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chapter9 - CHAPTER 9 TWO-SAMPLE PROBLEMS 9.1 Comparing Two...

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1 CHAPTER 9: TWO-SAMPLE PROBLEMS 9.1 Comparing Two Population Means n 1 , 1 x n 2 , 2 x The means μ 1 and μ 2 and standard deviations σ 1 and σ 2 are unknown. Problem: Compare the means μ 1 and μ 2 of two normal populations based on sample data (inferences about μ 1 - μ 2 ). Estimate of μ 1 - μ 2 : 1 2 x x . The sampling distribution of 1 2 x x is normal with the mean at μ 1 - μ 2 and standard deviation σ ( 1 2 x x ) = 2 2 1 2 1 2 n n σ σ + . Normal Population 1 μ 1, 1 σ Normal Population 2 μ 2, 2 σ
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2 What can be inferred about μ 1 - μ 2 given 1 2 x x ? Example: Consider the sampling distributions of 1 x and 2 x . 1 x 2 x •••••••• •• ••• ••••••• μ 1 μ 2 1 x 2 x Conclusion: If 1 2 x x is far away from 0 and spreads are small, then strong evidence that μ 1 - μ 2 is not zero i.e. μ 1 μ 2 . •••••••••••••••• •• •••••••••••••••••• μ 1 μ 2 1 x 2 x Conclusion: If the spreads are large, 1 2 x x may be a good or poor estimate of μ 1 - μ 2 .
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3 Standard Error of 1 2 x x : SE( 1 2 x x ) = 2 2 1 2 1 2 s s n n + . It estimates the standard deviation σ ( 1 2 x x ). Define the t statistic as follows: 1 2 1 2 2 2 1 2 1 2 1 2 ( ) 0 . ( ) x x x x t SE x x s s n n = = + Then t follows approximately a t distribution with the number of degrees of freedom k which is the smaller of n 1 –1 and n 2 –1. Remark: If s 1 are similar s 2 , and n 1 and n 2 are close, then k is close to n 1 + n 2 -2. Confidence Interval for μ 1 - μ 2 t * 1- α Area = α /2 t(k)
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4 At least (1- α )•100% confidence interval for μ 1 - μ 2 : 2 2 1 2 1 2 1 2 ( ) . s s x x t n n ± + Test about μ 1 - μ 2 : To test H 0 : μ = μ 0 compute the two-sample t statistic 1 2 2 2 1 2 1 2 . x x t s s n n = + and use p-values for the t (k) distribution, where k is the smaller of n 1 –1 and n 2 –1. The true p-value is no higher than that calculated from t (k). Assumptions: 1. Independent random samples for the two groups, 2. Approximately normal population distribution for each group. Example: One study compared forest plots in Borneo that had never been logged with similar plots nearby that had been logged 8 years earlier. The data on the number of tree species in 12 unlogged plots and 9 logged plots is given below: Unlogged 22 18 22 20 15 21 13 13 19 13 19 15 Logged 17 4 18 14 18 15 15 10 12
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