Twelve runners are asked to run a 10-kilometer race on each of two consecutive weeks. In one of the races the runners wear one brand of shoe and in the other a second brand. The brand they wear in each race is determined at random. All runners are timed and are asked to run their best in each race. The results in minutes are given below. Assume normality of the populations. Is there sufficient evidence that the mean time using brand 1 is different than the mean times using brand 2? Runner Brand 1 Brand 2
1 31.23 32.02
2 29.33 28.98
3 30.50 30.63
4 32.20 32.67
5 33.08 32.95
6 31.52 31.53
7 30.68 30.83
8 31.05 31.10
9 33.00 33.12
10 29.67 29.50
11 30.55 30.57
12 32.12 32.30

Given: ad¯¯¯¯¯ is an altitude. prove: ab2+ac2=cb2 right triangle a b c with right angle a. point d lies on side b c and segment a d is drawn. angle a d c is a right angle. drag and drop a reason into each box to correctly complete the two-column proof. statement reason ad¯¯¯¯¯ is an altitude, and ∠bac is a right angle. given ∠adb and ∠adc are right angles. definition of altitude ∠bac≅∠bda ? ∠bac≅∠adc ? ∠b≅∠b ? ∠c≅∠c reflexive property of congruence △abc∼△dba ? △abc∼△dac aa similarity postulate abbd=cbab ? ab2=(cb)(bd) cross multiply and simplify. acdc=cbac polygon similarity postulate ac2=(cb)(dc) cross multiply and simplify. ab2+ac2=ab2+(cb)(dc) addition property of equality ab2+ac2=(cb)(bd)+(cb)(dc) substitution property of equality ab2+ac2=(cb)(bd+dc) ? bd+dc=cb segment addition postulate ab2+ac2=cb2 substitution property of equality