Research suggests that auditory environments can affect cognitive performance. A cognitive psychologist investigated this using an independent measures design. They compared the concentration scores of participants who completed a complex puzzle while listening to classical music to those who completed the puzzle in silence. Concentration was scored on a scale from 0 to 20 (where 0 = 'extremely poor concentration' and 20 = 'excellent concentration'). Some of the data collected is presented in the table below.
| Classical Music | Silence (No Music) | ||
|---|---|---|---|
| Concentration Score | Rank | Concentration Score | Rank |
| 5 | 3 | 2 | 1 |
| 8 | 5 | 4 | 2 |
| 12 | 8 | 7 | 4 |
| 13 | 9 | 10 | 6 |
| 16 | 12 | 11 | 7 |
| 17 | 13 | 14 | 10 |
| 19 | 15 | 15 | 11 |
| 20 | 16 | 18 | 14 |
Mann-Whitney U Formulas:
U=the smaller of U1 and U2U = \text{the smaller of } U_1 \text{ and } U_2U=the smaller of U1 and U2
Where:
U1=n1n2+n1(n1+1)2−∑R1 U_1 = n_1 n_2 + \frac{n_1(n_1 + 1)}{2} - \sum R_1 U1=n1n2+2n1(n1+1)−∑R1and
U2=n1n2+n2(n2+1)2−∑R2 U_2 = n_1 n_2 + \frac{n_2(n_2 + 1)}{2} - \sum R_2 U2=n1n2+2n2(n2+1)−∑R2Calculate the U U\,U value for the data collected in this study using the Mann-Whitney U U\,U formulas provided. Show your workings.
