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Chapter 25 1 Matched pairs procedures To compare responses to two treatments in a matched pairs design, apply the one-sample t procedures to the observed differences of the pairs: Assumptions: independently selected pairs of measurements (hence not more than 10% of the size of the population when selected without replacement) form a SRS of differences d that satisfy a Nearly Normal Condition. (View a histogram or normal probability plot of the data to check this condition.) A level C confidence interval for µ d (the mean difference) is d ± t* € n −1 sd n where s d is the sample standard deviation of € differences, and€t * is the appropriate critical n −1 value depending on the level of confidence C for the € t-distribution with n – 1 degrees of freedom € Chapter 25 2 Paired sample t hypothesis test • State hypotheses: Null hypothesis H0 : µ d = 0 Alternative hypothesis HA: µ > 0, or µ < 0, or µ ≠ 0 € • Choose the model: An independently selected SRS of differences drawn from a nearly normal population satisfying the 10% Condition, so Student’s t model applies to standardized sampling distribution for y • Mechanics: Compute t-statistic based on H€ 0: t = d −0 (s d / n ) . Probability associated with appropriate HA: P = P( T ≥ t ), or P = P( T ≤ t ), or P = 2P( T ≥ t ) € • Conclusion: Assess evidence against H0 in favor of HA depending on how small P is. [TI-83: STAT TESTS T-Test… ] Chapter 25 3 The Sign test A distribution-free method recharacterizes the hypothesized mean value 0 as a hypothesized median value, then ask what proportion of the differences exceed or fall shy of the median value 0? This replaces the original underlying statistic d with a proportion pˆ = proportion of the differences greater than (or less than) the € hypothesized median value 0 € We then carry out a one-proportion z test with null hypothesis H0: p = .50 and suitable alternative hypothesis. While this sign test dispenses with many of the conditions required by the t test, the t test is the more powerful when all its conditions are met.