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:
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:
Show 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.
Updated On: Jun 4, 2026
- \(P\) is correct and \(Q\) is NOT correct
- \(P\) is NOT correct and \(Q\) is correct
- Both \(P\) and \(Q\) are correct
- Neither \(P\) nor \(Q\) is correct
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Understand the concept of runs.
A run is a sequence of similar symbols occurring consecutively.
The given sequence is
\[ + \; + \; - \; + \; - \; - \; + \; + \; + \; + \; - \; - \; - \; + \; + \; - \]
Step 2: Count the runs.
Grouping consecutive similar signs, we get
\[ ++ \;|\; - \;|\; + \;|\; -- \;|\; ++++ \;|\; --- \;|\; ++ \;|\; - \] Therefore, the number of runs is
\[ R=8 \] Hence, statement \(P\) is correct.
Step 3: Test randomness using the runs test.
Here, the number of positive signs is
\[ 9 \] and the number of negative signs is
\[ 7 \] The observed value is
\[ R=8 \] For a two-sided runs test at level \(0.05\), \(H_0\) is rejected only when the observed number of runs is unusually small or unusually large.
Given that
\[ P(R\leq 6)=0.108 \] This probability is already greater than \(0.05\).
Also, for the observed value
\[ R=8, \] we have
\[ P(R\leq 8)=0.4271 \] which is not small.
Thus, the observed number of runs is not extreme enough to reject \(H_0\).
Therefore, statement \(Q\) is NOT correct.
Step 4: Final conclusion.
Statement \(P\) is correct and statement \(Q\) is NOT correct.
Hence,
\[ \boxed{(A)} \]
A run is a sequence of similar symbols occurring consecutively.
The given sequence is
\[ + \; + \; - \; + \; - \; - \; + \; + \; + \; + \; - \; - \; - \; + \; + \; - \]
Step 2: Count the runs.
Grouping consecutive similar signs, we get
\[ ++ \;|\; - \;|\; + \;|\; -- \;|\; ++++ \;|\; --- \;|\; ++ \;|\; - \] Therefore, the number of runs is
\[ R=8 \] Hence, statement \(P\) is correct.
Step 3: Test randomness using the runs test.
Here, the number of positive signs is
\[ 9 \] and the number of negative signs is
\[ 7 \] The observed value is
\[ R=8 \] For a two-sided runs test at level \(0.05\), \(H_0\) is rejected only when the observed number of runs is unusually small or unusually large.
Given that
\[ P(R\leq 6)=0.108 \] This probability is already greater than \(0.05\).
Also, for the observed value
\[ R=8, \] we have
\[ P(R\leq 8)=0.4271 \] which is not small.
Thus, the observed number of runs is not extreme enough to reject \(H_0\).
Therefore, statement \(Q\) is NOT correct.
Step 4: Final conclusion.
Statement \(P\) is correct and statement \(Q\) is NOT correct.
Hence,
\[ \boxed{(A)} \]
Was this answer helpful?
0
0
Top IIT JAM MS Statistics Questions
- A bag has 5 blue balls and 15 red balls. Three balls are drawn at random from the bag simultaneously. Then the probability that none of the chosen balls is blue equals
- Let \( Y \) be a continuous random variable such that \( P(Y > 0) = 1 \) and \( \mathbb{E}(Y) = 1 \). For \( p \in (0, 1) \), let \( \xi_p \) denote the \( p \)th quantile of the probability distribution of the random variable \( Y \). Then which of the following statements is always correct?
- Let 𝑋 be a continuous random variable having the 𝑈(−2, 3) distribution. Then which of the following statements is correct?
- Let \( X \) be a random variable having the Poisson distribution with mean \( 1 \). Let \( g: \mathbb{N} \cup \{0\} \to \mathbb{R} \) be defined by
\[g(x) = \begin{cases} 1 & \text{if } x \in \{0, 2\} \\ 0 & \text{if } x \notin \{0, 2\} \end{cases}\]
Then \( E(g(X)) \) is equal to: - For \( n \in \mathbb{N} \), let \( Z_n \) be the smallest order statistic based on a random sample of size \( n \) from the \( U(0,1) \) distribution. Let \( nZ_n \xrightarrow{d} Z \) as \( n \to \infty \) for some random variable \( Z \). Then \( P(Z \leq \ln 3) \) is equal to:
View More Questions
Top IIT JAM MS Testing of Hypotheses Questions
- Let \( X_1, X_2, \ldots, X_n \) be a random sample from an \( \text{Exp}(\lambda) \) distribution, where \( \lambda \in \{1, 2\} \). For testing \( H_0: \lambda = 1 \) against \( H_1: \lambda = 2 \), the most powerful test of size \( \alpha \), \( \alpha \in (0,1) \), will reject \( H_0 \) if and only if
- Let \( X_1, X_2, \ldots, X_{10} \) be a random sample from a \( N(0, \sigma^2) \) distribution, where \( \sigma > 0 \) is unknown. For testing \( H_0: \sigma^2 \leq 1 \) against \( H_1: \sigma^2 > 1 \), a test of size \( \alpha = 0.05 \) rejects \( H_0 \) if and only if \( \sum_{i=1}^{10} X_i^2 > 18.307 \). Let \( \beta \) be the power of this test, at \( \sigma^2 = 2 \). Then \( \beta \) lies in the interval
(You may use \( \chi^2_{10,0.05} = 18.307 \), \( \chi^2_{10,0.1} = 15.9872 \), \( \chi^2_{10,0.25} = 12.5489 \), \( \chi^2_{10,0.5} = 9.3418 \), \( \chi^2_{10,0.75} = 6.7372 \), \( \chi^2_{10,0.9} = 4.8652 \), \( \chi^2_{10,0.95} = 3.9403 \), \( \chi^2_{10,0.975} = 3.247 \)) - Let X1, X2, X3, X4 be a random sample from a distribution with the probability mass function
\(f(x) = \begin{cases} \theta^x(1-\theta)^{1-x}, & x=0,1 \\ 0, & \text{otherwise}, \end{cases}\)
where θ ∈ (0, 1) is unknown. Let 0 < α ≤ 1. To test the hypothesis \(H_0:\theta=\frac{1}{2}\) against \(H_1:\theta>\frac{1}{2},\), consider the size α test that rejects H0 if and only if \(\sum^4_{i=1} 𝑋𝑖 ≥ k_α\), for some 𝑘α ∈ {0, 1, 2, 3, 4}. Then for which one of the following values of α, the size α test does NOT exist ? - Let X1, X2, X3, X4 be a random sample from a Poisson distribution with unknown mean λ > 0. For testing the hypothesis
H0 : λ = 1 against H1 : λ = 1.5,
let β denote the power of the test that rejects H0 if and only if \(∑_{i=1}^4 𝑋_𝑖 ≥ 5\). Then which one of the following statements is true ? - Let X1, X2 , … , Xn (n > 1) be a random sample from a N(μ, 1) distribution, where μ ∈ \(\R\) is unknown. Let 0 < α < 1. To test the hypothesis H0 : μ = 0 against H1 : μ = δ, where δ > 0 is a constant, let β denote the power of the size α test that rejects H0 if and only if \(\frac{1}{n}\sum^n_{i=1}X_i > c_{\alpha}\) , for some constant cα. Then which of the following statements is/are true ?
View More Questions
Top IIT JAM MS Questions
- Let \( a_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \) and \( b_n = \sum_{k=1}^{n} \frac{1}{k^2} \) for all \( n \in \mathbb{N} \). Then
- Let \( f(x,y) = x^2 + 3y^2 - \frac{2}{3} xy \) at \( (x,y) \in \mathbb{R}^2 \). Then
- Let \( A = \begin{pmatrix} a & 0 \\ c & d \end{pmatrix} \) be a real matrix, where \( ad = 1 \) and \( c \ne 0 \). If \( A^{-1} + (\text{adj} \, A)^{-1} = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} \), then \( (\alpha, \beta, \gamma, \delta) \) is equal to
- A bag has 5 blue balls and 15 red balls. Three balls are drawn at random from the bag simultaneously. Then the probability that none of the chosen balls is blue equals
- Let \( Y \) be a continuous random variable such that \( P(Y > 0) = 1 \) and \( \mathbb{E}(Y) = 1 \). For \( p \in (0, 1) \), let \( \xi_p \) denote the \( p \)th quantile of the probability distribution of the random variable \( Y \). Then which of the following statements is always correct?
View More Questions