In: Statistics and Probability
Professor Bert Forman believes the students who complete his examinations in the shortest time receive the highest grades and those who take the longest to complete them receive the lowest grades. To verify his suspicion, he assigns a rank to the order of finish and then grades the examinations. The results are shown below:
Student | Order of Completion | Score (50 possible) | Student | Order of Completion | Score (50 possible) | |||
Gromney | 1 | 84 | Smythe | 7 | 59 | |||
Bates | 2 | 81 | Arquette | 8 | 79 | |||
MacDonald | 3 | 82 | Govito | 9 | 61 | |||
Sosa | 4 | 67 | Gankowski | 10 | 62 | |||
Harris | 5 | 66 | Bonfigilo | 11 | 74 | |||
Cribb | 6 | 48 | Matsui | 12 | 65 |
Convert the test scores to a rank and find the coefficient of rank correlation between the order of completion and the rank of the test score. At the 0.05 significance level, can Professor Forman conclude there is a positive association between the order of finish and the test scores? (Round your answers to 3 decimal places.)
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Using Excel :
Student | Order of Completion | Score (50 possible) | Rank | d | d^2 |
Gromney | 1 | 84 | 12 | -11 | 121 |
Bates | 2 | 81 | 10 | -8 | 64 |
MacDonald | 3 | 82 | 11 | -8 | 64 |
Sosa | 4 | 67 | 7 | -3 | 9 |
Harris | 5 | 66 | 6 | -1 | 1 |
Cribb | 6 | 48 | 1 | 5 | 25 |
Smythe | 7 | 59 | 2 | 5 | 25 |
Arquette | 8 | 79 | 9 | -1 | 1 |
Govito | 9 | 61 | 3 | 6 | 36 |
Gankowski | 10 | 62 | 4 | 6 | 36 |
Bonfigilo | 11 | 74 | 8 | 3 | 9 |
Matsui | 12 | 65 | 5 | 7 | 49 |
440 |
Spearman's Rank Correlation Coefficient:
=1-1.5384
=-0.5385
The value -0.5384 indicates there exists negative relation between two variables.
H0: The rank correlation in the population is 0
H1: There is positive association between rankings
here alpha=0.05 and degrees of freedom=12-2=10 so t critical value is 1.182
Test statistic
Since the computed t test statistic < 1.182, we do not reject the null hypothesis. So it is reasonable to conclude that there is no positive association between rankings.
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