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STA 13 Lecture 14 Comparison of Two Independent Populations (continued) III. Inferences When σ1 and σ2 are Unknown A Practical Problem: Suppose we are interested in comparison average systolic blood pressures (SBP) between males and females 45-50 years old. We collected the following data Male: Female: 130 127 134 138 137 125 135 134 140 134 Question: Is SBP on average higher for males compared to females? a. If the two populations are normal with unknown variances, then b. A 100( 1- α) % confidence interval for ( µ1 - µ2 ) when the population standard deviations are unknown is: Example: Systolic Blood Pressure c. To test Ho : (µ1 - µ 2) = D0 , we compute t = and reject Ho in favor of Ha of Ha : (µ1 - µ 2) < D0 if t < - tα Ha : (µ1 - µ 2) > D0 if t > tα Ha : (µ1 - µ 2) ≠ D0 if t < - tα/2 or z > tα/2 2 Example: Systolic Blood Pressure 3 d. The Pooled Two-Sample Procedure: If σ1 and σ2 are unknown but σ1 = σ2 , then a pooled estimate of the common variance is: Using this estimate, a 100(1- α )% confidence interval for ( µ1 - µ2 ) is To test Ho : (µ1 - µ 2) = D0 , we compute t = and reject Ho in favor of Ha Ha : (µ1 - µ 2) < D0 if t < - tα Ha : (µ1 - µ 2) > D0 if t > tα Ha : (µ1 - µ 2) ≠ D0 if t < - tα/2 or z > tα/2 4 Example: The Orange Juice Study We would like to compare the taste of a “new” product to an “old” product based on samples of size 15 and 14. Suppose the mean of the first sample is 9.5 with a standard deviation of 1.3, and the mean of the second sample is 7.1 with a standard deviation 1.5. Is the new product better than the old product by one unit? Test at 0.05 level of significance. 5