Question

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:

• A. \(P\) is correct and \(Q\) is NOT correct
• B. \(P\) is NOT correct and \(Q\) is correct
• C. Both \(P\) and \(Q\) are correct
• D. Neither \(P\) nor \(Q\) is correct

Hint:

In a runs test, very small or very large values of the number of runs indicate non-randomness. A central value of \(R\) does not lead to rejection of randomness.

Answer 1

The Correct Option is A

Step 1: Group the signs into runs.
The sequence $++-+--++++---++-$ breaks into blocks: $++\,|\,-\,|\,+\,|\,--\,|\,++++\,|\,---\,|\,++\,|\,-$.

Step 2: Count the runs for $P$.
That is $8$ blocks, so $R=8$, and $P$ is correct.

Step 3: Judge randomness for $Q$.
With $9$ plus and $7$ minus signs, the runs test rejects only for unusually few or many runs.

Step 4: Compare with the cutoffs.
$R=8$ is a middling value ($P(R\le8)=0.4271$), nowhere near extreme, so we cannot reject randomness. $Q$ is not correct.

Step 5: Conclude.
$P$ right, $Q$ wrong, option (A).
\[ \boxed{(A)} \]