Confidence Intervals - Business Statistics - Homework, Exercises of Business Statistics

Download Confidence Intervals - Business Statistics - Homework and more Exercises Business Statistics in PDF only on Docsity! Source URL: http://cnx.org/content/m16966/latest/ Saylor URL: http://saylor.org/courses/bus204 Attributed to: [Susan Dean and Barbara Illowsky] Saylor.org Page 1 of 15 Confidence  Intervals:  Homework   Susan  Dean  and  Barbara  Illowsky  (2012)     NOTE:    If  you  are  using  a  student-­‐t  distribution  for  a  homework  problem  below,  you  may  assume  that   the  underlying  population  is  normally  distributed.    (In  general,  you  must  first  prove  that  assumption,   though.)   EXERCISE  1   Among  various  ethnic  groups,  the  standard  deviation  of  heights  is  known  to  be  approximately  3  inches.     We  wish  to  construct  a  95%  confidence  interval  for  the  average  height  of  male  Swedes.    48  male  Swedes   are  surveyed.    The  sample  mean  is  71  inches.    The  sample  standard  deviation  is  2.8  inches.   a. Complete  the  following:     i.  =  _____       ii. σ  =  _____     iii. sx  =  _____     iv. n  =  _____           v. n  -­‐  1  =  _____   b. Define  the  Random  Variables  X  and   ,  in  words.   c. Which  distribution  should  you  use  for  this  problem?    Explain  your  choice.   d. Construct  a  95%  confidence  interval  for  the  population  average  height  of  male  Swedes.   i. State  the  confidence  interval.           ii. Sketch  the  graph.   iii. Calculate  the  error  bound.   e. What  will  happen  to  the  level  of  confidence  obtained  if  1000  male  Swedes  are  surveyed  instead  of  48?     Why?   EXERCISE  2   In  six  packages  of  “The  Flintstones  Real  Fruit  Snacks”  there  were  5    Bam-­‐Bam  snack  pieces.  The  total   number  of  snack  pieces  in  the  six  bags  was  68.    We  wish  to  calculate  a  96%  confidence  interval  for  the   population  proportion  of  Bam-­‐Bam  snack  pieces.   a. Define  the  Random  Variables  X  and  P',  in  words.   b. Which  distribution  should  you  use  for  this  problem?    Explain  your  choice     c. Calculate  p’.   d. Construct  a  96%  confidence  interval  for  the  population  proportion  of  Bam-­‐Bam  snack  pieces  per   bag.   Source URL: http://cnx.org/content/m16966/latest/ Saylor URL: http://saylor.org/courses/bus204 Attributed to: [Susan Dean and Barbara Illowsky] Saylor.org Page 2 of 15 i. State  the  confidence  interval.         ii. Sketch  the  graph.   iii. Calculate  the  error  bound.   e. Do  you  think  that  six  packages  of  fruit  snacks  yield  enough  data  to  give  accurate  results?    Why  or   why  not?   EXERCISE  3   A  random  survey  of  enrollment  at  35  community  colleges  across  the  United  States  yielded  the  following   figures    (source:    Microsoft  Bookshelf):    6414;  1550;  2109;  9350;  21828;  4300;  5944;  5722;  2825;  2044;   5481;  5200;  5853;  2750;  10012;  6357;  27000;  9414;  7681;  3200;  17500;  9200;  7380;  18314;  6557;   13713;  17768;  7493;  2771;  2861;  1263;  7285;  28165;  5080;  11622.    Assume  the  underlying  population  is   normal.   a. Complete  the  following:     i.  =  _____         ii. sx  =  _____     iii. n  =  _____           iv. n  -­‐  1  =  _____   b. Define  the  Random  Variables  X  and   ,  in  words.   c. Which  distribution  should  you  use  for  this  problem?    Explain  your  choice.   d. Construct  a  95%  confidence  interval  for  the  population  average  enrollment  at  community  colleges  in   the  United  States.   i. State  the  confidence  interval.           ii. Sketch  the  graph.   iii. Calculate  the  error  bound.   e. What  will  happen  to  the  error  bound  and  confidence  interval  if  500  community  colleges  were   surveyed?    Why?   EXERCISE  4   From  a  stack  of  IEEE  Spectrum  magazines,  announcements  for  84    upcoming  engineering  conferences   were  randomly  picked.    The  average  length  of  the  conferences  was  3.94  days,  with  a  standard  deviation   of  1.28  days.  Assume  the  underlying  population  is  normal.   a. Define  the  Random  Variables  X  and   ,    in  words.   Source URL: http://cnx.org/content/m16966/latest/ Saylor URL: http://saylor.org/courses/bus204 Attributed to: [Susan Dean and Barbara Illowsky] Saylor.org Page 5 of 15 iii. Calculate  the  error  bound.   f. Construct  a  98%  confidence  interval  for  the  population  average  weight  of  the  candies.       i. State  the  confidence  interval.           ii. Sketch  the  graph.   iii. Calculate  the  error  bound.   g. In  complete  sentences,  explain  why  the  confidence  interval  in  f  is  larger  than  the  confidence   interval  in  e.   h. In  complete  sentences,  give  an  interpretation  of  what  the  interval  in  f  means.   EXERCISE  8   A  pharmaceutical  company  makes  tranquilizers.    It  is  assumed  that  the  distribution  for  the  length  of   time  they  last  is  approximately  normal.    Researchers  in  a  hospital  used  the  drug  on  a  random  sample  of   9  patients.    The  effective  period  of  the  tranquilizer  for  each  patient  (in  hours)  was  as  follows:    2.7;  2.8;   3.0;  2.3;  2.3;  2.2;  2.8;  2.1;  and  2.4  .       a. Complete  the  following   i.  =  __________     ii. sx  =  __________     iii. n  =  ________           iv. n  -­‐  1  =  ________   b. Define  the  Random  Variable  X,  in  words.   c. Define  the  Random  Variable   ,  in  words.   d. Which  distribution  should  you  use  for  this  problem?    Explain  your  choice.   e. Construct  a  95%  confidence  interval  for  the  population  average  length  of  time.   i. State  the  confidence  interval.           ii. Sketch  the  graph.   iii. Calculate  the  error  bound.   f. What  does  it  mean  to  be  “95%  confident”  in  this  problem?   EXERCISE  9   Suppose  that  14  children  were  surveyed  to  determine  how  long  they  had  to  use  training  wheels.    It  was   revealed  that  they  used  them  an  average  of  6  months  with  a  sample  standard  deviation  of  3  months.   Assume  that  the  underlying  population  distribution  is  normal.   a. Complete  the  following   Source URL: http://cnx.org/content/m16966/latest/ Saylor URL: http://saylor.org/courses/bus204 Attributed to: [Susan Dean and Barbara Illowsky] Saylor.org Page 6 of 15 i.  =  __________     ii. sx  =  __________     iii. n  =  ________         iv. n  -­‐  1  =  ________   b. Define  the  Random  Variable  X,  in  words.   c. Define  the  Random  Variable   ,  in  words.   d. Which  distribution  should  you  use  for  this  problem?    Explain  your  choice.   e. Construct  a  99%  confidence  interval  for  the  population  average  length  of  time  using  training  wheels.   i. State  the  confidence  interval.           ii. Sketch  the  graph.   iii. Calculate  the  error  bound.   f. Why  would  the  error  bound  change  if  the  confidence  level  was  lowered  to  90%?   EXERCISE  10   Insurance  companies  are  interested  in  knowing  the  population  percent  of  drivers  who  always  buckle  up   before  riding  in  a  car.   a. When  designing  a  study  to  determine  this  population  proportion,  what  is  the  minimum  number  you   would  need  to  survey  to  be  95%  confident  that  the  population  proportion  is  estimated  to  within  0.03?   b. If  it  was  later  determined  that  it  was  important  to  be  more  than  95%  confident  and  a  new  survey  was   commissioned,  how  would  that  affect  the  minimum  number  you  would  need  to  survey?    Why?   EXERCISE  11   Suppose  that  the  insurance  companies  did  do  a  survey.    They  randomly  surveyed  400  drivers  and  found   that  320  claimed  to  always  buckle  up.  We  are  interested  in  the  population  proportion  of  drivers  who   claim  to  always  buckle  up.   a. Complete  the  following   i. x  =  __________       ii. n  =  ________       iii. p'=  __________   b. Define  the  Random  Variables  X  and  P',  in  words.   c. Which  distribution  should  you  use  for  this  problem?    Explain  your  choice.   d. Construct  a  95%  confidence  interval  for  the  population  proportion  who  claim  to  always  buckle   up.   i. State  the  confidence  interval.           ii. Sketch  the  graph.   iii. Calculate  the  error  bound.   Source URL: http://cnx.org/content/m16966/latest/ Saylor URL: http://saylor.org/courses/bus204 Attributed to: [Susan Dean and Barbara Illowsky] Saylor.org Page 7 of 15 e. If  this  survey  were  done  by  telephone,  list  3  difficulties  the  companies  might  have  in  obtaining   random  results.   EXERCISE  12   Unoccupied  seats  on  flights  cause  airlines  to  lose  revenue.    Suppose  a  large  airline  wants  to  estimate  its   average  number  of  unoccupied  seats  per  flight  over  the  past  year.    To  accomplish  this,  the  records  of   225  flights  are  randomly  selected  and  the  number  of  unoccupied  seats  is  noted  for  each  of  the  sampled   flights.    The  sample  mean  is  11.6  seats  and  the  sample  standard  deviation  is  4.1  seats.   a. Complete  the  following   i.  =  __________     ii. sx  =  __________     iii. n  =  ________         iv. n  -­‐  1  =  ________   b. Define  the  Random  Variables  X  and   ,  in  words.   c. Which  distribution  should  you  use  for  this  problem?    Explain  your  choice.   d. Construct  a  92%  confidence  interval  for  the  population  average  number  of  unoccupied  seats  per   flight.   i. State  the  confidence  interval.           ii. Sketch  the  graph.   iii. Calculate  the  error  bound.   EXERCISE  13   According  to  a  recent  survey  of  1200  people,  61%  feel  that  the  president  is  doing  an  acceptable  job.  We   are  interested  in  the  population  proportion  of  people  who  feel  the  president  is  doing  an  acceptable  job.   a. Define  the  Random  Variables  X  and  P',  in  words.   b. Which  distribution  should  you  use  for  this  problem?    Explain  your  choice.   c. Construct  a  90%  confidence  interval  for  the  population  proportion  of  people  who  feel  the   president  is  doing  an  acceptable  job.   i. State  the  confidence  interval.           ii. Sketch  the  graph.   iii. Calculate  the  error  bound.     EXERCISE  14   A  survey  of  the  average  amount  of  cents  off  that  coupons  give  was  done  by  randomly  surveying  one   coupon  per  page  from  the  coupon  sections  of  a  recent    San  Jose  Mercury  News.    The  following  data  were   collected:    20¢;  75¢;  50¢;  65¢;  30¢;  55¢;  40¢;  40¢;  30¢;  55¢;  $1.50;  40¢;  65¢;  40¢.    Assume  the   underlying  distribution  is  approximately  normal.   a. Complete  the  following   Source URL: http://cnx.org/content/m16966/latest/ Saylor URL: http://saylor.org/courses/bus204 Attributed to: [Susan Dean and Barbara Illowsky] Saylor.org Page 10 of 15   i. State  the  confidence  interval.           ii. Sketch  the  graph.   iii. Calculate  the  error  bound.   d. Explain  what  a  “97%  confidence  interval”  means  for  this  study.   EXERCISE  19   In  a  recent  sample  of  84  used  cars  sales  costs,  the  sample  mean  was  $6425  with  a  standard  deviation  of   $3156.    Assume  the  underlying  distribution  is  approximately  normal.   a. Which  distribution  should  you  use  for  this  problem?    Explain  your  choice.   b. Define  the  Random  Variable   ,  in  words.   c. Construct  a  95%  confidence  interval  for  the  population  average  cost  of  a  used  car.   i. State  the  confidence  interval.           ii. Sketch  the  graph.   iii. Calculate  the  error  bound.   d. Explain  what  a  “95%  confidence  interval”  means  for  this  study.   EXERCISE  20   A  telephone  poll  of  1000  adult  Americans  was  reported  in  an  issue  of  Time  magazine.    One  of  the   questions  asked  was  “What  is  the  main  problem  facing  the  country?”    20%  answered  “crime”.    We  are   interested  in  the  population  proportion  of  adult  Americans  who  feel  that  crime  is  the  main  problem.   a. Define  the  Random  Variables  X  and  P',  in  words.   b. Which  distribution  should  you  use  for  this  problem?    Explain  your  choice.   c. Construct  a  95%  confidence  interval  for  the  population  proportion  of  adult  Americans  who  feel   that  crime  is  the  main  problem.   i. State  the  confidence  interval.           ii. Sketch  the  graph.   iii. Calculate  the  error  bound.   d. Suppose  we  want  to  lower  the  sampling  error.    What  is  one  way  to  accomplish  that?   e. The  sampling  error  given  by  Yankelovich  Partners,  Inc.  (which  conducted  the  poll)  is    ±  3%.    In  1-­‐ 3  complete  sentences,  explain  what  the  ±  3%  represents.   Source URL: http://cnx.org/content/m16966/latest/ Saylor URL: http://saylor.org/courses/bus204 Attributed to: [Susan Dean and Barbara Illowsky] Saylor.org Page 11 of 15 EXERCISE  21     Refer  to  (20).    Another  question  in  the  poll  was  “[How  much  are]  you  worried  about  the  quality  of   education  in  our  schools?”    63%  responded  “a  lot”.  We  are  interested  in  the  population  proportion  of   adult  Americans  who  are  worried  a  lot  about  the  quality  of  education  in  our  schools.   a. Define  the  Random  Variables  X  and  P',  in  words.   b. Which  distribution  should  you  use  for  this  problem?    Explain  your  choice.   c. Construct  a  95%  confidence  interval  for  the  population  proportion  of  adult  Americans  worried  a  lot   about  the  quality  of  education  in  our  schools.   i. State  the  confidence  interval.       ii. Sketch  the  graph.   iii. Calculate  the  error  bound.   d. The  sampling  error  given  by  Yankelovich  Partners,  Inc.  (which  conducted  the  poll)  is    ±  3%.    In  1-­‐3   complete  sentences,  explain  what  the  ±  3%  represents.     Source URL: http://cnx.org/content/m16966/latest/ Saylor URL: http://saylor.org/courses/bus204 Attributed to: [Susan Dean and Barbara Illowsky] Saylor.org Page 12 of 15 EXERCISE  22   Six  different  national  brands  of  chocolate  chip  cookies  were  randomly  selected  at  the  supermarket.    The   grams  of  fat  per  serving  are  as  follows:  8;  8;  10;  7;  9;  9.  Assume  the  underlying  distribution  is   approximately  normal.   a. Calculate  a  90%  confidence  interval  for  the  population  average  grams  of  fat  per  serving  of   chocolate  chip  cookies  sold  in  supermarkets.   i. State  the  confidence  interval.           ii. Sketch  the  graph.   iii. Calculate  the  error  bound.   b. If  you  wanted  a  smaller  error  bound  while  keeping  the  same  level  of  confidence,  what   should  have  been  changed  in  the  study  before  it  was  done?   c. Go  to  the  store  and  record  the  grams  of  fat  per  serving  of  six  brands  of  chocolate  chip   cookies.   d. Calculate  the  average.   e. Is  the  average  within  the  interval  you  calculated  in  part  (a)  ?    Did  you  expect  it  to  be?    Why   or  why  not?   EXERCISE  23   A  confidence  interval  for  a  proportion  is  given  to  be  (–  0.22,  0.34).    Why  doesn’t  the  lower  limit  of  the   confidence  interval  make  practical  sense?    How  should  it  be  changed?    Why?     Try  these  multiple  choice  questions.   Questions  24  –  26  refer  to  the  following:  According  a  Field  Poll  conducted  February  8  –  17,  2005,  79%  of   California  adults  (actual  results  are  400  out  of  506  surveyed)  feel  that  “education  and  our  schools”  is  one   of  the  top  issues  facing  California.    We  wish  to  construct  a  90%  confidence  interval  for  the  true   proportion  of  California  adults  who  feel  that  education  and  the  schools  is  one  of  the  top  issues  facing   California.     EXERCISE  24   A  point  estimate  for  the  true  population  proportion  is: