# 9 - STAT E-50 Introduction to Statistics Comparing Means...

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STAT E-50 - Introduction to Statistics Comparing Means; Paired Samples Sampling distribution for the difference between two means When the conditions are met, the standardized sample difference between the means of two independent groups, 1 2 1 2 1 2 (y - y )- ( μ - μ ) t = SE(y - y ) can be modeled by a Student’s t-model with a number of degrees of freedom found with a special formula. We estimate the standard error with 2 2 1 2 1 2 1 2 s s SE(y - y ) = + n n

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Assumptions and Conditions Independence assumption Randomization condition 10% condition Normal population assumption Nearly normal condition - for both samples n<15 in either group Be sure there is no severe skewness n<40 in either group Some skewness OK, but not any outliers both samples >40 Watch out for outliers and for multiple modes Independent groups assumption The two groups must be independent of each other; think carefully about how the data was gathered. 2
Two-sample t-interval When the conditions are met, the confidence interval for the difference between the means of two independent groups, μ 1 - μ 2 , is: ( ) * df t ± × 1 2 1 2 y - y SE(y - y ) where the standard error of the difference of the means, 2 2 1 2 1 2 1 2 s s SE(y - y ) = + n n . The critical value of t depends on the particular confidence level you specify, and on the number of degrees of freedom, which is determined by the sample sizes and a special formula. 3

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Testing the difference between two means The conditions for the two-sample t-test for the difference between the means of two independent groups are the same as for the two-sample interval. We test the hypothesis H 0 : μ 1 - μ 2 = Δ 0 where the hypothesized difference is almost always 0, using the statistic ( ) 1 2 0 1 2 y - y - Δ t = SE(y - y ) The standard error of 1 y - y 2 is 2 2 1 2 1 2 1 2 s s SE(y - y ) = + n n . When the conditions are met and the null hypothesis is true, this statistic can be closely modeled by a Student’s t-model with a number of degrees of freedom determined by the sample sizes. 4
1. Tennis elbow is thought to be aggravated by the impact experienced when hitting the ball. The article “Forces on the Hand in the Tennis One-Handed Backhand” ( Int. J. of Sport Biomechanics (1991) reported the force (N) on the hand just after impact on a one-handed backhand drive for six advanced players and for eight intermediate players. Advanced 44.70 26.31 55.75 28.54 46.99 39.46 Intermediate 15.58 19.16 24.13 10.56 32.88 21.47 14.32 33.09 Does the data shown above indicate that the mean force after impact is greater for advanced tennis players than it is for intermediate players? a) What do you want to know? b) What is the parameter you want to test? 5

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