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# Business Statistics STAT 2160

WMU

GPA 3.54

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This 25 page Class Notes was uploaded by Alia Gerhold on Wednesday September 30, 2015. The Class Notes belongs to STAT 2160 at Western Michigan University taught by Jung Wang in Fall. Since its upload, it has received 18 views. For similar materials see /class/216952/stat-2160-western-michigan-university in Statistics at Western Michigan University.

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Date Created: 09/30/15

Chapter 7 Confidence Intervals JC Wang Department of Statistics Western Michigan University Goal and Objectives Goal to learn confidence intervals Objectives gt To understand that each interval has two endpoints lower and upper bound and Interpret the confidence interval gt To compute the confidence interval a point estimate it the margin of error gt To determine the sample size Department of Statistics noquot Outline Introduction Applications Definations and Notation Confidence Intervals Computation zConfidence Intervals tConfidence Intervals Sample Size Determination An Example Comparing Two Populations Comparing Means of Two Independent Populations Comparing Means of Two Dependent Populations Confidence Interval for Proportion Confidence Interval for Population Proportion Sample Size Determination Applications of Estimation in Business examples Store inventory value Manufacture process Distribution process Drug delivery Auditor VVVVV Department of Statistics mm m m mu Definitions gt Sample statistic a value computed from the sample ie from data gt Point estimate ptest a single sample statistic that estimates the population parameter such as the mean or proportion gt Interval estimate of the true population parameter takes into account the sampling distribution of the point estimate where we have an upper bound and a lower bound cpartmcnt of Statistics mmmm m nmmn Notations to be discussed and used later gt Cl confidence interval gt CVal critical value gt ME margin of error gt SE standard error gt SD standard deviation gt ptest point estimate gt Zaz normal distribution critical value use invnorm gt 771 students t distribution critical value with n1 degrees of freedom use math solver or the ian Department of Statistics quotor r Computation of confidence intervals gt ptest in ME gt Where the point estimate estimates population mean u by Y or population proportion p by b gt marginOfError criticalValue x standardError gt In other words ME CValSE partmcnt of Statistics mmn ammimmn rm Standard Error gt Most of the time we will not have the SD of population mean but we can compute sample SE of the mean 3 SE7 7 X W gt Also we will not have the SD of population proportion but we can compute sample proportion SE SE M n D partmentof Statistics rm 7 Critical Value gt z for normal distribution gt t for students tdistribution gt The students tdistribution has n 7 1 degrees of freedom df n 7 1 partmcnt of Statistics wmmun v v mmmn zCritical Value gt Notation zag upper 100 x a2th standard normal percentile gt That is PZ gt zaz 042 E PZ g Zaz1ia2 SO Zaz inVNorm1 7 a 2 gt Example 95 confidence interval will give 25 in each tail of the bellshaped curve therefore the zCVaI ch Z025 invNorm1i025 invNorm975 196 Department of Statistics rm 39 zCritical Value con nued or area to the le ofzgis1rE Dcpartmcnt of Statistics Jpnhfwvbmn nyviiinwtun tCritical Value using TI calculators 1 math gt solver gt tcdfL U D 7 AT where y L ta to be solved U 9999 gt D df n 71 gt A a error rate gt T number of tails 2 for ci 2 oruse ian1 ia2df Department of Statistics Cereal Box Packaging Example Consider a cereal packaging plant in Battle Creek that is concerned with putting 368 gram of cereal into a box gt What are the costs associated with putting too much cereal in a box gt What are the costs associated with putting too little cereal in a box Cereal Box Packaging Example con nued gt Suppose sample size n 25 gt Suppose sample average x 365 grams gt Suppose SD is a process SD therefore a 15 grams gt Suppose we want a 95 confidence interval gt Therefore the critical value is 20V 196 Department of Statistics noquot Cereal Box Packaging Example continued margin of error gt Recall ME CVal x SE gt The critical value CVal for 95 Cl means that the area under the curve of one tail is 5 2 or 0025 therefore ch invNorm1 7 025 invNorm975 196 gt s 15 SE773 W 25 gt ME196gtlt3588 partmcnt of Statistics mmun m m mmmn rm Cereal Box Packaging Example continued confidence interval gt Since the confidence interval is the ptest in ME gt CI 365 i 588 35912 37088 gt Therefore we are 95 confident that the population mean is between 359 and 371 gt Since 368 the value that is printed on the box indicates the manufacturing process is working properly is within the interval there is no reason to conclude that anything is wrong with the process D partmentof Statistics rm 39 zConfidence Interval Using Tl Calculators example Let39s use Tl calculator gt Do this STAT gt TESTS gt Zinterval gt STATS 1015 1365 1 n 25 l C Level 95 l CALCULATE gt READOUT Zinterval 35912 37088 Y 365 n 25 Since 368 the target of the package is within the interval production should continue partmcnt of Statistics wmmun ammlmmu Note on zConfidence Intervals gt The value of 2 selected for constructing such a confidence interval is called the critical value for the distribution gt There are different critical values for each level of confidence or confidence level CL 1 7 a where a significance level SL or error rate gt Frequently Used 20V SL CL 2tailed CVaI 1 0 90 1 645 5 95 1 96 1 99 258 Note There is a trade off between the width of the confidence interval and the level of confidence Department of Statistics rm 7 Problem When SD is Unknown We have been dealing with My 0 where a population or process SD is known What happens when standard deviation 0 is not from a population or process SD Is this requirement rigid Can we compute standard deviation from the sample Let us review some history first Department of Statistics x hmrmwp aam nmmn History of the Student t Distribution William Gosset an employee of Guinness Breweries in Ireland had a preoccupation with making statistical inferences about the mean when SD was unknown Since the employees of the company were not allowed to publish their scientific work under their own name He chose the pseudonym Student Therefore his contribution is still known as Student39s tDistribution D partmentof Statistics quotmum an Comparing Standard Normal Curve with 1 curves Comparison of Standard Normal with t Curves N01 ti tin density tConfidence Interval for the Mean summer ll quiz example Construct a 95 CI for the mean score for Summer Quiz Data of 14 students Given 95 CL Y 25 s 10777 n 14 SE 0 Val ME pt est 95 0 10777 28803 m tazynil 11025713 iDVT1 7 21604 gtlt 28803 62225 25 ptest in ME 1877831222 Department of Statistics may nu mmmm tConfidence Interval for the Mean using Tl calculators gt Do this STAT gt TESTS l TInterval gt STATS LY25LSX10777Lnl4iC Level 95i CALCULATE gt READOUT Tinterval 18778 31222 Y 25 n 14 gt We are 95 confident that the true mean quiz score is between 188 and 312 Sample Size Determination based on confidence intervals gt What sample size should we use for the average quiz score determination if we want 95 confidence ME 5 and a 10777 2 2 2 2 20 196 gtlt10777 178g18 quotW 52 Department of Statistics mm m m mu Slow Wave Sleep Example page 100 problem 1 21202279142391025151711 Y156154ands61310 a b population average and SD not possible gt c the sample average will miss the population average by V V the i gt d SE sf 6131 oN 17 gt e ME CVal gtlt SE 19757131 X 17 21733 X 17 3704 f 95 CI is 156154 1 3704 1 191 1932 rm partmcnt of Statistics mmn ammrmmn Slow Wave Sleep Example con nued gt f continued can also do this assuming data have been entered into list 1 L1 STAT gt TESTS l tTntervaT gt DATA lListL1 l CALCULATE gt g If the confidence level is reduced to 90 the new interval will be shorter h 90 Cigt 1258518646 gt i Interpret the 95 Cl We are 95 percent confident that the true population average is between 12 and 19 D partmentof Statistics rm 7 Slow Wave Sleep Example con nued gt j Does the 95 Cl suggest that elderly men over 60 spend 20 of their sleep in REM No since 20 o is not in the 95 Cl gt k What sample size should we use if we change the ME to 25 n i CVa2 X 802 i 1962 X 6132 ME2 7 T 2310 is 24 partmcnt of Statistics mmun m m mmmn rm Comparing Means atwoindependentpopwa ons gt We are not limited to comparing an average to a constant Suppose we want to compare the means of two independent populations gt Parameter of interest 6 M1 7 M2 gt Recall Cl is ptest in ME ptest d Y1 7 Y2 ME CVal x SE where CVal tn1n22 SE lSE12 SE22 D partmentof Statistics rm 39 Example battery example A statistics student designed an experiment to see if there was any real difference in battery life between brandname AA batteries and generic AA batteries He used six pairs of AA alkaline batteries from two major battery manufactures a well known brand name and a generic brand He measured the length of battery life while playing a CD player continuously He recorded the time in minutes when the sound stopped Department of Statistics mmmummmuz gm nmmu Battery Example con nued Generic Brand Name x 206 1874 8 103 146 n 6 6 Want 95 Cl gt a What is the standard error b What is the 95 CVal c What is the ME d What is the 95 Cl e Does this confidence interval suggest that generic AA batteries will last longer than brandname AA batteries gt f Interpret the 95 Cl D partmentof Statistics mv mn Battery Example continued answers gt a 71 2 i1 2 pooed 50 W 12 n1 172 7 2 1262 1262 SE 727 6 6 gt b CVal 1 an ian975 10 22281 0 ME CVal gtlt SE 22281 X 727 162527 d 95 Cl gt 235 3485 partmcnt of Statistics mmun m m mmmn rm Battery Example continued answers gt e Does this confidence interval suggest that generic AA batteries will last longer than brandname AA batteries Yes because zero is not Within the interval gt f Interpret the 95 CI We are 95 confident that the true mean difference is between 2 and 35 D partmentof Statistics rm 39 Battery Example continued using Tl calculator gt Do this STAT gt TESTS l 2 SampTInt gt STATS l Y1206OlellO3ln16lY2l874le22146l 726 l C Level 95 lPooleszes l CALCULATE gt READOUT 2 sampTInt 23471 34853 dflO Sxp 12 6342788 gt Zero is not within this interval we can conclude that there is a difference between the two means Comparing Means of two related groups gt We are not limited to comparing two averages of independent populations Suppose we want to compare the means of two related populations gt Recall CI is ptest in ME ptest Y1 7Y2 ME CVal x SE where CVal ray27H ian1 e n e 1 Sdiff SE 7 W Department of Statistics rm 39 Example computer stock prices We want to compare January 2002 prices vs January 2003 prices of computer companies see page 92 Computer Stock Prices Jan 02 Jan 03 Diff X 2591 1796 7946 s 634 565 61426 n 5 5 5 partmcnt of Statistics wmmun v v mmmn Computer Stock Prices Example con nued gt What is Standard Error gt What is 95 Critical Value gt What is 95 Margin of Error gt What is a 95 Confidence Interval gt Does this confidence interval suggest a difference in stock prices between Jan 2002 and Jan 2003 gt Interpret the 95 Cl Department of Statistics rm 39 Computer Stock Prices Example answers Sdiff SE i W i xB CVal 025721 ian1 i 025 4 27764 gt ME 27764 gtlt 27471 76271 95Cl gt 03189 15573 gt Does this confidence interval suggest a difference in stock prices between Jan 2002 and Jan 2003 Yes because zero is NOT Within Cl gt Interpret the 95 CI We are 95 confident that the true difference is between 3 and 156 27471 partmcnt of Statistics wmmun v v mmmn Computer Stock Prices Example answers using Tl calculators gt Do this STAT gt EDIT and Enter data into L1 and L2 then place cursor on L3 do 2nd2 4 2nd1 ieL2 7 L1 gt STAT gt TESTS l tInterval gt DATA l ListL3l C Level 95 l CALCULATE gt READOUT Tinterval 03189 15573 Y 7946 Sx 61426 n 5 gt Zero is not within this interval we can conclude that there is a difference between the two means Department of Statistics rm 39 West Michigan Telecom Example problem 13 on page 104 Some stock market analysts have speculated that parts of West Michigan Telecom might be worth more that the whole For example the company39s communication systems in Ann Arbor and Detroit can be sold to other communications companies Suppose that a stock market analyst chose nine 9 acquisition experts and asked each to predict the return in percent on investment ROI in the company held to the year 2003 if i it does business as usual or ii if it breaks up its communication system and sells all its parts Their predictions follow Department of Statistics mmtm agm nmmn West Michigan Telecom Example continued Expert123456789 NotBreak 12 21 8 20 16 5 18 21 10 Break Up 15 25 12 17 17 10 21 28 15 SE sdm 28626 09542 CVal amp 102578 1mm 7 0258 23060 gt ME 2306 X 9542 22004 gt 95CI gt 10218 54226 gt Does this confidence interval suggest a difference between breaking up the company or not Yes because zero is NOT Within Cl gt Interpret the 95 Cl We are 95 confident that the true difference among the experts is between 10 and 54 D partmentof Statistics nmmmun West Michigan Telecom Example continued using TI calculators gt STAT gt EDIT and Enter data into L1 and L2 and place cursor on L3 do 2nd2 7 2nd1 ie L2 7 L1 then do STAT a TESTS i tInterval a Data i ListL3i C Level 95 l CALCULATE gt READOUT Tinterval l O2 5 42 Y 322 Sx 2 8 626 n 9 gt Zero is not within this interval we can conclude that there is a difference between the two means epartmcntof Stati 39 s wmmm Confidence Interval for population proportion Suppose we want to estimate the population proportion using intervals gt Recall Cl is ptest in ME gt Therefore use success X test i p sampleSze n gt ME CVaI x SE gt This Cl works well if nxpgt5andnx1ipgt5 Note that is it woks well if the expected number of successes and the expected number of failures are both greater than 5 Department of Statistics rm 7 EAS Sensor Example If a sales clerk fails to remove the EAS sensor when an item is purchased it can result in an embarrassing situation for the customer A survey was conducted to study consumer reaction to such false alarms Of 250 customers surveyed 40 said that if they were to set off an EAS alarm because store personnel did not deactivate the merchandise then they would never shop at the store again partmcnt of Statistics mmwp aammtmmn EAS Sensor Example con nued gt ptest 016 SEf MW 002319 gt CVal az 2025 invNorm1 7 025 196 gt ME 196 X 002319 004544 gt 95CI 11456 20544 gt Interpret the 95 Cl We are 95 confident that the true proportion is between 011 and 021 Department of Statistics quot9439 r EAS Sensor Example continued using TI calculators gt Do this STAT gt TESTS L l PropZint L x40 L n250 L C Level 95 L CALCULATE gt READOUT Zinterval 11456 20544 gt We are 95 confident that the true proportion is between 1 1 and 21 partmcnt of Statistics mmn mmmmmn mu Exercise 17 on page 105 Given x 600 n2000 gt 600 i 2000 i 31737 SEP 7 0 2000 700102 CVal Zaz Z025 invNo rm1 7 025 196 ME 196 X 00102 00201 ptest 95CI 2799 3201 gt Interpret the 95 CI We are 95 confident that the true proportion is between 028 and 032 D partmentof Statistics nun 39 Sample Size Determination based on CI of proportion gt What is the true proportion of success p gt Decide which confidence level to use gt Determine margin of error that you re willing to accept partmcnt of Statistics wmmun v v mmmn EAS Example Suppose that you are a student with a grant to study this EAS issue and you realize that there are not enough funds to gather data on 250 subjects 80 you want to determine a new sample size by relaxing the confidence level to 90 and use p16 and the ME of 004544 what is the new sample size Department of Statistics rm 39 EAS Example continued answer 2 n xmoem gt CVal90 217 9V2 2 05 invNorm1 i 05 1645 gt ME 04544 gt f 16 note see discussions on next slide gt 2 i 1645 7 uie n 045442 gtlt161i16717617177 partmcnt of Statistics mmn mmmmmn mi Sample Size Determination for Proportion discussions 2 n xmoem gt When a rough estimate of p is available such as that from a pilot study or some educated guess use it for b above Otherwise use for b a conservative estimate gt It is recommended to always round it up to the next integer as n 177 here which is rounded up from 1761 D partmentof Statistics nun 39

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