Question

Candidate - Rank by Prof A - Rank by Prof B
1 6 5
2 10 11
3 2 6
4 1 3
5 5 4
6 11 12
7 4 2
8 3 1
9 7 7
10 12 10
11 9 8
12 8 9

What is the Spearman rank-correlation coefficient (rs)?

What is σrs?

What is the value of the test statistic z?

What is the p-value?

Using α = .02 what is your conclusion about how Professor A and Professor B rank the twelve students?

Answer 1

Answer on Question #57389- Math - Statistics and Probability

Candidate - Rank by Prof A - Rank by Prof B

$n = 12$

$d_{i} = x_{i} - y_{i}$ is the difference between ranks;

$$\sum d _ {i} ^ {2} = 1 + 1 + 1 6 + 4 + 1 + 1 + 4 + 4 + 0 + 4 + 1 + 1 = 3 8$$

Question

What is the Spearman rank-correlation coefficient (rs)?

Solution

$$r_s = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)} = 1 - \frac{6 \cdot 38}{12(12^2 - 1)} = 0.8671.$$

Answer: 0.8671.

Question

What is $\sigma_s$?

Solution

$$\sigma_{rs} = \frac{1}{\sqrt{12 - 1}} = 0.3015$$

Answer: 0.3015.

Question

What is the value of the test statistic $z$?

Solution

$$z = \frac{0.8671 - 0}{0.3015} = 2.88$$

Answer: 2.88.

Question

What is the p-value?

Solution

$$p - value = 2(1 - P(z < 2.88)) = 2(1 - 0.9980) = 0.004$$

Answer: 0.004.

Question

Using $\alpha = .02$ what is your conclusion about how Professor A and Professor B rank the twelve students?

Solution

The null hypothesis: there is no association between the two variables.

P-value is less than $\alpha = .02$, thus we reject the null hypothesis that population rank-correlation coefficient is zero. There is significant rank-correlation between rank of the twelve students by Professor A and Professor B. If we set $\alpha = 0.02$, achieving a statistically significant Spearman rank-order correlation means that we can be sure that there is less than a $2\%$ chance that the strength of the relationship we found (our coefficient) happened by chance if the null hypothesis were true.

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