CHAPTER 11: Hypothesis Testing Involving two Sample means or Proportions to accompany Introduction to business statistics fourth edition, by ronald m. Weiers Presentation by priscilla Chaffe -Stengel Donald n. Stengel o 2002 The Wadsworth Group
CHAPTER 11: Hypothesis Testing Involving Two Sample Means or Proportions to accompany Introduction to Business Statistics fourth edition, by Ronald M. Weiers Presentation by Priscilla Chaffe-Stengel Donald N. Stengel © 2002 The Wadsworth Group
l Chapter 11-Learning objectives Select and use the appropriate hypothesis test In comparing Means of two independent samples Means of two dependent samples Proportions of two independent samples Variances of two independent samples Construct and interpret the appropriate confidence interval for differences in Means of two independent samples Means of two dependent samples Proportions of two independent samples o 2002 The Wadsworth Group
Chapter 11 - Learning Objectives • Select and use the appropriate hypothesis test in comparing – Means of two independent samples – Means of two dependent samples – Proportions of two independent samples – Variances of two independent samples • Construct and interpret the appropriate confidence interval for differences in – Means of two independent samples – Means of two dependent samples – Proportions of two independent samples © 2002 The Wadsworth Group
l Chapter 11-Key Terms Independent vs dependent samples Pooled estimate of the common varlance common standard deviation population proportion Standard error of the estimate for the difference of two population means difference of two population proportions Matched, or paired, observations Average difference o 2002 The Wadsworth Group
Chapter 11 - Key Terms • Independent vs dependent samples • Pooled estimate of the – common variance – common standard deviation – population proportion • Standard error of the estimate for the – difference of two population means – difference of two population proportions • Matched, or paired, observations • Average difference © 2002 The Wadsworth Group
l Independent us dependent samples Independent Dependent Samples: Samples Samples taken from two Samples taken from two populations where either different populations, (1) the element sampled is a member of both populations where the selection p or (2) the element sampled process for one sample is independent of the in the second population is selected because it is similar selection process for the on all other characteristics other sample or"matched, to th element selected from the first population o 2002 The Wadsworth Group
Independent vs Dependent Samples • Independent Samples: Samples taken from two different populations, where the selection process for one sample is independent of the selection process for the other sample. • Dependent Samples: Samples taken from two populations where either (1) the element sampled is a member of both populations or (2) the element sampled in the second population is selected because it is similar on all other characteristics, or “matched,” to the element selected from the first population © 2002 The Wadsworth Group
I Examples: Independent versus Dependent samples Independent Dependent Samples: Samples Testing a companys Testing the relative claim that its peanut fuel efficiency of 10 butter contains less fat trucks that run the than that produced by same route twice a competitor once with the current air filter installed and once with the new filter o 2002 The Wadsworth Group
Examples: Independent versus Dependent Samples • Independent Samples: – Testing a company’s claim that its peanut butter contains less fat than that produced by a competitor. • Dependent Samples: – Testing the relative fuel efficiency of 10 trucks that run the same route twice, once with the current air filter installed and once with the new filter. © 2002 The Wadsworth Group
l Identifying the appropriate Test Statistic Ask the following questions Are the data from measurements(continuous variables or counts(discrete variables)? Are the data from independent samples? Are the population variances approximately equal? Are the populations approximately normally distributed? What are the sample sizes? o 2002 The Wadsworth Group
Identifying the Appropriate Test Statistic Ask the following questions: • Are the data from measurements (continuous variables) or counts (discrete variables)? • Are the data from independent samples? • Are the population variances approximately equal? • Are the populations approximately normally distributed? • What are the sample sizes? © 2002 The Wadsworth Group
Ⅷ Test of(1-u2)a=a2 Populations normal Test statistic x1-x2)]-[1-1 hc24+)2+(2)2 and df=n,+n2-2 o 2002 The Wadsworth Group
Test of (µ1 – µ2 ), s1 = s2 , Populations Normal • Test Statistic and df = n1 + n2 – 2 –2 1 2 2 2 –1) 2 2 ( 1 –1) 1 ( where 2 2 1 1 2 1 0 ] 2 – 1 ] – [ 2 – 1 [ n n n s n s p s p n n s x x t + × + × = + = ÷÷ ÷ ÷ ÷ m m © 2002 The Wadsworth Group
I Example: Equal-Variances t-Test Problem 11.2: An educator is considering two different videotapes for use in a half-day session designed to introduce students to the basics of economics Students have been randomly assigned to two groups and they all take the same written examination after viewing the videotape. The scores are summarized below assuming normal populations with equal standard deviations, does it appear that the two videos could be equally effective? What is the most accurate statement that could be made about the value for the test? Videotape1:x1=77.1,s1=78,11=25 Videotape2:x=80.0,s2=8.1,m2=25 C 2002 The Wadsworth Group
Example: Equal-Variances t-Test • Problem 11.2: An educator is considering two different videotapes for use in a half-day session designed to introduce students to the basics of economics. Students have been randomly assigned to two groups, and they all take the same written examination after viewing the videotape. The scores are summarized below. Assuming normal populations with equal standard deviations, does it appear that the two videos could be equally effective? What is the most accurate statement that could be made about the p-value for the test? Videotape 1: = 77.1, s1 = 7.8, n1 = 25 Videotape 2: = 80.0, s2 = 8.1, n2 = 25 x 1 x 2 © 2002 The Wadsworth Group
lt- Test, Two Independent means I. Ho: u1-u2=0 The two videotapes are equally effective. There is no difference in student erformance H: u1-u2*0 The two videotapes are not equally effective. There is a difference in student performance. I. Rejection region Reject H Do Not 0/Reject H Reject H 0 =0.05 0.025 0.95 0.025 df=25+25-2=48 Reject Ho if t>2.011 or t <-2.011 t-2011 o 2002 The Wadsworth Group
t-Test, Two Independent Means • I. H0 : µ1 – µ2 = 0 The two videotapes are equally effective. There is no difference in student performance. H1 : µ1 – µ2 0 The two videotapes are not equally effective. There is a difference in student performance. • II. Rejection Region a = 0.05 df = 25 + 25 – 2 = 48 Reject H0 if t > 2.011 or t < –2.011 t=-2.011 t=2.011 Do Not Reject H 0 0 0 Reject H Reject H © 2002 The Wadsworth Group
l t-Test, Problem 11.2 cont III. Test statistic s2=248)+248.1)2=146016+1564646325 25+25-2 48 77.1-80.0 t= 11 1.289 632251+ 25257 P o 2002 The Wadsworth Group
t-Test, Problem 11.2 cont. • III. Test Statistic 63.225 48 1460.16 1564.64 25 25 – 2 2 24 (8.1) 2 24 (7.8) 2 = + = + = × + × p s –1.289 25 1 25 1 63.225 77.1–80.0 2 1 1 1 2 2 – 1 = + = + = ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ n n p s x x t © 2002 The Wadsworth Group
