Question

4. In some firefighting organizations, you must serve as a firefighter for some period of time before you can become part of the emergency medical service arm of the organization. Does EMS workers are older, on average, than traditional firefighters? Use the data and rank shown and a 05 to test whether EMS workers are significantly older than firefighters. Assume the two groups are independent. (12.25 pts) Firefighter EMS workers 27 39 29 23 32 37 24 28 21 25 27 41 36 30 28 28 36 27 30 Rank: rank group age 30 EMS workers 30 EMS workers 30 EMS workers rank group age 13 21 Firefighter 23 Firefighter 24 Firefighter 25 Firefighter 27 Firefighter 27 EMS workers 13 13 15 16.5 16.5 18.5 18.5 20 32 Firefighter 33 EMS workers 33 EMS workers 36 Firefighter 36 EMS workers 27 EMS workers 28 Firefighter 28 EMS workers 97 Firefeghter 39 EMS workers 41 Firefighter 28 EMS workers 29 EMS workers

Answer 1

Null Hypothesis H0: EMS workers have same average than firefighters.
Alternative Hypothesis H1: EMS workers are older than firefighters.

We will use Wilcoxon Rank Sum Test.

The test statistic for the Wilcoxon Rank Sum Testis denoted U and is the smaller of U₁ and U₂,

where

U₁ = n1 n2 + n1(n1 + 1)/2 - R1

U₂ = n1 n2 + n2(n2 + 1)/2 - R2

where n1, n2 are the sample size of the two groups

and R1 and R2 are the sum of the ranks of the firefighters and EMS workers respectively.

n1 = 10, n2 = 12

R1 = 1 + 2 + 3 + 4 + 6 + 9 + 15 + 18.5 + 20 + 22 = 100.5

R2 = 6 + 6 + 9 + 9 + 11 + 13 + 13 + 13 + 16.5 + 16.5 + 18.5 + 21 = 152.5

U₁ = 10 * 12 + 10 (10 + 1)/2 - 100.5 = 74.5

U₂ = 10 * 12 + 12 (12 + 1)/2 - 152.5 = 45.5

U = Minimum of U1 and U2 is 45.5

Critical value of U at a = 0.05 and n1 = 10 and n2 = 12 for one-tailed test is 34.

Since the observed U (45.5) is larger than the critical value (34), we fail to reject H0 and conclude that there is no significant evidence that EMS workers are older than firefighters.