# OSU BUSMGT 2320 - s_4_Comparisons [mu] Autumn 2014 (37 pages)

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## s_4_Comparisons [mu] Autumn 2014

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- Pages:
- 37
- School:
- Ohio State University
- Course:
- Busmgt 2320 - Decision Sciences: Statistical Techniques

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Lecture 4 Comparing Two Groups If you don t know where you re going any road will take you there C S Lewis 1898 1963 1 Learning Objectives 1 Be able to identify the various approaches for comparing two populations with respect to the mean Paired matched Samples Independent Samples Population variances known Population variances unknown Assumed equal Assumed unequal Learning Objectives 2 Know the parameter and the corresponding sample statistic for each of the approaches for comparing two populations with respect to the mean 3 Understand the sampling distribution for each of the sample statistics used to compare two populations with respect to the mean 4 Understand and be able to calculate a pooled variance estimate of 2 3 Learning Objectives 5 Be able to construct the appropriate 1 confidence interval estimate of both D and 1 2 6 Be able to conduct appropriate hypothesis tests for both D and 1 2 4 Terminology Matched paired Samples D sD nD X D Independent Samples 1 2 and X 1 X 2 Pooled Variance Estimate sp2 Satterthwaite Approximation 5 INTRODUCTION 6 Selecting the Case All Classical applications require SRS and Normal distribution for the sample statistic Type of data Quantitative use means and standard deviations Nominal use proportions Number of populations groups How was the data collected For example Paired or Independent Samples What assumptions can be made For example equal or unequal population variances 7 Selecting the Case Means quantitative data One Group See lectures 1 3 Population Variances Unknown Independent Samples Parameter 1 2 Statistic X 1 X 2 Two Groups Paired Samples Use Di X1i X2i Parameter D XD Statistic Population Variances known Use Z X1 X 2 Assumed Use Satterthwaite t Approximation S X1 X 2 s12 s22 n1 n2 Assume Use t n1 n2 2 SX 1 X2 s 2p n1 s 2p n2 12 22 n1 n2 8 Matched Samples v Independent Samples Nike wants to see if there is a difference in the durability of 2 sole materials 1 One type of material is placed on one shoe the other type on the shoe of the same pair Twenty five people are given a pair of the shoes to wear for 3 months 2 Twenty five people are given a pair of shoes soled with material 1 and twenty five people are given a pair of shoes soled with material 2 The fifty people wear the shoes for 3 months 9 The Repeating Logic Reasonable sampling error critical st dev critical st error Two sided symmetric problem 1 2 4 3 2 m 1 Parameter 2 0 m1 2 3 4 Sample Statistic Z z z 2 t t 2 z z 2 t t 2 t or 10 EXAMPLE 1 11 Mini Case Delivery Service Problem A Chicago based firm has documents that must be distributed to district offices throughout the United States Because of the critical information contained in the documents quick deliveries to the district offices are essential The firm has decided to select one of two express delivery services that in many instances can provide next day deliveries to the district offices 12 Mini Case Delivery Service Problem In testing the delivery times of the two services the firm sends two documents to a sample of ten district offices with one document carried by one delivery service and the other document carried by the second delivery service Is one service able to deliver in less time than the other on average 13 Mini Case Delivery Service Data Plan District Office Seattle Los Angeles Boston Cleveland New York Houston Atlanta St Louis Milwaukee Denver Overnight Courier 32 30 19 16 15 18 14 10 7 16 Flight Express 25 24 15 15 13 15 15 8 9 11 Difference 77 6 4 1 2 3 1 2 2 5 14 Mini Case Delivery Service Plan Summarize Data One Variable Summary Mean Variance Std Dev Median Minimum Maximum Count 1st Quartile 3rd Quartile Interquartile Range OC 17 700 62 011 7 875 16 000 7 000 32 000 10 14 000 19 000 5 000 FE 15 000 31 778 5 637 15 000 8 000 25 000 10 11 000 15 000 4 000 D 2 700 8 456 2 908 2 000 2 000 7 000 10 1 000 5 000 4 000 15 Mini Case Delivery Service Plan Check for Normality outliers Q Q Normal Plot of D 3 5 2 5 Standardized Q Value 1 5 0 5 3 5 2 5 1 5 0 5 0 5 0 5 1 5 2 5 3 5 1 5 2 5 3 5 Z Value 16 Mini Case Delivery Service Calculate 95 Confidence Interval to Estimate the Mean Difference in Delivery Times x D t 2 nD 1 s D n D 0 62 4 78 17 Mini Case Delivery Service Report With 95 confidence we can state that Overnight Courier will take between 0 62 hours and 4 78 hours longer on average to make the deliveries than Flight Express 18 Conf Intervals One Sample Sample Size Sample Mean Sample Std Dev Confidence Level Mean Degrees of Freedom Lower Limit Upper Limit StatTools Statistical Inference Hypothesis Test Mean Std Dev Analysis Type Paired Sample Variables OC and FE D 10 2 700 2 908 95 0 9 0 620 4 780 StatTools Results StatTools Statistical Inference Hypothesis Test Mean Std Dev Analysis Type One Sample Variable D Conf Intervals Paired Sample Sample Size Sample Mean Sample Std Dev Confidence Level Degrees of Freedom Lower Limit Upper Limit OC FE 10 2 7 2 908 95 0 9 0 620 4 780 19 EXAMPLE 2 20 Mini Case Par Golf Inc Problem Par Inc is a manufacturer of golf equipment and has recently developed a new golf ball that has been designed for extra distance In a test of driving distance a sample of par golf balls is compared with a sample of golf balls manufactured by Par s competitor A mechanical driving device is used to create a constant driving force and the distance for each sample ball hit is recorded Par Inc desires a 95 confidence interval estimate of the difference in mean driving distance for their golf ball and the competitor s golf ball Mini Case Par Golf Inc Data Plan PAR Competitor n1 15 balls n2 10 balls Sample Mean x1 235 yards x2 218 yards Sample Stdev s1 15 yards s2 20 yards Sample Size Mini Case Par Golf Inc Plan Independent Samples SRS Normal Assume population variances are equal Use pooled sample variance 2 2 n 1 s n 1 s 1 2 2 s 2p 1 n1 n2 2 23 Mini Case Par Golf Inc Calculate 95 CI for 1 2 17 2 069 6 994 2 5 yds 31 5 yds Mini Case Par Golf Inc Report With 95 confidence we can report that Par s new golf ball will travel between 2 5 and 31 5 yards farther than the competitor s golf ball on average Larger sample sizes would allow for smaller margin of error in the estimate currently at 14 5 yds 25 Mini Case Par Golf …

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