in   the   Seminar   you   will   then   know   what   your   weaknesses   are   and   be   able   to   pay   special   attention   to   these  


Introduction    

 

Uniseminar  –  Quantitative  Methods  I  

sections   or   ask   questions   about   it.   Our   Summary   Script   and   Learning   Cards   are  

updated  every  year  according  to  the  current  course’s  content  and  we  are  always   trying  to  optimize  the  folder  as  much  as  possible.    

A b o u t    U U s  

 

Uniseminar   was   founded   5   years   ago   by   two   students   at   the   University   of  

St.Gallen   in   order   to   make   Exam   Preparation   more   efficient   and   coherent.   Since   2005  we  have  expanded  our  vision  and  are  now  offering  seminars  and  material  

for   an   efficient   exam   preparation   in  Switzerland,  the   Netherlands,   Italy   and   Germany.  

Thanks  to  this  longstanding  experience,  we  were  able  to  build  up  a  team  of  highly  

qualified   tutors   and   editors   and   are   therefore   able   to   guarantee   high   quality   of   exam  preparation.  

The   team   of   Uniseminar   is   grown   strongly   over   the   years   and   comprehends  

several   mathematicians,   statisticians   and   economists,   who   all   bring   a   great   didactical  experience.  All  tutors  of  Uniseminar  have  been  teaching  their  field  for   years  and  know  exactly  what  is  important  in  order  to  optimally  prepare  and  pass   the  exam.    


Seminar

Extras

Exams

Practice

Theory

T


Theory  

    Quantitative  Methods  I   Academic  Year  2012/2013,  Block  1  

 

 

 


Theory  

T h e o r y  

 

Uniseminar  –  Quantitative  Methods  I  

The   Theory   Script   summarizes   the   whole   theory   of   the   course   in   a   simple   and  

understandable   way.   Concepts   are   explained   with   the   help   of   demonstrative   examples.  It  is  structured  according  to  the  seven  weeks  of  the  course  and  is  one  of   the   most   important   parts   of   your   exam   preparation.   Although   practice   is   very  

important,  it  is  even  more  crucial  to  understand  the  basic  concepts  of  the  course  

in  order  to  be  able  to  calculate  and  understand  all  different  kinds  of  exercises  and  

exam  questions.    

 

T a b l e    o o f    C C o n t e n t s    

M a t h e m a t i c s  

0   Prior  Knowledge  

1   Functions  of  one  variable  

2   Properties  of  functions  

1  

1  

16  

28  

3   Differentiation  

32  

5   Single  Variable  Optimization  

45  

4   Derivatives  in  use  

6   Functions  of  two  variables  

39  

51  

7   Two  Variable  Optimization  

56  

S t a t i s t i c s  

6 2  

 

1   Sampling  &  Types  of  Data  

62  

3   Probability  Theory  

73  

5   Confidence  Intervals  

88  

2   Descriptive  Statistics  

4   Sampling  Distributions  &  Normal  Model   6   Hypothesis  Testing  

7   Inference  for  means    

66  

83  

92   97  


Uniseminar  –  Quantitative  Methods  I  

Mathematics   0  

 

   Theory  -­�  Mathematics  

Prior  Knowledge  

In   order   to   understand   the   whole   theory,   you   should   make   sure   to   know   the  

following   basics   by   heart.   You   should   never   again   forget   these   basic   definitions   and  rules  so  that  you  can  easily  solve  mathematical  problems  by  making  simple  

transformations.    

0 . 1   R e a l    N N u m b e r s  

I n t e g e r    N N u m b e r s  –  all  whole  numbers:   

 

ℤ = ‌ , −3, −2, −1,0,1,2,3, ‌    

The   integer   numbers   can   be   divided   into   two   groups,   namely   odd   and   even   numbers:   

  

đ?‘œđ?‘œđ?‘œđ?‘œđ?‘œđ?‘œ = 2đ?‘˜đ?‘˜ + 1, đ?‘¤đ?‘¤đ?‘¤đ?‘¤đ?‘¤đ?‘¤â„Ž  đ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Ž  đ?‘˜đ?‘˜   ∈ ℤ = ‌ , −5, −3, −1,1,3,5, ‌  

đ?‘’đ?‘’đ?‘’đ?‘’đ?‘’đ?‘’đ?‘’đ?‘’ = 2đ?‘˜đ?‘˜, đ?‘¤đ?‘¤đ?‘–đ?‘–đ?‘–đ?‘–â„Ž  đ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Ž  đ?‘˜đ?‘˜   ∈ ℤ = ‌ , −4, −2, −0,2,4,6, ‌  

N a t u r a l    N N u m b e r s  –  all  non-­�negative  and  nonzero  integer  numbers:   

 

ℕ = 1,2,3,4,5,6, ‌    

R a t i o n a l   N u m b e r s   –   all   real   numbers   that   can   be   written   as   a   fraction   of  

integer  numbers:   

â„š=

 

, đ?‘¤đ?‘¤đ?‘¤đ?‘¤đ?‘¤đ?‘¤â„Ž  đ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Ž  đ?‘šđ?‘š, đ?‘›đ?‘›   ∈  ℤ  đ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Ž  đ?‘›đ?‘› ≠ 0  

Examples  are:  1 3 , − 2 7 , 17 101 , 4 2 , 5 1 , − 11 1 , 0 2    

 

I r r a t i o n a l   N u m b e r s  –  all  real  numbers  that  cannot  be  written  as  a  fraction  of  

integer  numbers:  

1  


Theory  -­�  Mathematics    

 

Uniseminar  –  Quantitative  Methods  I  

Examples  are:   2, đ?œ‹đ?œ‹, đ?‘’đ?‘’, 5  

Please  note  that  terms  which  are  mathematically  not  defined,  such  as   −1  1,   1 0   or  ln 0 ,  are  not  part  of  the  irrational  numbers!        

 

0 . 2   B a s i c    R R u l e s   A l g e b r a   

             

đ?‘Žđ?‘Ž + đ?‘?đ?‘? = đ?‘?đ?‘? + đ?‘Žđ?‘Ž  

đ?‘Žđ?‘Ž + đ?‘?đ?‘? + đ?‘?đ?‘? = đ?‘Žđ?‘Ž + đ?‘?đ?‘? + đ?‘?đ?‘?  

đ?‘Žđ?‘Ž − đ?‘?đ?‘? + đ?‘?đ?‘? = đ?‘Žđ?‘Ž − đ?‘?đ?‘? − đ?‘?đ?‘?  

�� + 0 = ��  

đ?‘Žđ?‘Ž + −đ?‘Žđ?‘Ž = 0   đ?‘Žđ?‘Ž ∙ đ?‘?đ?‘? = đ?‘?đ?‘? ∙ đ?‘Žđ?‘Ž  

đ?‘Žđ?‘Ž ∙ đ?‘?đ?‘? ∙ đ?‘?đ?‘? = đ?‘Žđ?‘Ž ∙ đ?‘?đ?‘? ∙ đ?‘?đ?‘?  

đ?‘Žđ?‘Ž ∙ 1 = đ?‘Žđ?‘Ž   đ?‘Žđ?‘Ž ∙ đ?‘Žđ?‘Ž =

 

= 1                  (������  �� ≠ 0)  

(– đ?‘Žđ?‘Ž) ∙ đ?‘?đ?‘? = đ?‘Žđ?‘Ž ∙ −đ?‘?đ?‘? = −(đ?‘Žđ?‘Ž ∙ đ?‘?đ?‘?)   – đ?‘Žđ?‘Ž ∙ −đ?‘?đ?‘? = đ?‘Žđ?‘Ž ∙ đ?‘?đ?‘?  

đ?‘Žđ?‘Ž ∙ đ?‘?đ?‘? + đ?‘?đ?‘? = đ?‘Žđ?‘Ž ∙ đ?‘?đ?‘? + đ?‘Žđ?‘Ž ∙ đ?‘?đ?‘?  

đ?‘Žđ?‘Ž + đ?‘?đ?‘? ∙ đ?‘?đ?‘? + đ?‘‘đ?‘‘ = đ?‘Žđ?‘Ž ∙ đ?‘?đ?‘? + đ?‘Žđ?‘Ž ∙ đ?‘‘đ?‘‘ + đ?‘?đ?‘? ∙ đ?‘?đ?‘? + đ?‘?đ?‘? ∙ đ?‘‘đ?‘‘  

F a c t o r i n g  

Factoring   is   a   very   common   operation,   but   unfortunately   not   always   easy   to  

identify.   Factoring   is   simply   the   result   of   the   last   two   Algebra   rules   above.   Looking  at  the  following  two  examples,  the  principle  of  factoring  should  get  clear:     10đ?‘Ľđ?‘Ľ  + 2đ?‘Ľđ?‘Ľ = 2đ?‘Ľđ?‘Ľ ∙ 5đ?‘Ľđ?‘Ľ  + 1  

−8đ?‘Žđ?‘Ž + 12đ?‘Žđ?‘ŽÂ˛ = 4đ?‘Žđ?‘Ž ∙ −2 + 3đ?‘Žđ?‘Ž                                                                                                                  

1  The  author  of  this  script  acknowledges  that  there  are  mathematically  valid  definitions  of  this  

expression.  These,  however,  are  not  covered  within  the  scope  of  this  class.   2  


Theory  -­�  Mathematics    

1  

 

F u n c t i o n s    o o f    o o n e    v v a r i a b l e  

Uniseminar  –  Quantitative  Methods  I  

1 . 1   G e n e r a l    D D e f i n i t i o n ,    D D o m a i n    & &    R R a n g e   G e n e r a l    D D e f i n i t i o n  

If   one   variable   depends   on   another   variable   it   is   said   that   the   variable   is   a  

function   of   the   other.   With   a   function   it   is   always   possible   to   identify   an  

independent   variable   and   a   dependent   variable.   In   the   classical   case   ��   is   the  

independent  variable  and  ��  is  the  dependent  variable.  Then  it  holds  that:   

 

�� = ��(��)    

–  in  words:  f  is  function  of  ��.  

D o m a i n    aa n d    R R a n g e  

Depending   on   the   function   the   domain   and   the   range   of   the   function   differ.   The  

domain  marks  the  numbers  that  can  be  put  into  the  function  as  a  value  of  ��.  The  

range  then  describes  which  set  of  numbers  can  result  from  the  domain  if  inserted   in   the   function.   Therefore   the   domain   defines   the   possible   ��   values,   while   the   range  defines  the  possible  ��  values.  

To  make  this  point  clear  consider  the  following  function:   �� = �� �� = ��   

The  domain  of  the  function  is  restricted  to  only  positive  numbers  and  zero,  as  the   square  root  is  not  defined  for  negative  values,  therefore  the  domain  is  �� ≼ 0.  The  

range,   in   turn,   is   generally   also   restricted   to   values   larger   than   zero,   as   an   exponential   function   can   never   turn   negative.   However   there   is   a   further  

restriction,  which  follows  from  the  domain.  Inserting  �� = 0  in  the  function  yields  

�� 0 = 1,  which  is  the  smallest  value  which  can  be  achieved,  due  to  the  restricted  

domain,  caused  by  the  square  root.  The  range  is  therefore  �� �� ≼ 1.  

     

16  

 


Uniseminar  –  Quantitative  Methods  I  

Some  common  domains  &  ranges:  

 

   Theory  -­�  Mathematics  

 

D o m a i n  

1 �� �� =   x

�� ≠ 0  

all  real   numbers  ≠ 0  

�� ≼ 0  

��(��) ≼ 0  

�� �� = x  

�� �� = x   

�� �� = ��   �� �� = ��   

�� �� = ln  (��)  

all  real   numbers   all  real   numbers   all  real   numbers   �� > 0    

R a n g e  

all  real   numbers  

�� �� ≼ 0   �� �� > 0   all  real   numbers  

Often   you   will   encounter   problems   with   complicated   functions   (mostly   a  

combination   of   the   above   functions),   where   you   will   need   to   define   the   domain  

and  range.  Stay  calm  with  these  exercises  and  check  the  domains  of  the  different   parts  as  if  they  were  single  functions.    

1 . 2   I m p o r t a n t    F F u n c t i o n s   G r a p h s    o o f    aa    ff u n c t i o n  

It  is  of  great  importance  to  have  a  broad  picture  in  mind  of  the  different  kinds  of  

functions,   because   it   will   be   a   lot   easier   to   make   statements   about   and   calculations  with  a  function  if  you  can  imagine  it.  Therefore  the  main  purpose  of   this   section   is   to   give   you   such   an   understanding   how   functions   looks   like.   The   following  sections  will  then  dig  deeper  into  the  different  types  of  functions.  

 

17  


Theory  -­�  Mathematics    

L i n e a r    F F u n c t i o n  

 

The   easiest   relationship   between   an   inde-­�

Uniseminar  –  Quantitative  Methods  I  

 

pendent   and   a   dependent   variable   is   a   linear  

relationship.   The   graph   of   the   linear   function  

�� �� = ��  is  displayed  by  a  straight  line.  The  only  

thing  that  can  happen  to  this  line  is  that  is  can  be   shifted  up  or  down  or  that  the  slope  can  change.  

 

 

 

Q u a d r a t i c    F F u n c t i o n  

 

The   second   function   is   a   quadratic   function   of     the   type  �� �� = ��².   The   quadratic   relationship   between   the   ��   and   the   ��   variable   can   be   seen  

from   the   fact   that   the   ��   values   increase   much  

faster   than   the   ��   values   which   leads   to   this  

parabola.   Because   of   the   square   term,   ��   can  

never  turn  negative,  which  restricts  the  range  to  

 

�� �� ≼ 0  in  this  case.    

 

S q u a r e    R R o o t    F F u n c t i o n  

In   the   third   graph   a   square   root   function   of   the  

   

   

format  �� �� = ��  is  displayed.  The  domain  of  ��  

values   is   restricted   to   �� ≼ 0   as   the   square   root   only  exists  for  positive  values.  As  a  result  of  this  

the  range  is  then  limited  to  �� �� ≼ 0.  Because  of  

the   square   root   the   ��   values   increase   slower  

than  the  ��  values.   18  

 


Theory  -­�  Mathematics    

3  

D i f f e r e n t i a t i o n  

 

Uniseminar  –  Quantitative  Methods  I  

3 . 1   C h a r a c t e r i s t i c s    o o f    F F u n c t i o n s   T h e    ss l o p e    o o f    aa    cc u r v e  

The  easiest  way  to  think  of  the  slope  is  to  think  about  the  absolute  speed  at  which   a  function  is  increasing  in  value  from  left  to  right.  Measuring  the  slope  of  a  line  is   very   easy   when   linear   functions   are   considered.   The   slope   of   a   line   does   not  

change  along  the  function  and  can  simply  be  read  off  its  equation.  However,  when  

non-­�linear  functions  are  considered,  this  does  no  longer  hold  true.  Neither  is  the   slope  constant  nor  can  it  be  read  off  easily.  

To  get  a  more  intuitive  understanding  of  the  slope   it  is  easiest  to  think  about  a  linear  function  which   is   tangent   to   a   non-­�linear   function   in   a   certain  

point.   In   this   point   the   non-­�linear   function   will   have   exactly   the   same   slope   as   the   straight   line.  

The  picture  shows  this  for  a  quadratic  function.  It   also   displays   three   linear   functions   with   slope  

�� = 0, �� = 2   and   �� = 4   which   are   tangent   to   the   quadratic   function.   In   the  

vertex  of  the  parabola  the  slope  is  zero.  It  is  then  easy  to  verify  that  the  speed  at  

which   the   quadratic   function   increases   becomes   larger.   Thus   also   the   slope   becomes  larger  as  we  move  along  the  parabola.      

I n c r e a s i n g    aa n d    d d e c r e a s i n g    ff u n c t i o n s  

There   are   four   different   categories   which   can   determine   the   behavior   of   a   function  on  a  certain  interval:    

(strictly)  increasing  

(strictly)  decreasing  

A  function  is  called   increasing  or   decreasing  if  its  slope  is  always  bigger  or  equal  

to   zero.   The   following   two   graphs   represent   an   increasing   and   a   decreasing   function.   32  


Uniseminar  –  Quantitative  Methods  I  

 

 

   Theory  -­�  Mathematics  

           

The   difference   between   these   two   functions   and   a   strictly   increasing   or   strictly  

decreasing  function  is  that  the  term  strictly  rules  out  the  case  in  which  the  slope   can   be   zero.   Both   of   the   functions   above   have   a   part   in   which   they   are   neither  

increasing  nor  decreasing  but  just  horizontal.  A  strictly  increasing  or  decreasing   function   is   not   allowed   to   have   such   a   part   but   needs   to   always   increase   or   decrease.   It   can   therefore   be   said   that   the   two   functions   above   are   strictly   increasing  or  strictly  decreasing  on  the  interval  �� > 0  only.    

 

L i m i t s  

The   book   and   the   course   teach   how   to   calculate   the  derivative  of  a  function  using  limits.  As  there  

are   much   more   efficient   ways   of   calculating   derivatives   we   will   skip   this.   Unfortunately   this  

does  not  mean  that  we  can  skip  the  whole  section   concerning  limits.  

A  limit  is  defined  for  a  term  if  one  variable  of  the  

term   is   pushed   to   a   certain   limit.   This   is   easy   to   illustrate   for   a   rectangular  

hyperbola.  The  function  value  of  the  hyperbola  ��(��)  never  actually  reaches  zero.  

However,  when  ��  increases  it  approaches  zero  more  and  more.  To  illustrate  this,   the  graph  of  a  hyperbola  has  been  included  again.   

Now  the  limit  of  this  hyperbola  function  đ?‘“đ?‘“ đ?‘Ľđ?‘Ľ =  for  đ?‘Ľđ?‘Ľ → ∞  is  defined  as  follows:   

33  


Theory  -­�  Mathematics    

 

1 = 0   → ��

Uniseminar  –  Quantitative  Methods  I  

lim

This   limit   is   now   actually   equal   to   zero   even   though   the   hyperbola   never   really   hits   the   axis.   However,   with   ��   approaching   infinity   ��(��)   approaches   0   or  

converges  to  0.  Sticking  to  this  function  we  can  also  take  the  limit  in  the  opposite   direction   and   let   đ?‘Ľđ?‘Ľ   approach   0,   which   lets   the   function   value   to   increase   to   virtually  infinity:   1 = ∞   → đ?‘Ľđ?‘Ľ lim

Here  are  some  rules  for  limits:  

If  it  holds  that  lim→ đ?‘“đ?‘“(đ?‘Ľđ?‘Ľ) = đ??´đ??´  and  lim→ g x = B,  then  it  also  holds  that:   

    

lim→ đ?‘“đ?‘“ đ?‘Ľđ?‘Ľ Âą đ?‘”đ?‘” đ?‘Ľđ?‘Ľ = đ??´đ??´ Âą đ??ľđ??ľ  

lim→ đ?‘“đ?‘“ đ?‘Ľđ?‘Ľ ∗ đ?‘”đ?‘” đ?‘Ľđ?‘Ľ = đ??´đ??´ ∗ đ??ľđ??ľ   lim→

 

 

lim→ �� ��

Example:  



=      đ?‘–đ?‘–đ?‘–đ?‘–  đ??ľđ??ľ ≠ 0   



= đ??´đ??´  

lim→ ��(��)  and  �� �� =





+ ��   

1 1 1 1 →   lim +  = lim + lim = ∞ + 1 = ∞   → 2đ?‘Ľđ?‘Ľ → 2đ?‘Ľđ?‘Ľ → đ?‘’đ?‘’ đ?‘’đ?‘’

 

3 . 2   D i f f e r e n t i a t i o n   R u l e s  

In   order   to   be   able   to   take   the   derivative   of   a   certain   function   one   needs   to   be   capable   of   applying   a   broad   variety   of   rules.   Therefore   in   the   following   we   will  

first   present   general   rules   of   differentiation   and   give   examples   to   each   of   them.   Afterwards   we   will   identify   certain   special   cases   and   show   how   to   differentiate   those.   Finally   some   harder   cases   in   which   more   than   one   rule   is   needed   will   be   34  


Theory  -­�  Mathematics    

7  

 

T w o    V V a r i a b l e    O O p t i m i z a t i o n  

Uniseminar  –  Quantitative  Methods  I  

7 . 1   E x t r e m e    P P o i n t s  

F i r s t    o o r d e r    cc o n d i t i o n s    ff o r    ee x t r e m e    p p o i n t s  

So   far   we   learned   how   to   differentiate   functions   of   two   independent   variables.   However,  to  be  able  to  find  the  optimal  solutions  to  such  functions  we  again  need   to   introduce   some   conditions.   Just   as   in   the   single   variable   case   there   are   again  

first  order  conditions  which  need  to  be  met.  In  order  to  be  a  stationary  point,  the  

slope  of  the  function  in  that  point  has  to  be  zero  again.  However,  since  we  have   two   independent   variables   we   also   have   two   partial   derivatives   and   therefore  

also  two  measures  of  the  slope.  It  is  insufficient  if  the  slope  is  only  zero  in  either   the  ��  or  the  ��  direction  and  it  therefore  has  to  hold  that  both  partial  derivatives   equal  zero:   

and   

�� ��, �� = 0  

�� ��, �� = 0  

Contrary   to   the   simple   case   this   is   now   not   only   an   equation   but   much   rather   a   system  of  equations  which  needs  to  be  solved.  In  order  to  make  this  point  clear,   consider  the  following  function:   1 đ?‘“đ?‘“ đ?‘Ľđ?‘Ľ, đ?‘Śđ?‘Ś = − đ?‘Ľđ?‘ĽÂ˛ − 2đ?‘Śđ?‘ŚÂ˛   2

Taking  the  partial  derivatives  and  plugging  them  equal  to  zero  respectively:   đ?‘“đ?‘“ đ?‘Ľđ?‘Ľ, đ?‘Śđ?‘Ś = −đ?‘Ľđ?‘Ľ = 0  

đ?‘“đ?‘“ đ?‘Ľđ?‘Ľ, đ?‘Śđ?‘Ś = −4đ?‘Śđ?‘Ś = 0  

This  solution  is  obviously  trivial  and  can  directly  be  read  of.     �� = 0   and  

�� = 0   56  


Uniseminar  –  Quantitative  Methods  I  

 

   Theory  -­�  Mathematics  

Plugging  these  points  into  function  allows  to  solve  for  the  function  value  in  that   stationary  point:   �� 0,0 = 0  

We   therefore   find   a   stationary   point   at   (0,0,0).   The  

graph   displays   the   three   dimensional   graph   of   the   function   and   makes   clear   that   this   stationary   point   must  be  a  maximum.  This  is  exactly  what  also  intuition   should  lead  to  as  the  function  consists  of  a  hill  shaped  

parabola   in   both   the   ��   and   the   ��   variable.   However,  

giving  a  mathematical  derivation  for  this  requires  again  second  order  derivatives   which  are  the  topic  of  the  next  section.  

However,   the   solution   to   the   system   resulting   from   the   first   order   conditions   cannot  always  be  easily  solved.  As  an  additional  example,  consider  this  function:   đ?‘“đ?‘“ đ?‘Ľđ?‘Ľ, đ?‘Śđ?‘Ś = 2đ?‘Ľđ?‘ĽÂ˛ − 4đ?‘Ľđ?‘Ľđ?‘Ľđ?‘Ľ + 4đ?‘Śđ?‘Ś  − 2đ?‘Ľđ?‘Ľ − 3đ?‘Śđ?‘Ś  

Again  taking  the  partial  derivatives  to  construct  the  first  order  conditions  gives:   (I)

(II)

đ?‘“đ?‘“ đ?‘Ľđ?‘Ľ, đ?‘Śđ?‘Ś = 4đ?‘Ľđ?‘Ľ − 4đ?‘Śđ?‘Ś − 2 = 0  

đ?‘“đ?‘“ đ?‘Ľđ?‘Ľ, đ?‘Śđ?‘Ś = −4đ?‘Ľđ?‘Ľ + 8đ?‘Śđ?‘Ś − 3 = 0  

This  is  now  a  linear  system  of  two  equations  and  two  unknowns.  Rewriting  (I)  in  

terms  of  ��  yields:   (I)

4�� = 4�� + 2  

1 → �� = �� +   2

Plugging  this  into  (II):   (II)

−4 đ?‘Śđ?‘Ś +

 

+ 8đ?‘Śđ?‘Ś − 3 = 0  

→ −4đ?‘Śđ?‘Ś − 2 + 8đ?‘Śđ?‘Ś − 3 = 0  

→  4�� = 5  

5 → �� =   4  

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Theory  -­�  Mathematics    

Which  then  can  be  used  to  solve  for  ��:  

 

Uniseminar  –  Quantitative  Methods  I  

7 �� =   4

Plugging  the  two  solutions  into  the  function  yields  a  value  for  ��:  

7 5 7 đ?‘“đ?‘“ , = 2 4 4 4



7 5 5 −4 ∗ +4 4 4 4



−2

7 5 27 −3 = −   4 4 4

We  therefore  know  that  there  must  be  a  stationary  point  at   without   any   second   order   conditions   it   is   now   hard   to  

 

, ,−

 

 

.  However,  

judge  whether  this  is  a  minimum  or  maximum  location.   A   graph   has   been   included   which   gives   a   graphical  

argument  in  favor  of  a  minimum.  This  argument  can  also   be   produced   without   having   a   graph,   as   the   function   value   is   increasing   for   đ?‘Ľđ?‘Ľ, đ?‘Śđ?‘Ś →  ∞   and   decreasing   for  

đ?‘Ľđ?‘Ľ, đ?‘Śđ?‘Ś → −∞.    

S e c o n d    o o r d e r    cc o n d i t i o n s    ff o r    ee x t r e m e    p p o i n t s  

As   in   the   simple   case   the   second   order   conditions   can   be   applied   to   identify   a  

stationary   point   as   being   a   minimum   or   a   maximum.   Even   though   these   conditions  are  not  that  simple  anymore  they  are  still  based  on  the  second  order  

derivatives   of   the   function.   These   second   order   derivatives   are   now   the   derivatives   of   the   partial   derivatives.   As   there   are   two   first   order   derivatives   there  exist  four  second  order  derivatives.  However,  one  general  property  which   always  applies  is  that:   

  �� ��, �� = �� (��, ��)  

This  means  that  it  does  not  matter  whether  we  first  differentiate  with  respect  to  ��  

and  then  with  respect  to  ��  or  vice  versa,  the  result  will  be  the  same.  In  a  way  this   cuts  down  the  number  of  second  order  derivatives  to  three.  

In  order  for  a  stationary  point  to  be  a  maximum  it  must  hold  that:   

58  

     đ?&#x2018;&#x201C;đ?&#x2018;&#x201C; đ?&#x2018;Ľđ?&#x2018;Ľ. đ?&#x2018;Śđ?&#x2018;Ś < 0,    đ?&#x2018;&#x201C;đ?&#x2018;&#x201C; đ?&#x2018;Ľđ?&#x2018;Ľ, đ?&#x2018;Śđ?&#x2018;Ś < 0    and    đ?&#x2018;&#x201C;đ?&#x2018;&#x201C; đ?&#x2018;Ľđ?&#x2018;Ľ, đ?&#x2018;Śđ?&#x2018;Ś â&#x2C6;&#x2014; đ?&#x2018;&#x201C;đ?&#x2018;&#x201C; đ?&#x2018;Ľđ?&#x2018;Ľ, đ?&#x2018;Śđ?&#x2018;Ś â&#x2C6;&#x2019; đ?&#x2018;&#x201C;đ?&#x2018;&#x201C; đ?&#x2018;Ľđ?&#x2018;Ľ, đ?&#x2018;Śđ?&#x2018;Ś



> 0  


Theory  -­‐  Statistics    

S t a t i s t i c s   1  

 

Uniseminar  –  Quantitative  Methods  I    

S a m p l i n g    aa n d    T T y p e s    o o f    D D a t a  

1 . 1   T y p e s    o o f    d d a t a  

Without   data,  statistics  would  be  impossible.  Whenever  a  statistical  statement  is  

made   this   is   based   on   some   form   of   data.   Basically   statistics   is   the   science   that   analyses  data  in  order  to  be  able  to  make  statements  about  it.  We  can  therefore   see  data  as  some  form  of   raw  information  about  anything.  The  main  purpose  of  

this  course  is  then  to  teach  you  how  to  make  use  of  this  information.  The  first  part   of   this   course   takes   a   rather   qualitative   standpoint,   and   teaches   how   to  

graphically  display  data  to  make  it  easier  to  understand  this  data.  Later  parts  then   take   a   rather   quantitative   approach   and   formally   analyse   data   by   applying  

different  tests.  However,  all  this  is  only  possible  if  you  have  a  clear  understanding   of  what  type  of  data  you  have  available  for  the  analysis.  In  general  two  different   types  can  be  observed:    

Q u a n t i t a t i v e    D D a t a  

Quantitative   data   can   in   most   cases   be   identified   by   the   fact   that   it   consists   of  

numbers.  Easy  examples  are  the  height  of  a  certain  person  or  the  maximum  speed   of   a   car.   However,   you   have   to   watch   out,   as   sometimes   numbers   are   used   to  

identify  certain  categories.  This  is  then  no  quantitative  data,  but  falls  in  the  next  

category.   A   good   way   to   distinguish   is   always   to   try   to   identify   a   certain   scale  

which   makes   intuitive   sense.   This   is   mostly   the   case   when   the   numbers   carry   a  

certain  type  of  unit  like  meters   or   kilograms.   If   all   this   is   the   case   then  the   data   type  is  quantitative.    

C a t e g o r i c a l    D D a t a  

Categorical  data  is  data  that  answers  rather  qualitative  questions  or  distinguishes  

between  certain  groups  or  categories.  Examples  can  be  the  tutorial  group  you  are  

in   or   the   whether   you   passed   an   exam.   Even   though   your   tutorial   group   is  

62  

 


Theory  -­‐  Statistics    

2  

D e s c r i p t i v e    SS t a t i s t i c s  

 

Uniseminar  –  Quantitative  Methods  I    

2 . 1   C a t e g o r i c a l    D D a t a  

The  previous  section  taught  you  how  to  discriminate  between  different  types  of   data.  This  knowledge  is  crucial  to  be  able  to  graph  data  the  appropriate  way.  The  

second   part   of   the   first   section   described   the   idea   of   sampling.   Whenever   we   work  with  data  here,  this  data  comes  from  sampling.  The  idea  is  again  to  describe   the   population   by   graphing   the   sample.   However,   this   issue   will   become   truly   important  in  later  sections  when  we  take  an  analytical  approach.  

Throughout   this   section   we   will   use   the   same   type   of   data   and   show   different  

ways  to  display  this  data.  For  this  purpose  consider  the  following  frequency  table  

which   gives   data   about   the   origin   of   Maastricht   University   first   year   students.   Next  to  this  table  you  find  another  table,  which  is  called  a  relative  frequency  table  

and   gives   percentages.   Further   note   that   all   data   is   completely   made   up   for   the   purpose  of  your  understanding.    

C o u n t r y    o o f   O r i g i n  

N e t h e r l a n d s   G e r m a n y   B e l g i u m   O t h e r    

T o t a l  

F r e q u e n c y  

581     501     73    

112    

1 2 6 7      

C o u n t r y    o o f   O r i g i n  

N e t h e r l a n d s   G e r m a n y   B e l g i u m   O t h e r   T o t a l  

R e l a t i v e   F r e q u e n c y  

45,86%   39,54%   5,76%   8,84%  

1 0 0 %  

While  those  tables  give  a  good  overview  concerning  the  actual  data,  they  do  not  

give   a   good   graphical   intuition   about   it.   This   can   be   achieved   by   converting   the   data  into  charts.  The   absolute  frequency  table  can  easily  be  converted  into  a   bar  

chart,  while  the   relative  frequency  table  is  better  represented  by  a  pie  chart.  The   two  graphs  are  displayed  below.  

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Uniseminar    –  Quantitative  Methods  I      

 

Theory  -­‐  Statistics    

 

A  last  type  of  table  combines  two  different  types  of  data  and  is  called  contingency  

table.  Especially  in  the  later  section  concerning  probabilities  this  table  will  prove  

handy.   By   adding   data   about   the   study   track   of �� the   students   to   the   data   about   their  country  of  origin  allows  us  to  construct  a  contingency  table.            

N L  

G E R   B E L  

O t h e r    

T o t a l  

E c o n o m i c s   I B   211  

317  

34  

37  

182  

61  

4 8 8  

F i s c .   E c o n o m e t r i c s   E c o n o m i c s  

297  

44  

6 9 5  

22  

31  

2  

0  

21   7  

5 2  

1   0  

3 2  

T o t a l  

5 8 1   5 0 1  

7 3  

1 1 2  

1 2 6 7  

For  sure  there  is  also  a  relative  contingency  table  in  which  all  the  data  sums  up  to   100%  instead  of  1267  observations.    

2 . 1   Q u a n t i t a t i v e    D D a t a   N u m e r i c a l    A A n a l y s i s  

Displaying   quantitative   data   is   a   bit   more   involving   than   it   was   with   categorical   data.   First   of   all   there   are   certain   measures   that   are   important   for   quantitative   data.  The  first  thing  that  needs  to  be  defined  is  the   centre  of  the  data.  The  best-­‐

known  concept  here  is  probably  that  of  the  m m e a n ,  which  is  defined  as:  

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Uniseminar    â&#x20AC;&#x201C;  Quantitative  Methods  I      

3  

P r o b a b i l i t y    tt h e o r y  

 

Theory  -­â&#x20AC;?  Statistics    

3 . 1   R u l e s    ff o r    P P r o b a b i l i t i e s  

So   far   you   know   about   the   different   kinds   of   data   and   how   to   display   these  

properly.   However,   whenever   any   statistical   statement   is   made   this   is   based   on  

some  form  of  calculation.  In  order  to  be  able  to  produce  these  calculations  basic   knowledge  of  probabilities  is  essential.  A  probability  is  defined  as  the  chance  that  

a  certain  event  occurs:   

0 â&#x2030;¤ đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x2018;&#x2039;đ?&#x2018;&#x2039; â&#x2030;¤ 1  

This   basically   means   that   every   probability   will   be   between   zero   and   one.   This   statement   should   make   intuitive   sense   as   the   chance   that   something   occurs   can   neither   be   smaller   than  0%  (not   going   to   happen)   nor   larger   than  100%  (will   happen  for  sure).  Additionally  we  can  define  that:   

đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x2018;&#x2039;đ?&#x2018;&#x2039; + đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x2018;&#x2039;đ?&#x2018;&#x2039; + â&#x2039;Ż đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x2018;&#x2039;đ?&#x2018;&#x2039; = 1  

which  means  that  the  sum  of  all  probabilities  must  be  equal  to  one.  In  everyday  

language   this   means   that   something   must   happen.   If   you   know   all   possible   options  than  one  of  these  must  occur  making  the  sum  of  all  probabilities  equal  to   one.  Therefore  we  also  know  that:   

đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x2018;&#x2039;đ?&#x2018;&#x2039; + đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x2018;&#x2039;đ?&#x2018;&#x2039; = 1  

Where   we   define  đ?&#x2018;&#x2039;đ?&#x2018;&#x2039;  to   be   the   opposite   or   the  

complement  of  đ?&#x2018;&#x2039;đ?&#x2018;&#x2039; .  If  đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x2018;&#x2039;đ?&#x2018;&#x2039;  gives  for  example  the  

probability   that   the   sun   shines   tomorrow,   than  

đ?&#x2018;&#x192;đ?&#x2018;&#x192;(đ?&#x2018;&#x2039;đ?&#x2018;&#x2039; )  will  give  the  probability  that  the  sun  does  

not   shine   tomorrow.   Obviously   the   probability   that   the   sun   shines   tomorrow   plus   the  

probability   that   it   does   not   shine   is   equal   to   one.   The   main   advantage   of   this   is  

that  if  we  know  the  probability  that  the  sun  will  shine  also  allows  us  to  calculate   the  probability  of  the  complement  since:   

đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x2018;&#x2039;đ?&#x2018;&#x2039; = 1 â&#x2C6;&#x2019; đ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x2018;&#x2039;đ?&#x2018;&#x2039;  

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Uniseminar  â&#x20AC;&#x201C;  Quantitative  Methods  I    

 

đ?&#x2019;&#x2018;đ?&#x2019;&#x2018;-­â&#x20AC;?-­â&#x20AC;?v v a l u e    tt e s t  

To  perform  a  đ?&#x2018;?đ?&#x2018;?-­â&#x20AC;?value  test  we  do  not  need  to  fix  the  significance  level  right  away.  

All  we  have  to  do  is  find  the  area  to  the  extreme  of  our  đ?&#x2018;§đ?&#x2018;§-­â&#x20AC;?value.  As  we  have  found  

a  value  of  â&#x2C6;&#x2019;2.07  we  need  to  find  the  area  to  the  extreme  or  as  it  is  negative  to  the  

left  of  this.  A  look  in  the  đ?&#x2018;§đ?&#x2018;§-­â&#x20AC;?table  will  show  that  the  area  is  indeed  equal  to  0.0192.  

However,  as  we  are  applying  a  two-­â&#x20AC;?sided  test  there  is  also  an  area  to  the  extreme   of  2.07  which   needs   to   be   considered.   Therefore   the  đ?&#x2018;?đ?&#x2018;?-­â&#x20AC;?value   is   equal   to  2 â&#x2C6;&#x2014;

0.0192 = 0.0384.   In   a   last   step   this  đ?&#x2018;?đ?&#x2018;?-­â&#x20AC;?value   can   then   be   compared   to   any  

significance   level  đ?&#x203A;źđ?&#x203A;ź  we   like.   Whenever   the  đ?&#x2018;?đ?&#x2018;?-­â&#x20AC;?value   is   larger   than  đ?&#x203A;źđ?&#x203A;ź  we   cannot  

reject   at   that   significance   level  đ?&#x203A;źđ?&#x203A;ź,   whenever   it   is   smaller   we   can   reject   at   that  đ?&#x203A;źđ?&#x203A;ź .   Again  this  is  summarized  by  a  graph.  Note  that  the  fact  that  the  đ?&#x2018;?đ?&#x2018;?-­â&#x20AC;?value  is  smaller  

than  the  significance  level  đ?&#x203A;źđ?&#x203A;ź  follow  directly  from  the  fact  that  the  critical  value  is   smaller   than   the  đ?&#x2018;§đ?&#x2018;§-­â&#x20AC;?score   as   the   areas   become   smaller   if   the  đ?&#x2018;§đ?&#x2018;§-­â&#x20AC;?values   become  

larger.  

     

94  

 


Seminar

Extras

Exams

Practice

P


Practice Exercises Quantitative Methods I Academic Year 2012/2013, Block 1


Practice Exercises This part contains practice exercises to each week and therefore to each chapter of the theory script. By this, you can deepen your theoretical knowledge with practical exercises and you can go through the exercises of these topics again, which you have not understood so well until now. Although you may think that you already have done enough exercises during the weeks, these exercises are tailored specifically to your needs and try to teach you the most important topics of the exam in a practical manner.

Table of Contents

Mathematics

1

Exercises

1

Solutions

13

Statistics

63

Exercises

63

Solutions

78


Uniseminar – Quantitative Methods I

Practice ‐ Mathematics

Mathematics ‐ Exercises 1

Week 1

Functions of one Variable 1.

For the following functions, a find the domain b solve the equations for

f x

i.

f x

ii.

f x

iii.

f x

iv.

f x

0 4x

√x

3

2.

Determine the straight line that satisfies the following conditions

i.

L : passes through point P 1/3 with slope m 2

ii.

L : passes through points P1 ‐2/2 and 3/3

iii.

L : passes through the origin with slope m ‐0.5

iv.

L : passes through points P1 a/0 and 0/b

3.

Sketch the graphs of the following functions

i.

y

x

ii.

y

x

iii.

y

x

iv.

y

x

v.

y

√x

vi.

y

√ 2x

4x

4

9 3 4

4.

Solve the following logarithmic equations for x

i.

2

1024

ii.

5

1


Practice ‐ Mathematics

5

Uniseminar – Quantitative Methods I

Week 5

Single Variable Optimization 1. i.

Find the maximum and minimum point s to the following functions 3

2

4

ii.

iii.

2

iv.

1

v. 2.

Economic Examples

i.

The price of a commodity is dependent on the quantity produced. A firm

has the following price and cost functions:

100

2 and

4 . Which quantity should the firm produce in order to maximize its

profits?

ii.

A firm produces a quantity Q of a product. The quantity is dependent on

the amount of labor L employed. The output is given by the function

4√ , with L being measured in hours. The price per unit of output is

120 Euro, and the cost of each hour of labor is 60 Euro. What are the profit

maximizing hours of Labor and corresponding quantities produced?

3.

Find possible inflection points to the following functions

i. ii. iii. 10


Practice ‐ Mathematics

6

Uniseminar – Quantitative Methods I

Week 6

Functions of two Variables 1.

Find the domain for which the function

i.

,

ii.

,

iii.

,

iv.

,

,

is defined

4 ln

ln

2

3

2.

Find all first‐ and second‐order partial derivatives for the following

functions

i.

,

ii.

,

iii.

,

iv.

,

v.

,

vi.

,

2 ln

ln

3.

Find the partial elasticities of z with respect to x and y; Remember:

iii. iv.

12

i. ii.


Uniseminar – Quantitative Methods I

7

Practice ‐ Mathematics

Week 7

Two Variable Optimization 1.

Find the minimum point to the following functions

i.

,

ii.

,

8 3

6

12

4

8

2.

A firm produces two different products A and B. The daily cost of

producing x units of A and y units of B is C x, y

40x

20y

4x

2xy

4y

256

i.

Suppose that the firm sells all its output at a unit price of €12 for A and €

16 for B. Find the production volume that maximizes profit

ii.

The firm is required to produce exactly 60 units of the two kinds combined.

Find the new optimal production plan

3.

Find x, y and z with 3

4

108 so that

is maximized

4.

For the following functions, find all stationary points and classify them

with the help of the second order conditions1 for extreme points.

i.

,

2

ii.

,

12

iii.

,

iv.

,

8

ln 1

1 See Math Theory Chapter 7

13


Uniseminar â&#x20AC;&#x201C; Quantitative Methods I

Practice â&#x20AC;? Mathematics

Mathematics â&#x20AC;&#x201C; Solutions 1

Solutions Week 1

1.

a The domain of the function

i.

f x

4x 2 | set the denominator equal to zero and solve 4 x

4 x

x

0 | Find all values of x for which the denominator is 0 4 | the domain are all values x

4

ii.

x x x

6x

x

f x

4

x 4

8x

0 | beware, the denominator contains the 3rd binomial formular

2 x 2 and x

2

0 2 | the domain are all values x

2 and x

2

iii. f x x

2x

16x x

32x

0 | the domain are all values x

0

15


Practice â&#x20AC;? Mathematics

iv.

Uniseminar â&#x20AC;&#x201C; Quantitative Methods I

f x x

4x

3 | the expression under the squareroot always needs to be larger 0

3

0 | factorize

4x

x x

3 x

1

0 | satisfied if either both terms are positive or both terms are

negative x

3 and x

1

1.

b Solve the functions for f x

i.

4x 2 4 x

0

0 | Setting the numerator of the equation equal to 0

and solving for x 4x

2

x

0

2

ii.

x

6x 4

x x

8x

6x

x x

6x

x x

2 x

x

8x

0 and x

8

0 | Generally, set the numerator equal to zero. Beware of den. 0 | Fatorize 0

4

0 4 | Note, that if 2, then the function is not defined see above

16


Practice – Mathematics

Uniseminar – Quantitative Methods I

3

Solutions Week 3

1.

Find the derivative of the following functions using the Newton Quotient

i.

2

2 2

,

0

2

ii.

2 2

3

3

3

3

2 ,

2

2

2

0

2 2

2

3

3

2.

Find ′ and ′′ of the following functions

i.

2 2

| 3

4

6

4

1 2

4

ii. 3

2

4

6 iii. 1 2

34

|

√ 4

1

2√

4

2


Uniseminar – Quantitative Methods I

12

1 4

1

Practice ‐ Mathematics

12

4√

3.

Find the line that is tangent to the graph in point P

i.

, 2 ,

,

3, 9

6 :

Using the point‐slope formula 9

6

6

3 9

2

ii. 2 ,

3 2

2

2 , ,

0, 2

2

0

2 2 iii.

2 ,

√ 1

2 ,

2√ 3 2

1

,

1, 1

1.5

1 1 2

3 2 4.

Examine where the following functions are increasing and decreasing 2

i.

2

2

First, find the derivative in order to find the domain s where is positive or negative respectively 4

3 4

3

2 4

,

2 | 0 |

4 ∙ 3 ∙ 2 √4 4∙3

0

35


Uniseminar â&#x20AC;&#x201C; Quantitative Methods I

3

Practice â&#x20AC;&#x201C; Statistics

Week 3

Probability Theory 1.

There is a bowl with 7 black balls and 4 red balls.

i.

What is the probability of drawing a red ball?

ii.

What is the probability of first drawing a red ball, and then without

replacing drawing a black ball.

iii.

What is the probability of drawing two black balls, if you place the black

ball you drew after the first draw?

2.

Consider the following table

Male

Female

Marketing

40

80

Engineering

35

7

Management

15

6

i.

What is the probability that a randomly selected employee is working in

Marketing?

ii.

Given that a randomly selected employee is a male, what is the probability

that he is working in Engineering?

iii.

Are gender and job category mutually exclusive?

3.

Consider a regular deck of cards 52 cards

i.

What is the probability that a randomly selected card is either a jack, queen

or king?

ii.

What is the probability that a randomly selected card is either a king or a

black card?

iii.

What is the probability that you draw 4 of the same kind on consecutive

draws

iv.

What is the probability that you draw 4 aces? 67


Practice ‐ Statistics

Uniseminar – Quantitative Methods I

4.

Evaluate the following probabilities for events, if A and B are not

independent.

i.

0.4,

ii.

0.3,

iii.

0.6,

0.7, ∩

∩ 0.15,

0.7

|

|

0.2 What is |

0.2 What is

0.36 What is

? ? ?

5.

You are playing a lottery. What is your expected outcome if

i.

A fair coin is tossed twice. Your payoff looks as follows: heads twice: win

10 Euro; tail twice: lose 4 Euro; tail and heads: lose 2 Euro

ii.

A fair coin is tossed three times: Your payoff looks as follows: For each

head, you receive 5 Euro. For each tail you lose 3 Euro.

iii.

For both cases i and ii above, what is the standard deviation of your

payoffs?

6.

Consider insurance claims. Of a 100 insurance claims, 40 relate to fire

insurance. However, the insurance knows that 16% of all insurance claims

are fraudulent. Construct a contingency table

i.

Which proportion of fire claims is not fraudulent

ii.

Are fraudulent and fire claim independent?

7.

Binomial Model: A baseball player has a batting average of 0.25 meaning,

he hits 1 out of 4 balls . Given a scenario of 3 attempts

i.

What is the probability that he hits 2 balls

ii.

What is the probability that he hits either 1 ball or 2 balls

iii.

What is the probability that he hits at least 2 balls?

68


Practice ‐ Statistics

Uniseminar – Quantitative Methods I

Statistics – Solutions 1

Solutions Week 1

1.

Classify the following data according to their type of scale, nominal ordinal,

interval or ratio.

i.

Is an example of a nominal scale, because you cannot associate any order or ranks with the outcomes of the game. ii.

Is an example of a nominal scale, because you cannot associate any order or ranks with the outcomes of the exam. iii.

Is an example of a ratio scale, because length as measured in meters is defined by a distinct zero value 0m and the distance between 1m and 2m is, for instance the same as the distance between 105.5m and 106.5m. iv.

This is an example of an interval scale: There is no meaningful zero‐value to temperature zero degree Celsius does not imply absence of temperature but the incremental heat change between different temperatures is meaningful, i.e. 10°C to 11°C or from 31°C to 32°C. v.

This is an example of an ordinal scale, because low, medium or high have meaningful ranks that distinguish them. 78


Uniseminar – Quantitative Methods I

vi.

Practice – Statistics

This is an example of a ratio scale. vii.

This is an example of an ordinal scale, because the stones are arranged in a meaningful order. 2.

The following group of data is?

i.

The data is cross‐sectional, as all respondents have filled in one questionnaire at one point in time. ii.

The data is an example of time series data, as a single item is observed at different points in time iii.

The data is cross‐sectional, as a selection of items is observed at the same point in time. iv.

The data is panel data as multiple observations are made at multiple points in time.

79


Uniseminar – Quantitative Methods I

Practice – Statistics

2

Solutions Week 2

1.

You read a magazine and come across the following table about starting

salaries of university graduates in the Netherlands, Germany and Belgium:

Min

Q1

Median

Q3

Max

Mean

€ 22,000 € 36,900 €41,533 €44,750 €51,000 €39,342 i.

IQR:

3

1

44750

36900

7850

ii.

Range:

51000

22000

29000

iii.

Skewness: Since the Mean is less than then Median and the distance between Q1 and the minimum is larger than the distance between Q3 and the maximum value, we can safely assume that the distribution is skewed with a tail to the left. iv.

The most appropriate measure of the central tendency is the median. The mean could easily be biased through extreme outliers, and thus is not the most appropriate measure. v.

Outliers are defined as being: 1 minimum: 36900

1.5

7850

1.5 25125

3

1.5

. Thus, for the

22000. Accordingly, the minimum is

an outlier For the maximum: 44700

1.5

7850

56245

51000. Accordingly, the

maximum is not an outlier 81


Practice ‐ Statistics

Uniseminar – Quantitative Methods I

vi.

Suppose that the data has been obtained from a sample of 40 graduates.

Now, responses of another 5 graduates are added, whose salaries are all in

the range of €39,000; 41,000 .

a.

The IQR shrinks, because both, Q1 and Q3 will be affected. Q1 will increase while Q3 will decrease. b.

The range is not affected, as minimum and maximum remain unchanged c.

Since all values added to the distribution are less than the median, the median will decrease. d.

We cannot be sure whether the mean will increase or decrease since the mean lies within the range of the newly added salaries. e.

Since the Median moves closer to the mean, the distribution will be less skewed, but still, it will be skewed with a tail to the left.

82


Seminar

Extras

Exams

E


Exams  

    Quantitative  Methods  I   Academic  Year  2012/2013,  Block  1    

   

 


Exams  

E x a m s  

 

 Uniseminar  –  Quantitative  Methods  I  

You   should   start   early   with   the   calculation   of   exams,   because   you   need   to   get   a   general   feeling   of   how   the   exams   are   built   up.   You   will   soon   discover   how   the  

exams  are  constructed  and  that  there  are  general  tendencies,  which  repeat  from   exam  to  exam.  In  this  part  you  will  find  old  exams  of  the  Maastricht  University,  as  

well   as   two   practice   exams   constructed   by   Uniseminar.   During   the   seminar   you   will  then  receive  a  third  practice  exam.  

 

T a b l e    o o f    C C o n t e n t s    

P r a c t i c e    E E x a m    1 1    (( i n c l .    ss o l u t i o n s )  

1  

 

 

P r a c t i c e    E E x a m    2 2    (( i n c l .    ss o l u t i o n s )   O l d    E E x a m s  

  11/12  Resit  

35 6 1  

  11/12  First  Sit     10/11  Resit  

  10/11  First  Sit     09/10  Resit  

  09/10  First  Sit     08/09  Resit  

  08/09  First  Sit     07/08  Resit  

  07/08  First  Sit  

 


Uniseminar  –  Quantitative  Methods  I    

P r a c t i c e    E E x a m    1 1   M a t h e m a t i c s   1.)

a.)

b.)

c.)

d.)

 

2.)

The  limit  of  the  function  f x =

Undefined.  

The  domain  of  the  function  f x = ln  (e − 1)  is  

(−∞,0)  

3.)

[0,∞)  

(−∞,0]  

Consider  the  function  f x = −x² + 3x + a.  For  which  value  of  a,  is  the  line    

 

y  =  −x  +  3  a  tangent  of  the  function?  

b.)

a  =  0  

a.)   c.)

d.)  

,  when  x → 0  

∞    

c.)  



1/6  

(0,∞)  

d.)

 

Practice  Exam  1  

6.  

a.)

b.)

 

a  =  2   a  =  1  

a  =  −1  

4.)

A  certain  radioactive  substance  decays  according  to  the  formula  

 

after  t  days.  The  half-­‐life,  the  time  at  which  half  of  the  mass  of  t=0  is  left,  of  

 

m t = 70e. ,   where   m   represents   the   mass   in   grams   that   remains  

 

this  substance  is  reached  after:  

b.)

11.2  days  

a.)

4.9  days  

1  


Practice  Exam  1    

c.)

d.)  

5.)

a.)

b.) c.)

 

48.6  days  

Uniseminar  –  Quantitative  Methods  I  

5.6  days  

What  is  the  inverse  of  the  function:  f x =

f  y =







f  y =    f  y =







 

   



 

 

d.)

f  y =   

6.)

Consider   the   two   points   A=(3,0)   and   B=(0,3).   The   point   P   =   (x,   y)   has  

 

 

equal    distance  to  both  A  and  B  exactly  when  

b.)

x² + y² = 9  

a.)

c.)

d.)  

x  +  y  =  0  

x  −  y  =  0   x−3



+ y−3



= 9  

7.)

The  derivative  of  the  function  f x = 4  at    x  =  2  is  equal  to  

b.)

4  ln  4  

d.)

4  

a.)

c.)

 

2  

ln  2  

ln  4  

 


Uniseminar  –  Quantitative  Methods  I    

S t a t i s t i c s  

 

Practice  Exam  1  

2 1 . ) Which  of  the  following  statements  are  true?  

I.  Categorical  variables  are  the  same  as  qualitative  variables.  

II.  Categorical  variables  are  the  same  as  quantitative  variables.   III.  Quantitative  variables  can  be  continuous  variables.   a.)

I  only  

c.)

I  and  III  

b.) d.)  

II  only  

II  and  III  

2 2 . ) Which  of  the  following  statements  are  true?  

I.  Random  sampling  is  a  good  way  to  reduce  response  bias.  

II.  To  guard  against  bias  from  undercoverage,  use  a  convenience  sample.   III.  To  guard  against  nonresponse  bias,  use  a  mail-­‐in  survey.  

a.)

I  only  

c.)

III  only  

b.) d.)

 

II  only  

None  of  the  above  

2 3 . ) Which   of   the   following   combinations   of   summary   statistics   are   most    

appropriate    for  describing  a  skewed  distribution?  

a.)

median  and  standard  deviation  

c.)

median  and  IQR  

b.) d.)

 

mean  and  range  

 mean  and  standard  deviation    

7  


Practice  Exam  1    

 

Uniseminar  –  Quantitative  Methods  I  

2 4 . ) The   local   park   district   is   planning   to   build   a   recreation  center.  The  park  district  conducted  a  poll  to   find   out   the   types   of   physical   activities   the   local  

populations  would  be  interested  in.  The  poll  was  based  

on  telephone  responses  from  1013  randomly  selected  adults.  The  table  shows  the   percentages  of  people  who  espressed  interest  in  various  activities  

Which  of  the  following  displays  is/  are  appropriate  for  these  data?  

 

a.)

b.)

c.)

d.)

 

8  

 

I  

I  and  II  

I,  II  and  III  

None  of  the  above  

 


Uniseminar  â&#x20AC;&#x201C;  Quantitative  Methods  I    

 

Practice  Exam  1  

2 5 . ) Four   symmetrical   distributions   are   represented   below   with   their    

 

histograms.    Without   performing   any   calculations   order   their   respective  

standard  deviations.    

a.)

đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; < đ?&#x153;&#x17D;đ?&#x153;&#x17D; < đ?&#x153;&#x17D;đ?&#x153;&#x17D; < đ?&#x153;&#x17D;đ?&#x153;&#x17D;  

c.)

đ?&#x153;&#x17D;đ?&#x153;&#x17D; < đ?&#x153;&#x17D;đ?&#x153;&#x17D; < đ?&#x153;&#x17D;đ?&#x153;&#x17D; < đ?&#x153;&#x17D;đ?&#x153;&#x17D;  

b.) d.)  

 

đ?&#x153;&#x17D;đ?&#x153;&#x17D; < đ?&#x153;&#x17D;đ?&#x153;&#x17D; < đ?&#x153;&#x17D;đ?&#x153;&#x17D; < đ?&#x153;&#x17D;đ?&#x153;&#x17D;   đ?&#x153;&#x17D;đ?&#x153;&#x17D; < đ?&#x153;&#x17D;đ?&#x153;&#x17D; < đ?&#x153;&#x17D;đ?&#x153;&#x17D; < đ?&#x153;&#x17D;đ?&#x153;&#x17D;  

2 6 . ) Consider  the  boxplot  below.  Which  of  the  following  statements  are  true?   I.  The  distribution  is  skewed  to  the  right.   II.  The  median  is  about  10.  

III.  The  interquartile  range  is  about  8.            

a.)

I  only    

c.)

III  only    

b.)

II  only  

9  


Uniseminar  â&#x20AC;&#x201C;  Quantitative  Methods  I  

 

Practice  Exam  1  

P r a c t i c e    E E x a m    1 1    -­â&#x20AC;?-­â&#x20AC;?    SS o l u t i o n s   M a t h e m a t i c s   1 -­â&#x20AC;?-­â&#x20AC;?C C :  

With  this  function  it  is  impossible  to  let  x  approach  zero  as  the  denominator   would  then  become  zero  as  well.  Therefore  some  rewriting  is  necessary:     đ?&#x2018;&#x201C;đ?&#x2018;&#x201C; đ?&#x2018;Ľđ?&#x2018;Ľ = =

 

đ?&#x2018;Ľđ?&#x2018;Ľ â&#x2C6;&#x2014;

x+9â&#x2C6;&#x2019; 9 x+9â&#x2C6;&#x2019; 9 x+9+ 9 đ?&#x2018;Ľđ?&#x2018;Ľ + 9 â&#x2C6;&#x2019; 9 = â&#x2C6;&#x2014; =   x x x + 9 + 9 đ?&#x2018;Ľđ?&#x2018;Ľ â&#x2C6;&#x2014; x + 9 + 9 đ?&#x2018;Ľđ?&#x2018;Ľ

x+9+ 9

1

=

x+9+ 9

Now  it  is  possible  to  take  the  limit:   lim

â&#x2020;&#x2019;

đ?&#x;?đ?&#x;?

C :     Which  gives  answer  C đ?&#x;&#x201D;đ?&#x;&#x201D;

 

 

1

x+9+ 9

=

1   3+3

2 -­â&#x20AC;?-­â&#x20AC;?A A :  

The  natural  logarithm  is  only  defined  for  values  larger  than  zero,  therefore     đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;  â&#x2C6;&#x2019; 1 > 0  

â&#x2020;&#x2019; đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;  > 1  

â&#x2020;&#x2019; ln đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;  > ln 1  

â&#x2020;&#x2019; đ?&#x2018;Ľđ?&#x2018;Ľ > 0  

There  is  no  upper  bound  and  therefore  the  domain  of  the  function  is  A :    (đ?&#x;&#x17D;đ?&#x;&#x17D;, â&#x2C6;&#x17E;)..      

 

17  


Practice  Exam  1    

3 -­â&#x20AC;?-­â&#x20AC;?D D :  

 

Uniseminar  â&#x20AC;&#x201C;  Quantitative  Methods  I    

In  order  to  be  tangent  two  conditions  need  to  be  satisfied.  First  the  line  and  the   function  need  to  have  the  same  slope  and  second  they  need  to  cross.  From  the   first  one,  it  follows:    

â&#x2C6;&#x2019;2đ?&#x2018;Ľđ?&#x2018;Ľ + 3 = â&#x2C6;&#x2019;1     â&#x2020;&#x2019;    đ?&#x2018;Ľđ?&#x2018;Ľ = 2  

The  second  condition  can  be  expressed  as  follows:   â&#x2C6;&#x2019;đ?&#x2018;Ľđ?&#x2018;Ľ  + 3đ?&#x2018;Ľđ?&#x2018;Ľ + đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; = â&#x2C6;&#x2019;đ?&#x2018;Ľđ?&#x2018;Ľ + 3   â&#x2020;&#x2019; đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; = đ?&#x2018;Ľđ?&#x2018;Ľ  â&#x2C6;&#x2019; 4đ?&#x2018;Ľđ?&#x2018;Ľ + 3  

Plugging  in  the  value  of  đ?&#x2018;Ľđ?&#x2018;Ľ,  which  has  been  derived  from  the  first  condition,  allows  

to  solve  for  đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;:  

đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; = 4 â&#x2C6;&#x2019; 8 + 3 = â&#x2C6;&#x2019;1  

Therefore  the  right  answer  is  D D :    đ?&#x2019;&#x201A;đ?&#x2019;&#x201A; = â&#x2C6;&#x2019;đ?&#x;?đ?&#x;?..    

4 -­â&#x20AC;?-­â&#x20AC;?B B :  

First  of  all  we  need  to  know  the  amount  of  the  radioactive  substance  at  t=0,   which  is:    

đ?&#x2018;&#x161;đ?&#x2018;&#x161; 0 = 70đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;  = 70  

At  half-­â&#x20AC;?life  the  mass  of  the  substance,  is  therefore  equal  to  35.  As  a  consequence,   it  must  hold  that:   35 = 70e.  

1 = đ?&#x2018;&#x2019;đ?&#x2018;&#x2019; .   2

ln

1 = â&#x2C6;&#x2019;0.062đ?&#x2018;Ąđ?&#x2018;Ą   2

1 ln 2   đ?&#x2018;Ąđ?&#x2018;Ą = â&#x2C6;&#x2019; 0.062

B :    1 1 1 . 2    d d a y s   Which  solves  exactly  for  answer  B  

 

18  


Seminar

Extras

E


Extras  

    Quantitative  Methods  I   Academic  Year  2012/2013,  Block  1  

               

   

 

 


Extras    

E x t r a s  

 

Uniseminar  –  Quantitative  Methods  I  

In   general,   the   Extras   part   will   supply   you   with   several   extras   that   will   be   very   helpful   for   your   exam   preparation.   In   the   script   of   this   course,   you   will   find   an   explanation   of   how   to   read-­‐off   the   critical   values   on   the   statistics   tables   and   an  

extensive  formula  sheet  for  mathematics  as  well  as  statistics.  

 

T a b l e    o o f    C C o n t e n t s    

F o r m u l a    SS h e e t s  

1  

  Statistics  

5  

  Mathematics    

H o w    tt o    rr e a d    ss t a t i s t i c s    tt a b l e s      

1  

9  

 


Uniseminar  –  Quantitative  Methods  I  

 

M a t h e m a t i c s   0  

Formula  Sheet  

P r e v i o u s    K K n o w l e d g e  

B i n o m i a l    F F o r m u l a s :   

   



𝑎𝑎 + 𝑏𝑏



𝑎𝑎 − 𝑏𝑏

= 𝑎𝑎 + 2𝑎𝑎𝑎𝑎 + 𝑏𝑏    = 𝑎𝑎 − 2𝑎𝑎𝑎𝑎 + 𝑏𝑏   

𝑎𝑎 + 𝑏𝑏 𝑎𝑎 − 𝑏𝑏 = 𝑎𝑎 − 𝑏𝑏   

P Q -­‐-­‐F F o r m u l a :    



 𝑥𝑥  + 𝑝𝑝𝑝𝑝 + 𝑞𝑞 = 0       →         𝑥𝑥, = − ± 

  

− 𝑞𝑞  

P o w e r s :  

 𝑎𝑎 = 1  

 𝑎𝑎 ∗ 𝑎𝑎 = 𝑎𝑎   

 

= 𝑎𝑎  

 𝑎𝑎 ∙ 𝑏𝑏  = 𝑎𝑎𝑎𝑎   

  

  



=  = 𝑎𝑎 𝑏𝑏   

w i t h    p p o w e r s :   G r o w t h    w 𝐾𝐾 = 𝐾𝐾 ∙ 1 + 𝑔𝑔     

L o g a r i t h m s :  

 ln 𝑥𝑥𝑥𝑥 = ln 𝑥𝑥 + ln 𝑦𝑦    ln





 ln 𝑥𝑥



= ln 𝑥𝑥 − ln 𝑦𝑦   = 𝑡𝑡 ∙ ln 𝑥𝑥  

 ln 1 = 0  

 ln 𝑒𝑒  = 𝑎𝑎    𝑒𝑒   = 𝑎𝑎  

1  


Formula  Sheet    

 

 𝐸𝐸 𝑋𝑋 ± 𝑌𝑌 = 𝐸𝐸 𝑋𝑋 + 𝐸𝐸 𝑌𝑌  

Uniseminar  –  Quantitative  Methods  I  

 𝑉𝑉𝑉𝑉𝑉𝑉 𝑋𝑋 ± 𝑐𝑐 = 𝑉𝑉𝑉𝑉𝑉𝑉 𝑋𝑋    𝑆𝑆𝑆𝑆 𝑋𝑋 ± 𝑐𝑐 = 𝑆𝑆𝑆𝑆(𝑋𝑋)  

 𝑉𝑉𝑉𝑉𝑉𝑉 𝑐𝑐 ∗ 𝑋𝑋 = 𝑐𝑐  ∗ 𝑉𝑉𝑉𝑉𝑉𝑉 𝑋𝑋    

 𝑉𝑉𝑉𝑉𝑉𝑉 𝑋𝑋 ± 𝑌𝑌 = 𝑉𝑉𝑉𝑉𝑉𝑉 𝑋𝑋 ± 𝑉𝑉𝑉𝑉𝑉𝑉 𝑌𝑌          

B i n o m i a l    M M o d e l :    𝑃𝑃 𝑋𝑋 = 𝑥𝑥 =

 

 𝐸𝐸 𝑋𝑋 = 𝑛𝑛 ∗ 𝑝𝑝  

 

 𝑆𝑆𝑆𝑆 𝑋𝑋 =

𝑝𝑝  ∗ 1 − 𝑝𝑝



                   𝑤𝑤𝑤𝑤𝑤𝑤ℎ  

 

=

!

!∗  !

 

𝑛𝑛 ∗ 𝑝𝑝 ∗ (1 − 𝑝𝑝)  

P o i s s o n    D D i s t r i b u t i o n :    𝑃𝑃 𝑋𝑋 = 𝑥𝑥 =  𝐸𝐸 𝑋𝑋 =  𝜆𝜆  

   

3    

!

 

 𝑆𝑆𝑆𝑆 𝑋𝑋 = 𝜆𝜆  

S a m p l i n g    d d i s t r i b u t i o n s    aa n d    tt h e    n n o r m a l    m m o d e l  

T e s t    SS t a t i s t i c :    𝑧𝑧 =



  



=

 ()

 

P a r a m e t e r  

S t a t i s t i c  

𝝁𝝁    

𝑦𝑦  

𝝆𝝆    

6  

  

𝑝𝑝  

S D ( s t a t i s t i c )   𝑝𝑝𝑝𝑝   𝑛𝑛 𝜎𝜎   𝑛𝑛

S D ( s t a t i s t i c )   𝑝𝑝𝑞𝑞   𝑛𝑛

𝑠𝑠

𝑛𝑛

 


Uniseminar  –  Quantitative  Methods  I    

 

Extras  

H o w    tt o    rr e a d    tt h e    SS t a t i s t i c s    tt a b l e s   z -­‐-­‐tt a b l e  

The   𝑧𝑧-­‐scores   are   given   on   the   outside   of   the   table.   The   numbers   in   the   inside   correspond  to  the  area  under  the  curve,  which  represents  the  probability.          

z  

0 . 0 0  

0 . 0 1  

0 . 0 2  

0 . 0 3  

0 . 0 4  

0 . 0 5  

0 . 0 6  

0 . 0 7  

0 . 0 8  

0 . 0 9  

0 . 0   0.5000   0.5040   0.5080   0.5120   0.5160   0.5199   0.5239   0.5279   0.5319   0.5359   0 . 1   0.5398   0.5438   0.5478   0.5517   0.5557   0.5596   0.5636   0.5675   0.5714   0.5753   0 . 2   0.5793   0.5832   0.5871   0.5910   0.5948   0.5987   0.6026   0.6064   0.6103   0.6141   0 . 3   0.6179   0.6217   0.6255   0.6293   0.6331   0.6368   0.6406   0.6443   0.6480   0.6517   0 . 4   0.6554   0.6591   0.6628   0.6664   0.6700   0.6736   0.6772   0.6808   0.6844   0.6879   0 . 5   0.6915   0.6950   0.6985   0.7019   0.7054   0.7088   0.7123   0.7157   0.7190   0.7224   0 . 6   0.7257   0.7291   0.7324   0.7357   0.7389   0.7422   0.7454   0.7486   0.7517   0.7549   0 . 7   0.7580   0.7611   0.7642   0.7673   0.7704   0.7734   0.7764   0.7794   0.7823   0.7852   0 . 8   0.7881   0.7910   0.7939   0.7967   0.7995   0.8023   0.8051   0.8078   0.8106   0.8133   0 . 9   0.8159   0.8186   0.8212   0.8238   0.8264   0.8289   0.8315   0.8340   0.8365   0.8389   1 . 0   0.8413   0.8438   0.8461   0.8485   0.8508   0.8531   0.8554   0.8577   0.8599   0.8621   1 . 1   0.8643   0.8665   0.8686   0.8708   0.8729   0.8749   0.8770   0.8790   0.8810   0.8830   1 . 2   0.8849   0.8869   0.8888   0.8907   0.8925   0.8944   0.8962   0.8980   0.8997   0.9015   1 . 3   0.9032   0.9049   0.9066   0.9082   0.9099   0.9115   0.9131   0.9147   0.9162   0.9177   1 . 4   0.9192   0.9207   0.9222   0.9236   0.9251   0.9265   0.9279   0.9292   0.9306   0.9319   1 . 5   0.9332   0.9345   0.9357   0.9370   0.9382   0.9394   0.9406   0.9418   0.9429   0.9441   1 . 6   0.9452   0.9463   0.9474   0.9484   0.9495   0.9505   0.9515   0.9525   0.9535   0.9545   1 . 7   0.9554   0.9564   0.9573   0.9582   0.9591   0.9599   0.9608   0.9616   0.9625   0.9633   1 . 8   0.9641   0.9649   0.9656   0.9664   0.9671   0.9678   0.9686   0.9693   0.9699   0.9706   1 . 9   0.9713   0.9719   0.9726   0.9732   0.9738   0.9744   0.9750   0.9756   0.9761   0.9767   2 . 0   0.9772   0.9778   0.9783   0.9788   0.9793   0.9798   0.9803   0.9808   0.9812   0.9817   2 . 1   0.9821   0.9826   0.9830   0.9834   0.9838   0.9842   0.9846   0.9850   0.9854   0.9857   2 . 2   0.9861   0.9864   0.9868   0.9871   0.9875   0.9878   0.9881   0.9884   0.9887   0.9890   2 . 3   0.9893   0.9896   0.9898   0.9901   0.9904   0.9906   0.9909   0.9911   0.9913   0.9916   2 . 4   0.9918   0.9920   0.9922   0.9925   0.9927   0.9929   0.9931   0.9932   0.9934   0.9936   2 . 5   0.9938   0.9940   0.9941   0.9943   0.9945   0.9946   0.9948   0.9949   0.9951   0.9952   2 . 6   0.9953   0.9955   0.9956   0.9957   0.9959   0.9960   0.9961   0.9962   0.9963   0.9964   2 . 7   0.9965   0.9966   0.9967   0.9968   0.9969   0.9970   0.9971   0.9972   0.9973   0.9974   2 . 8   0.9974   0.9975   0.9976   0.9977   0.9977   0.9978   0.9979   0.9979   0.9980   0.9981  

 

2 . 9   0.9981   0.9982   0.9982   0.9983   0.9984   0.9984   0.9985   0.9985   0.9986   0.9986   3 . 0   0.9987   0.9987   0.9987   0.9988   0.9988   0.9989   0.9989   0.9989   0.9990   0.9990  

9  



UM_QM1_1213_Issuu