QUANT ANALYSIS IN BUS DECISION MAKING
QUANT ANALYSIS IN BUS DECISION MAKING DIS 651
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Date Created: 10/23/15
Chapter 9 Confidence Intervals Confidence intervals provide an interval estimate ofthe population parameter using the information in a sample Confidence interval for the population mean Confidence interval for the difference between two means Related or paired samples Unrelated or independent samples Population proportion Example Assume thatthere is a population with a mean of 100 and standard deviation of 25 Assume that we take a sample of size 100 The distribution of the sample means will be approximately normally distributed with a mean of 100 and a standard deviation of 25 100 25 Sampling Distribution of the Sample Mean Since this distribution is normal we can determine the upper and lower limit values Within which 95 ofthe sample mean values will occur Sampling Distribution of the Sample Mean Finding the Limits Mean 100 Standard deviation 25 The lower value represents the 25 h percentile The upper value represents the 975 h percentile Lower value NORMINV002510025 Upper value NORMINVO97510025 Lower value 951 Upper value 1049 The interval 100 49 includes 95 of all possible sample means Conversely if we consider a sample mean in this interval and construct the interval X r 4 9 then such an interval will contain the population mean By contrast if we consider a sample mean outside this interval and construct the same interval then such an interval will not contain the population mean Let us say thatthe sample mean resulting from our sample was 102 Is this value in the 95 interval Construct the interval 102 149 971 to 1069 Does this interval contain the population mean value of 100 Sampling Distribution of the Sample Mean Now consider a sample mean value of 94 Is this value Within the 95 limits Construct the interval 94 149 891 to 989 Does this interval contain the population mean value of 100 Sampling Distribution of the Sample Mean If we construct an interval using every possible sample mean from this population as 7i49 then what percentage of these intervals would contain the population mean In most cases we do not know what the population mean is Hence we rely on the above finding to estimate the population mean We simply construct the interval using the information we have from the sample Example 64 Sample mean Y 6 25 Sample Standard deviation s i 597 Sample size n 40 Standard error Standard deviation of the sampling distribution i 597N40 02525 Con dence level 95 or 0 95 i Con dence level 0 05 39 0 052 0 025 39 LowerLirnlt NORMlNVU 0256 250 2525 5 755 39 UpperLirnlt NORMlNVU 9756 250 2525 6 744 95 confidence interval forpopvla lon mean 5 755 e 744 we 25104949 Please note that our answer is different from that in the text book The interval in the text book is 5739 to 6761 The reason for this difference is that the text book uses the t distributionquot while we used the 2 distributionquot Except for very small samples the difference as it is in this case will be very small For our purposes you can use either one o Alternative Approach using Descriptive Statistics from Data Analysis You can also use Descriptive Statistics from Data Analysis to compute con dence interval m W We 1m Weed e e 9W5 ceneei r Bows new 0 Labels in rst row Output options r Qutvut Range 2 a New WorksheetElv l New waymaak r summevv statistics is aeriaenee Level for Mean of a r m Largest l r Kt smallest F The resulting output is Satisfaction Confidence Leel95 0 0 510832287 The con dence interval is 625105103 3 57392 to 6760 This interval is the same as that in the book Example 64 Extension nstmct a 90 con dence interval instead of a 95 con dence interval The resulting 90 con dence interval is 533 to 666 or 625104153 95 con dence interval was 625104949 Note that increasing the con dence interval also increases the idthquotolthe con dence interval This is the tradeoff between efliciencyand el e iveness A more effective cor dence interval ef cient0 4949 A more efficient confidence interval 0 M53 is less effec ive 90 The 5 is less Confidence Interval for the Difference Between Two Means Related or paired samples We can consider tw amples to be related or m le can 0 s paired if every observation In one sa b Unrelated or Independent samples When the paired sample condition is not satis ed Example 99 Since the data was collected from a Husband and erquot combination for each household there is speci c reason to pair the observations We are interested in constmcting a confidence interval for the erence Hence takethe difference between the responses from the Husband and Now you have only one variable t in the values Wife he difference Construct a con dence interval just like you did before Resun The mean ofthe differences is 16286 Analysis using Descriptive Statistics in Data Analysis yields a result of Difference Con dence Level950 0571692 The 95 con dence interval is 16286 1 05717 Independent Samples Example 97 When you have two unrelated independent samples the construction ofthe con dence interval gets a little more complicated We rst have to compute the standard errorquot In order to compute the standard error we have to compute the pooled variancesp1 as 2 2 2 n1 151 n2 152 p 111 112 2 Alternative Meth Data od Use ttest Assuming Equal Variancesquot from Analysis we m me lineage fwa p mlnq Unequal Variances zrfest Two Sample for Means Cancel help l One of the outputs 39om this procedure is the Men Twnrsample Assuming Equal Variances Pooled Variancequot m eresl TwoSampis Assuming Equal Vailances presume rm VariableZRanqe 555 5535 1 44cm SugglierA SUMi9 a harp Me 74 555 5557 HYDathgsized Mean Difference Vauame EUEBE SE W552 71 ii Labels oh 3 alpha 0 us Fooled Variance 74D9U ES WWW Hypnlhasrzed Mean ll erk 50 u u an a i 39 39 32W F l ievt il El 095157 New WWW l Critical one iaii I B7i553 ltt Wutall D 1903M 25 leIlical lwuvtail 2 DDWlE 25 Computing Standard Error Standard Error Sp Confidence Interval Mean Mean of SupplierA Mean of Supplier B 7488 65567 9313 Standard error V7409065 30 30 Lower lim39 NORMINV0025931 3 702807 U er Iim ORMINV09759313702807 Z7 The interval is 4416 to 23088 75 Good News Confidence Interval for Proportions I will not ask you to compute the Typically used for YesNoquot responses con dence interval for two independent Example 95 samples on the exam 7 Number or invoices checked 93 7 Number or invoices Wim errors 2 7 Proportion or invoices Wim errors 13 293 0 7 Standard error is 1713 00215 1700215 n 93 0 015042 E a Confidence level 90 or 090 Mean 00215 Standard error 0015042 Upper Limit NORMINV095002150015042 00462 Lower Limit NORMINV005002150015042 0003 Since proportions cannot be less than 0 the confidence interval is 0 to 0046 It is likely that the maximum proportion of errors in all invoices is not more than 0046 Sample Size Determination for Proportions The sample size necessary for estimating a certain proportion can be easily determined Specify I would like to estimate the proportion of people who I ke my product to within 12 or 2 I would like a confidence level of 90 Determining Sample Size First compute the Z value as follows Determine 1 Con dence level 1 090 010 Determine 12gtlt010 005 Z value NORMSINV1 005 1645 Please note that you are using NORM INV 1690964 1691 H7 05gtltZvaluezi 05x1645Z e 002 Why If we do not have any information about proportion then what is the best we can assume Also the maximum variance for any sample is 025 If we select the sample in the manner specified above then the resulting confidence interval will be within 002 This is the reason you see a lot of surveys where the number of people surveyed is some odd number like 1543 etc Skip the Following Sections Section 94 Confidence interval for a total Section 96 Confidence interval for a standard deviation we will coverthis later Section 98 Confidence interval for difference between proportions Section 99 You are responsible for sample size determination for proportions Suggested Problems 8 15 17 19 23 24 25 27 33 35 38 39 42 43 44 45 46 59 6o 61 62 63 64 73 74 75 76 For any problem that requires you to take a sample just pick the required number at random