Chapter 11 Section 1 Inference about Two Means: Dependent Samples Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 1 of 26 Chapter 11 – Section 1 ● Learning objectives 1 Distinguish between independent and dependent sampling 2 Test hypotheses made regarding matched-pairs data 3 Construct and interpret confidence intervals about the population mean difference of matched-pairs data Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 2 of 26 Chapter 11 – Section 1 ● Learning objectives 1 Distinguish between independent and dependent sampling 2 Test hypotheses made regarding matched-pairs data 3 Construct and interpret confidence intervals about the population mean difference of matched-pairs data Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 3 of 26 Chapter 11 – Section 1 ● Chapter 10 covered a variety of models dealing with one population The mean parameter for one population The proportion parameter for one population The standard deviation parameter for one population ● However, there are many real-world applications that need techniques to compare two populations Our Chapter 10 techniques do not do these Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 4 of 26 Chapter 11 – Section 1 ● Examples of situations with two populations We want to test whether a certain treatment helps or not … the measurements are the “before” measurement and the “after” measurement We want to test the effectiveness of Drug A versus Drug B … we give 40 patients Drug A and 40 patients Drug B … the measurements are the Drug A and Drug B responses Two precision manufacturers are bidding for our contract … they each have some precision (standard deviation) … are their precisions significantly different Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 5 of 26 Chapter 11 – Section 1 ● In certain cases, the two samples are very closely tied to each other ● A dependent sample is one when each individual in the first sample is directly matched to one individual in the second ● Examples Before and after measurements (a specific person’s before and the same person’s after) Experiments on identical twins (twins matched with each other Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 6 of 26 Chapter 11 – Section 1 ● On the other extreme, the two samples can be completely independent of each other ● An independent sample is when individuals selected for one sample have no relationship to the individuals selected for the other ● Examples Fifty samples from one factory compared to fifty samples from another Two hundred patients divided at random into two groups of one hundred Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 7 of 26 Chapter 11 – Section 1 ● The dependent samples are often called matched-pairs ● Matched-pairs is an appropriate term because each observation in sample 1 is matched to exactly one in sample 2 The person before the person after One twin the other twin An experiment done on a person’s left eye the same experiment done on that person’s right eye Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 8 of 26 Chapter 11 – Section 1 ● Learning objectives 1 Distinguish between independent and dependent sampling 2 Test hypotheses made regarding matched-pairs data 3 Construct and interpret confidence intervals about the population mean difference of matched-pairs data Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 9 of 26 Chapter 11 – Section 1 ● The method to analyze matched-pairs is to combine the pair into one measurement “Before” and “After” measurements – subtract the before from the after to get a single “change” measurement “Twin 1” and “Twin 2” measurements – subtract the 1 from the 2 to get a single “difference between twins” measurement “Left eye” and “Right eye” measurements – subtract the left from the right to get a single “difference between eyes” measurement Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 10 of 26 Chapter 11 – Section 1 ● Specifically, for the before and after example, d1 = person 1’s after – person 1’s before d2 = person 2’s after – person 1’s before d3 = person 3’s after – person 1’s before ● This creates a new random variable d ● We would like to reformulate our problem into a problem involving d (just one variable) Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 11 of 26 Chapter 11 – Section 1 ● How do our hypotheses translate? The two means are equal … the mean difference is zero … μd = 0 The two means are unequal … the mean difference is non-zero … μd ≠ 0 ● Thus our hypothesis test is H0: μd = 0 H1: μd ≠ 0 The standard deviation σd is unknown ● We know how to do this! Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 12 of 26 Chapter 11 – Section 1 ● To solve H0: μd = 0 H1: μd ≠ 0 The standard deviation σd is unknown ● This is exactly the test of one mean with the standard deviation being unknown ● This is exactly the subject covered in section 10.2 Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 13 of 26 Chapter 11 – Section 1 ● In order for this test statistic to be used, the data must meet certain conditions The sample is obtained using simple random sampling The sample data are matched pairs The differences are normally distributed with no outliers, or the sample size is (n at least 30) ● These are the usual conditions we need to make our Student’s t calculations Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 14 of 26 Chapter 11 – Section 1 ● An example … whether our treatment helps or not … helps meaning a higher measurement ● The “Before” and “After” results Before After Difference 7.2 6.6 6.5 5.5 8.6 7.7 6.2 5.9 1.4 1.1 – 0.3 0.4 5.9 7.7 1.8 Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 15 of 26 Chapter 11 – Section 1 ● Hypotheses H0: μd = 0 … no difference H1: μd > 0 … helps (We’re only interested in if our treatment makes things better or not) α = 0.01 ● Calculations n=5 d = .88 sd = .83 Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 16 of 26 Chapter 11 – Section 1 ● Calculations n=5 d = 0.88 sd = 0.83 ● The test statistic is d d 0.88 0 t0 2.36 s/ n 0.83 / 5 ● This has a Student’s t-distribution with 4 degrees of freedom Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 17 of 26 Chapter 11 – Section 1 ● Use the Student’s t-distribution with 4 degrees of freedom ● The right-tailed α = 0.01 critical value is 3.75 ● 2.36 is less than 3.75 (the classical method) ● Thus we do not reject the null hypothesis ● There is insufficient evidence to conclude that our method significantly improves the situation ● We could also have used the P-Value method Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 18 of 26 Chapter 11 – Section 1 ● Matched-pairs tests have the same various versions of hypothesis tests Two-tailed tests Left-tailed tests (the alternatively hypothesis that the first mean is less than the second) Right-tailed tests (the alternatively hypothesis that the first mean is greater than the second) Different values of α ● Each can be solved using the Student’s t Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 19 of 26 Chapter 11 – Section 1 ● Each of the types of tests can be solved using either the classical or the P-value approach Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 20 of 26 Chapter 11 – Section 1 ● A summary of the method For each matched pair, subtract the first observation from the second This results in one data item per subject with the data items independent of each other Test that the mean of these differences is equal to 0 ● Conclusions Do not reject that μd = 0 Reject that μd = 0 ... Reject that the two populations have the same mean Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 21 of 26 Chapter 11 – Section 1 ● Learning objectives 1 Distinguish between independent and dependent sampling 2 Test hypotheses made regarding matched-pairs data 3 Construct and interpret confidence intervals about the population mean difference of matched-pairs data Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 22 of 26 Chapter 11 – Section 1 ● We’ve turned the matched-pairs problem in one for a single variable’s mean / unknown standard deviation We just did hypothesis tests We can use the techniques in Section 9.2 (again, single variable’s mean / unknown standard deviation) to construct confidence intervals ● The idea – the processes (but maybe not the specific calculations) are very similar for all the different models Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 23 of 26 Chapter 11 – Section 1 ● Confidence intervals are of the form Point estimate ± margin of error ● This is precisely an application of our results for a population mean / unknown standard deviation The point estimate d and the margin of error t / 2 sd n for a two-tailed test Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 24 of 26 Chapter 11 – Section 1 ● Thus a (1 – α) • 100% confidence interval for the difference of two means, in the matched-pair case, is sd d t / 2 n where tα/2 is the critical value of the Student’s t-distribution with n – 1 degrees of freedom Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 25 of 26 Summary: Chapter 11 – Section 1 ● Two sets of data are dependent, or matchedpairs, when each observation in one is matched directly with one observation in the other ● In this case, the differences of observation values should be used ● The hypothesis test and confidence interval for the difference is a “mean with unknown standard deviation” problem, one which we already know how to solve Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 26 of 26