Hypothesis about Matched pairs 1.Claiming a “before and after” relationship about a population. 2.The sample data consist of matched pairs. E.g., before and after measurements The samples are simple random samples. 3.If the number of pairs of sample data is small (n  30), then the population of differences in the paired values must be approximately normally distributed.

Testing Matched Pair Claims µ d = mean of the differences between the pairs of data for the population ‘after’ minus ‘before’ d = mean of the differences between the pairs of data for the sample s d = standard deviation of the differences between the pairs of data for the sample

For example SAT prep course increases SAT scores BeforeAfterd

Procedure: Classic value 1.Write claim –No change between ‘before’ and ‘after’: µ d = 0 –Increase from ‘before’ to ‘after’ : µ d > 0 –Decrease from ‘before’ to ‘after’ : µ d < 0 2.Write hypothesis 3.Test Statistic: 4.Critical value from z- or t table

Testing Matched Pairs Original claimPick null and alternate hypothesis Opposite claim Degree of confidence Sampled-bar =s =n = Test statistic Two tailed (=) Right tailed () Left tailed Critical value P-value Conclusion 5

P-value method Z-Test μ 0 : 0 σ: s d x-bar: d-bar n: Sample size or T-Test μ 0 : 0 x-bar: d-bar Sx: s d n: Sample size

Example Claim: men exaggerate their heights Degree of confidence: 95% Before:men's’ estimated heights After:men's’ measured heights Mathematical claim:

Testing Matched Pairs Original claimPick null and alternate hypothesis Opposite claim Degree of confidence Sampled-bar =s =n = Test statistic Two tailed (=) Right tailed () Left tailed Critical value P-value Conclusion 8

Your Turn A low-sodium diet lowers blood pressure BeforeAfter

Testing Matched Pairs Original claimPick null and alternate hypothesis Opposite claim Degree of confidence Sampled-bar =s =n = Test statistic Two tailed (=) Right tailed () Left tailed Critical value P-value Conclusion 10

Homework Before After The preceding table lists individual’s systolic blood pressure taken before and after trying a new medication. Using a significance level, test the claim that the medication helps reduce blood pressure. 2.The table below lists gas mileage of a sample of cars after switching gasoline brands. Test the claim that switching brands had no effect on mileage Before After

More Homework Authorized repair center$797$571$904$1147$418 Independent repair center$523$488$875$911$297 3.In the preceding table, cars subjected to crash test were taken to two shops for repair estimates. Using a significance level of 0.05, test the claim that the independent center is cheaper than the authorized center. 4.The table below lists the measured pain scale for several subjects before and after undergoing hypnosis. Using a 99% degree of confidence, test the claim that hypnotism relieves pain Before After

Problem #1 Original claim μ d < 0 H 1 Pick null and alternate hypothesis Opposite claim μ d ≥ 0 H 0 Degree of confidence 95% Sampled-bar = -5.4s = 7.89n = 5 Test statistict = Two tailed (=) Right tailed () Left tailed Critical value P-value 0.1 > 0.05 ConclusionFTR H 0, FTR original claim 13

Problem #2 Original claim μ d = 0 H 0 Pick null and alternate hypothesis Opposite claim μ d ≠ 0 H 1 Degree of confidence 95% Sampled-bar = -1.1s = 3.54n = 9 Test statistict = Two tailed (=) Right tailed () Left tailed Critical value P-value 0.38 > 0.05 ConclusionFTR H 0, FTR original claim 14

Problem #2 fixed Original claim μ d = 0 H 0 Pick null and alternate hypothesis Opposite claim μ d ≠ 0 H 1 Degree of confidence 95% Sampled-bar = -0.01s = 1.05n = 9 Test statistict = Two tailed (=) Right tailed () Left tailed Critical value P-value 0.97 > 0.05 ConclusionFTR H 0, FTR original claim 15

Testing Matched Pairs Original claim μ d < 0 H 1 Pick null and alternate hypothesis Opposite claim μ d ≥ 0 H 0 Degree of confidence 95% Sampled-bar = s = n = 5 Test statistict = Two tailed (=) Right tailed () Left tailed Critical value P-value 0.02 < 0.05 ConclusionReject H 0, Accept original claim 16

Testing Matched Pairs Original claim μ d < 0 H 1 Pick null and alternate hypothesis Opposite claim μ d ≥ 0 H 0 Degree of confidence 99% Sampled-bar = s = 2.911n = 8 Test statistict = Two tailed (=) Right tailed () Left tailed Critical value P-value < 0.01 ConclusionReject H 0, Accept original claim 17