Question

Many fast-food restaurants have soft drink dispensers with preset amounts, so that when the operator merely pushes a button for the desired rink the cup is automatically filled. This method apparently saves time and seems to increase worker productivity. A researcher randomly selects 9 workers from a restaurant with automatic dispensers and 9 works from a restaurant with manual dispensers. At a 1% significance level, use the Mann-Whitney U Test to test whether workers with automatic dispensers are significantly more productive.

Automatic (Group 1): 153, 128, 143, 110, 152, 168, 144, 137, 118
Manual (Group 2): 105, 118, 129, 114, 125, 117, 106, 92, 126

What rank will be given to the observation value, 118 that is in both the automatic and manual groups? (Round answer to 1 decimal). ___________.


When rounding the U test statistic up to the next value, what is the p-value from the Mann Whitney Table of p-values? (Round to 4 decimal places) ___________.

Answer 1

The ranks (given in brackets) are

Automatic (Group 1): 153 (17) , 128 (11) , 143 (14) , 110 (4) , 152 (16), 168 (18) , 144 (15), 137 (13) , 118 (7.5)
Manual (Group 2): 105 (2) , 118 (7.5) , 129 (12) , 114 (5),  125 (9) , 117 (6) , 106 (3) , 92 (1) , 126 (10)

The observation value, 118 that is in both the automatic and manual groups share the rank 7 and 8.

Rank = (7 + 8)/2 = 7.5

R1 = (17 + 11 + 14 + 4 + 16 + 18 + 15 + 13 + 7.5) = 115.5

R2 = (2 + 7.5 + 12 + 5 + 9 + 6 + 3 + 1 + 10) = 55.5

n1 = n2 = 9

U1 = n1n2 + n1(n1 + 1)/2 - R1 = 9 * 9 + 9 * 10 / 2 - 115.5 = 10.5

U2 = n1n2 + n2(n2 + 1)/2 - R2 = 9 * 9 + 9 * 10 / 2 - 55.5 = 70.5

U = min(10.5 , 70.5) = 10.5   11 (Rounding up to the next value)

E(U) = n1n2/2 = (9 * 9)/2 = 40.5

= 11.32475

Test statistic, z = (11 - 40.5)/11.32475 = -2.605

p-value = 2 * P(z < -2.605) =  0.0092