This question concerns the following primal linear program: Minimize Subject to

ID: 446160 • Letter: T

Question

This question concerns the following primal linear program: Minimize Subject to a: Prove, by computing the appropriate reduced costs, that the optimal primal solution is x*= (x*1,x*2, x*3, x*4)=(0, 2, 4, 0). To use the formulas derived in the class notes, you will have to first add some slack/surplus variables to make all the constraints into equalities. Which slack or surplus variable must be basic? b: Return to the above linear program as stated with the inequalities. Write down a linear program that is dual to this one. c: Using some of the values you have already computed in (a), write down an optimal solution pi* = (pi*1, pi*2, pi*3) for the dual linear program. d: Verify that the optimal objective value of the primal linear program equals the optimal objective value of the dual linear program. e: The complementary slackness property says that either pi*1 = 0, or 2x*1+6x*2 + 3x*3 + 5x*4 = 24, or both. In this case, you can verify it's the latter that holds. What does the complementary slackness property say about pi*3? About pi*1? About x*2? Verify that all these relationships do hold for the example at hand.

Explanation / Answer

Summary statistics for two samples

Female

Male

Sample sizes

Sample means

102.23

86.46

Sample standard deviations

93.393

59.695

Confidence interval for difference between means

Sample mean difference

15.77

Pooled standard deviation

79.466

Std error of difference

23.23