Discussion
The recorded air quality index (AQI) at a place in a city for 16 consecutive days for the current year is compared with the historical average AQI. The direction of the recorded AQI is noted as '+' if it is above the historical average and as '−' if it is below the historical average. The dataset is as below:
+ + − + − − + + + + − − − + + −
The total number of runs, \(R\), is being used to test
\(H_0\): The direction of the recorded AQI is random
against
\(H_1\): The direction of the recorded AQI is not random
Consider the following statements.
P: The observed value of total number of runs is 8.
Q: Based on the above dataset, \(H_0\) is rejected in favor of \(H_1\) at level 0.05.
Then which one of the following is true?
[You may use: Under \(H_0\), with 7 negative signs and 9 positive signs, \(P(R \leq 6) = 0.108, P(R \leq 7) = 0.231, P(R \leq 8) = 0.427\)]
Answer: A
P is correct and Q is NOT correct.
A run is a maximal block of identical signs. Splitting the sequence:
(+ +) (−) (+) (− −) (+ + + +) (− − −) (+ +) (−)
gives \(R = 8\) runs, with 9 plus and 7 minus signs, so P is correct.
The alternative 'not random' is two-sided: too few runs (clustering) or too many (alternation) both count against \(H_{0}\). Under \(H_{0}\) the expected number of runs is \(1 + \frac{2 \cdot 9 \cdot 7}{16} \approx 8.9\), and the observed 8 is close to it. The lower tail \(P(R \leq 8) = 0.427\) is far above the 0.025 a two-sided test at level 0.05 allows each side; even 6 runs would not be significant, since \(P(R \leq 6) = 0.108\). So \(H_{0}\) is not rejected at level 0.05, and Q is not correct.
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