Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 1 of 26 Chapter 11 Section 1 Inference about Two Means: Dependent Samples

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 2 of 26 Chapter 11 – Section 1 ●Learning objectives  Distinguish between independent and dependent sampling  Test hypotheses made regarding matched-pairs data  Construct and interpret confidence intervals about the population mean difference of matched-pairs data 1 2 3

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 3 of 26 Chapter 11 – Section 1 ●Learning objectives  Distinguish between independent and dependent sampling  Test hypotheses made regarding matched-pairs data  Construct and interpret confidence intervals about the population mean difference of matched-pairs data 1 2 3

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 4 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 ●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 – 2 nd Edition – Chapter 11 Section 1 – Slide 5 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 ●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 ●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 – 2 nd Edition – Chapter 11 Section 1 – Slide 6 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 ●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 – 2 nd Edition – Chapter 11 Section 1 – Slide 7 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 ●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 – 2 nd Edition – Chapter 11 Section 1 – Slide 8 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 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 – 2 nd Edition – Chapter 11 Section 1 – Slide 9 of 26 Chapter 11 – Section 1 ●Learning objectives  Distinguish between independent and dependent sampling  Test hypotheses made regarding matched-pairs data  Construct and interpret confidence intervals about the population mean difference of matched-pairs data 1 2 3

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 10 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 ●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 ●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 – 2 nd Edition – Chapter 11 Section 1 – Slide 11 of 26 Chapter 11 – Section 1 ●Specifically, for the before and after example,  d 1 = person 1’s after – person 1’s before  d 2 = person 2’s after – person 1’s before  d 3 = person 3’s after – person 1’s before ●Specifically, for the before and after example,  d 1 = person 1’s after – person 1’s before  d 2 = person 2’s after – person 1’s before  d 3 = 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 – 2 nd Edition – Chapter 11 Section 1 – Slide 12 of 26 Chapter 11 – Section 1 ●How do our hypotheses translate?  The two means are equal … the mean difference is zero … μ d = 0  The two means are unequal … the mean difference is non-zero … μ d ≠ 0 ●How do our hypotheses translate?  The two means are equal … the mean difference is zero … μ d = 0  The two means are unequal … the mean difference is non-zero … μ d ≠ 0 ●Thus our hypothesis test is  H 0 : μ d = 0  H 1 : μ d ≠ 0  The standard deviation σ d is unknown ●We know how to do this!

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 13 of 26 Chapter 11 – Section 1 ●To solve  H 0 : μ d = 0  H 1 : μ d ≠ 0  The standard deviation σ d 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 – 2 nd Edition – Chapter 11 Section 1 – Slide 14 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 – 2 nd Edition – Chapter 11 Section 1 – Slide 15 of 26 Chapter 11 – Section 1 ●An example … whether our treatment helps or not … helps meaning a higher measurement ●The “Before” and “After” results BeforeAfterDifference –

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 16 of 26 Chapter 11 – Section 1 ●Hypotheses  H 0 : μ d = 0 … no difference  H 1 : μ d > 0 … helps  (We’re only interested in if our treatment makes things better or not)  α = 0.01 ●Hypotheses  H 0 : μ d = 0 … no difference  H 1 : μ d > 0 … helps  (We’re only interested in if our treatment makes things better or not)  α = 0.01 ●Calculations  n = 5  d =.88  s d =.83

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 17 of 26 Chapter 11 – Section 1 ●Calculations  n = 5  d = 0.88  s d = 0.83 ●Calculations  n = 5  d = 0.88  s d = 0.83 ●The test statistic is ●Calculations  n = 5  d = 0.88  s d = 0.83 ●The test statistic is ●This has a Student’s t-distribution with 4 degrees of freedom

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 18 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 ●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 ●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 – 2 nd Edition – Chapter 11 Section 1 – Slide 19 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 – 2 nd Edition – Chapter 11 Section 1 – Slide 20 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 – 2 nd Edition – Chapter 11 Section 1 – Slide 21 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 ●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 μ d = 0  Reject that μ d = 0... Reject that the two populations have the same mean

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 22 of 26 Chapter 11 – Section 1 ●Learning objectives  Distinguish between independent and dependent sampling  Test hypotheses made regarding matched-pairs data  Construct and interpret confidence intervals about the population mean difference of matched-pairs data 1 2 3

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 23 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 ●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 – 2 nd Edition – Chapter 11 Section 1 – Slide 24 of 26 Chapter 11 – Section 1 ●Confidence intervals are of the form Point estimate ± margin of error ●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 for a two-tailed test

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 25 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 where t α/2 is the critical value of the Student’s t-distribution with n – 1 degrees of freedom

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 26 of 26 Summary: Chapter 11 – Section 1 ●Two sets of data are dependent, or matched- pairs, 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