Q.1 – Q.5 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).
Q.1
The current population of a city is 11,02,500. If it has been increasing at the rate of 5% per annum, what was its population 2 years ago?
(A)
9,92,500
(B)
9,95,006
(C)
10,00,000
(D)
12,51,506
Q.2
p and q are positive integers and pq+qp=3then, P2/q2 +q2/P2
(A)
3
(B)
7
(C)
9
(D)
11
Q.3
The least number of squares that must be added so that the line P-Q becomes the line of symmetry is _______.
(A)
4
(B)
3
(C)
6
(D)
7
Q.4
Nostalgia is to anticipation as_________is to__________Which one of the following options maintains a similar logical relation in the above sentence?
(A)
Present, past
(B)
Future, past
(C)
Past, future
(D)
Future, present
Q.5
Consider the following sentences:I woke up from sleep.I woked up from sleep.I was woken up from sleep.I was wokened up from sleep.Which of the above sentences are grammatically CORRECT?
(A)
(i) and (ii)
(B)
(i) and (iii)
(C)
(ii) and (iii)
(D)
(i) and (iv)
Q.6 – Q. 10 Multiple Choice Question (MCQ), carry TWO marks each (for each wrong answer: – 2/3).
Q.6
Given below are two statements and two conclusions. Statement 1: All purple are green.Statement 2: All black are green. Conclusion I: Some black are purple. Conclusion II: No black is purple.Based on the above statements and conclusions, which one of the following options is logically CORRECT?
(A)
Only conclusion I is correct.
(B)
Only conclusion II is correct.
(C)
Either conclusion I or II is correct.
(D)
Both conclusion I and II are correct.
Q.7
Computers are ubiquitous. They are used to improve efficiency in almost all fields from agriculture to space exploration. Artificial intelligence (AI) is currently a hot topic. AI enables computers to learn, given enough training data. For humans, sitting in front of a computer for long hours can lead to health issues.Which of the following can be deduced from the above passage?Nowadays, computers are present in almost all places.Computers cannot be used for solving problems in engineering.For humans, there are both positive and negative effects of using computers.Artificial intelligence can be done without data.
(A)
(ii) and (iii)
(B)
(ii) and (iv)
(C)
(i), (iii) and (iv)
(D)
(i) and (iii)
Q.8
Consider a square sheet of side 1 unit. In the first step, it is cut along the main diagonal to get two triangles. In the next step, one of the cut triangles is revolved about its short edge to form a solid cone. The volume of the resulting cone, in cubic units, is
(A)
𝜋/3
(B)
2𝜋/3
(C)
3𝜋/2
(D)
3π
Q.9
The number of minutes spent by two students, X and Y, exercising every day in a given week are shown in the bar chart above.The number of days in the given week in which one of the students spent a minimum of 10% more than the other student, on a given day, is
(A)
4
(B)
5
(C)
6
(D)
7
Q.10
Corners are cut from an equilateral triangle to produce a regular convex hexagon as shown in the figure above.The ratio of the area of the regular convex hexagon to the area of the original equilateral triangle is
(A)
2 : 3
(B)
3 : 4
(C)
4 : 5
(D)
5 : 6
Statistics
Q.1 – Q.9 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).
Q.1
Let 𝑿 be a non-constant positive random variable such that (𝑿) = 𝟗. Then which one of the following statements is true?
(A)
(B)
(C )
(D)
Q.2
Let {(𝒕)}𝒕≥𝟎be a standard Brownian motion. Then the variance of𝑾(𝟏)𝑾(𝟐) equals
(A)
1
(B)
2
(C)
3
(D)
4
Q.3
Let 𝑿𝟏, 𝑿𝟐, … , 𝑿𝒏be a random sample of size 𝒏 (≥ 𝟐) from a distribution having the probability density function
where 𝜃 ∈ (0, ∞). Then the method of moments estimator of 𝜃 equals
(A)
(B)
(C)
(D)
Q.4
Let {𝒙𝟏, 𝒙𝟐, … , 𝒙𝒏} be a realization of a random sample of size 𝒏 (≥ 𝟐) from a 𝑵(𝝁, 𝝈𝟐) distribution, where −∞ < 𝝁 < ∞ and 𝝈 > 𝟎. Which of the following statements is/are true?
P: 95% confidence interval of 𝜇 based on {𝑥1, 𝑥2, … , 𝑥𝑛} is unique when𝜎 is known
Q.: 95% confidence interval of 𝜇 based on {𝑥1, 𝑥2, … , 𝑥𝑛} is NOT unique when 𝜎 is unknown.
(A)
P only
(B)
Q only
(C)
Both P and Q
(D)
Neither P nor Q
Q.5
(A)
𝑆 has a monotone likelihood ratio in 𝑇1 and 𝐻0 is rejected if 𝑇1 > Xn,𝛼2
(B)
𝑆 has a monotone likelihood ratio in 𝑇1 and 𝐻0 is rejected if 𝑇1 > Xn,1-𝛼2
(C)
𝑆 has a monotone likelihood ratio in 𝑇2 and 𝐻0 is rejected if 𝑇2 > Xn,𝛼2
(D)
𝑆 has a monotone likelihood ratio in 𝑇2 and 𝐻0 is rejected if 𝑇2 > Xn,1- 𝛼2
Q.6
(A)
Under 𝐻0, the test statistic follows central chi-square distribution with one degree of freedom and the observed value of the test statistic is 20/3
(B)
Under 𝐻0, the test statistic follows central chi-square distribution with three degrees of freedom and the observed value of the test statistic is 20/3
(C)
Under 𝐻0, the test statistic follows central chi-square distribution with one degree of freedom and the observed value of the test statistic is 16/3
(D)
Under 𝐻0, the test statistic follows central chi-square distribution with three degrees of freedom and the observed value of the test statistic is 16/3
Q.7
(A)
4Φ𝜌(0,0) − 1
(B)
4Φ𝜌(0,0)
(C)
4Φ𝜌(0,0) + 1
(D)
Φ𝜌(0,0)
Q.8
(A)
(B)
(C)
(D)
Q.9
(A)
𝑍 follows 𝑊4(7, 𝐼4) distribution
(B)
𝑍 follows 𝑊4(4, 𝐼4) distribution
(C)
𝑍 follows 𝑊7(4, 𝐼7) distribution
(D)
𝑍 follows 𝑊7(7, 𝐼7) distribution
Q.10 – Q.25 Numerical Answer Type (NAT), carry ONE mark each (no negative marks).
Q.10
Q.11
Let
Then the value of 𝒆𝑰+𝝅equals (round off to 𝟐 decimal places).
Q.12
Q.13
Q.14
Let 𝑨 and 𝑩 be two events such that (𝑩) = ¾ and 𝑷(𝑨 𝖴 𝑩𝒄) = ½. If 𝑨 and 𝑩 are independent, then 𝑷(𝑨) equals_________(round off to𝟐 decimal places).
Q.15
A fair die is rolled twice independently. Let 𝑿 and 𝒀 denote the outcomes of the first and second roll, respectively. Then 𝑬(𝑿 + 𝒀 | (𝑿 − 𝒀)𝟐 = 𝟏) equals
Q.16
Q.17
If the marginal probability density function of the 𝒌𝒕𝒉order statistic of a random sample of size 8 from a uniform distribution on [𝟎, 𝟐] is then 𝒌 equals
Q.18
Q.19
Let {𝑿𝒏}≥𝟏be a sequence of independent and identically distributed random variables each having uniform distribution on [𝟎, 𝟐]. For 𝒏 ≥ 𝟏, letZn=-logei = 1n(2-Xi)1nThen, as 𝒏 → ∞, the sequence {𝒁𝒏}𝒏≥𝟏converges almost surely to ___________(round off to 𝟐 decimal places).
Q.20
Let {𝑿𝒏}𝒏≥𝟎be a time-homogeneous discrete time Markov chain with state space {𝟎, 𝟏} and transition probability matrix
Q.21
Let {𝟎, 𝟐} be a realization of a random sample of size 𝟐 from a binomial distribution with parameters 𝟐 𝐚𝐧𝐝 𝒑, where 𝒑 ∈ (𝟎, 𝟏). To test 𝑯1 : 𝒑 = ½ against 𝑯0 : 𝒑 ≠ ½, the observed value of the likelihood ratio test statisticequals_________(round off to 𝟐 decimal places).
Q.22
Q.23
Q.24
Let 𝑿𝟏, 𝑿𝟐and 𝑿𝟑be a random sample from a bivariate normal distribution with unknown mean vector 𝝁 and unknown variance- covariance matrix 𝚺, which is a positive definite matrix. The 𝒑-value corresponding to the likelihood ratio test for testing 𝑯𝟎: 𝝁 = 𝟎against 𝑯1: 𝝁 = 𝟎 based on the renzation 12, 4-2, -50sample equals_________(round off to 𝟐 decimal places).
Q.25
Let 𝒀𝒊 = + 𝒙𝒊 + 𝝐𝒊, 𝒊 = 𝟏, 𝟐, 𝟑, where 𝒙𝒊’s are fixed covariates, 𝑎 andare unknown parameters and 𝝐𝒊’s are independent and identically distributed random variables with mean zero and finite variance. Let and be the ordinary least squares estimators of and , respectively. Given the following observations.
the value of + equals__________(round off to 𝟐 decimal places).
Q.26 – Q.43 Multiple Choice Question (MCQ), carry TWO mark each (for each wrong answer: – 2/3).
Q.26
(A)
𝑓 is not continuous at 0
(B)
𝑓 is not differentiable at 0
(C)
𝑓 is differentiable at 0 and the derivative of 𝑓 at 0 equals 0
(D)
𝑓 is differentiable at 0 and the derivative of 𝑓 at 0 equals 1
Q.27
Let 𝒇: [𝟎, ∞) → ℝ be a function, where ℝ denotes the set of all real numbers. Then which one of the following statements is true?
(A)
If 𝑓 is bounded and continuous, then 𝑓 is uniformly continuous
(B)
If 𝑓 is uniformly continuous, then 𝑥∞ 𝑓(𝑥) exists
(C)
If 𝑓 is uniformly continuous, then the function 𝑔(𝑥) = 𝑓(𝑥) sin 𝑥 is also uniformly continuous
(D)
If 𝑓 is continuous and 𝑥∞ 𝑓(𝑥) is finite, then 𝑓 is uniformly continuous
Q.28
Let 𝒇: ℝ → ℝ be a differentiable function such that 𝒇(𝟎) = 𝟎 and 𝒇′(𝒙) + 𝟐𝒇(𝒙) > 𝟎 for all 𝒙 ∈ ℝ, where 𝒇′denotes the derivative of 𝒇 and ℝ denotes the set of all real numbers. Then which one of the following statements is true?
(A)
𝑓(𝑥) > 0, for all 𝑥 > 0 and 𝑓(𝑥) < 0, for all 𝑥 < 0
(B)
𝑓(𝑥) < 0, for all 𝑥 ≠ 0
(C)
𝑓(𝑥) > 0, for all 𝑥 ≠ 0
(D)
𝑓(𝑥) < 0, for all 𝑥 > 0 and 𝑓(𝑥) > 0, for all 𝑥 < 0
Q.29
Let 𝑴 be the collection of all × 𝟑 real symmetric positive definite matrices. Consider the set
where 𝟎 denotes the 𝟑 × 𝟑 zero matrix. Then the number of elements in𝑺 equals
(A)
0
(B)
1
(C)
8
(D)
∞
Q.30
Let 𝑨 be a 𝟑 × 𝟑 real matrix such that 𝑰𝟑 + 𝑨 is invertible and let 𝑩 = (𝑰𝟑 + 𝑨)−𝟏(𝑰𝟑 − 𝑨), where 𝑰𝟑denotes the 𝟑 × 𝟑 identity matrix. Then which one of the following statements is true?
(A)
If 𝐵 is orthogonal, then 𝐴 is invertible
(B)
If 𝐵 is orthogonal, then all the eigenvalues of 𝐴 are real
(C)
If 𝐵 is skew-symmetric, then 𝐴 is orthogonal
(D)
If 𝐵 is skew-symmetric, then the determinant of 𝐴 equals −1
Q.31
Let 𝑿 be a random variable having Poisson distribution such that 𝑬(𝑿𝟐) = 𝟏𝟏𝟎. Then which one of the following statements is NOT true?
(A)
𝐸(𝑋𝑛) = 10 𝐸[(𝑋 + 1)𝑛−1], for all 𝑛 = 1, 2, 3, …
(B)
𝑃(𝑋 is even) = ¼ (1 + 𝑒−20)
(C)
𝑃(𝑋 = 𝑘) < 𝑃(𝑋 = 𝑘 + 1), for 𝑘 = 0, 1, … , 8
(D)
𝑃(𝑋 = 𝑘) > 𝑃(𝑋 = 𝑘 + 1), for 𝑘 = 10, 11, …
Q.32
Let 𝑿 be a random variable having uniform distribution on –2, 2 Then which one of the following statements is NOT true?
(A)
𝑌 = cot 𝑋 follows standard Cauchy distribution
(B)
𝑌 = tan 𝑋 follows standard Cauchy distribution
(C)
(D)
Q.33
Let 𝛀 = {𝟏, 𝟐, 𝟑, … } represent the collection of all possible outcomes of a random experiment with probabilities 𝑷({𝒏}) = 𝑎𝒏for 𝒏 ∈ 𝛀. Then which one of the following statements is NOT true?
(A)
nan=0
(B)
(C)
(D)
Q.34
Let (𝑿, 𝒀) have the joint probability density function
Then which one of the following statements is NOT true?
(A)
(B)
𝑃(𝑋 + 𝑌 > 4) = ¾
(C)
𝐸(𝑋 + 𝑌) = 4 log𝑒 2
(D)
𝐸(𝑌 | 𝑋 = 2) = 4
Q.35
Let 𝑿𝟏, 𝑿𝟐and 𝑿𝟑be three uncorrelated random variables with common variance 𝝈𝟐< ∞. Let 𝒀𝟏 = 𝟐𝑿𝟏 + 𝑿𝟐 + 𝑿𝟑, 𝒀𝟐 = 𝑿𝟏 + 𝟐𝑿𝟐 + 𝑿𝟑and 𝒀𝟑 = 𝑿𝟏 + 𝑿𝟐 + 𝟐𝑿𝟑. Then which of the following statements is/are true?P : The sum of eigenvalues of the variance covariance matrix of (𝒀𝟏, 𝒀𝟐, 𝒀𝟑) is 𝟏𝟖𝝈𝟐.Q : The correlation coefficient between 𝒀𝟏 𝐚𝐧𝐝 𝒀𝟐equals that between 𝒀𝟐 𝐚𝐧𝐝 𝒀𝟑.
(A)
P only
(B)
Q only
(C)
Both P and Q
(D)
Neither P nor Q
Q.36
Let {𝑿𝒏}≥𝟎be a time-homogeneous discrete time Markov chain with either finite or countable state space 𝑺. Then which one of the following statements is true?
(A)
There is at least one recurrent state
(B)
If there is an absorbing state, then there exists at least one stationary distribution
(C)
If all the states are positive recurrent, then there exists a unique stationary distribution
(D)
If {𝑋𝑛}𝑛≥0 is irreducible, 𝑆 = {1, 2} and [𝜋1 𝜋2] is a stationary distribution, then lim 𝑃(𝑋𝑛 = 𝑖 | 𝑋0 = 𝑖) = 𝜋𝑖 for 𝑖 = 1, 2𝑛→∞
Q.37
Let customers arrive at a departmental store according to a Poisson process with rate 𝟏𝟎. Further, suppose that each arriving customer is either a male or a female with probability ½ each, independent of all other arrivals. Let (𝒕) denote the total number of customers who have arrived by time 𝒕.Then which one of the following statements is NOT true?
(A)
If 𝑆2 denotes the time of arrival of the second female customer, then𝑃(𝑆 ≤ 1) = 25 ∫1 𝑠𝑒−5𝑠𝑑𝑠2 0
(B)
If 𝑀(𝑡) denotes the number of male customers who have arrived by time 𝑡,then 𝑃 (𝑀 (⅓) = 0 | 𝑀(1) = 1) = ⅓
(C)
𝐸 [(𝑁(𝑡))2] = 100𝑡2 + 10𝑡
(D)
𝐸[𝑁(𝑡)𝑁(2𝑡)] = 200𝑡2 + 10𝑡
Q.38
Let 𝑿(𝟏) < 𝑿(𝟐) < 𝑿(𝟑) < 𝑿(𝟒) < 𝑿(𝟓)be the order statistics corresponding to a random sample of size 𝟓 from a uniform distribution on [𝟎, 𝜽], where𝜽 ∈ (𝟎, ∞). Then which of the following statements is/are true?P : 𝟑𝑿(𝟐)is an unbiased estimator of 𝜽.Q : The variance of 𝑬[𝟐𝑿(𝟑) | 𝑿(𝟓)] is less than or equal to the variance of 𝟐𝑿(𝟑).
(A)
P only
(B)
Q only
(C)
Both P and Q
(D)
Neither P nor Q
Q.39
(A)
𝑇/𝑛2
(B)
T/n
(C)
(𝑛 + 1)𝑇/2𝑛
(D)
(𝑛 + 1)2𝑇/4𝑛2
Q.40
Let 𝑿𝟏, 𝑿𝟐, … , 𝑿𝒏be a random sample of size 𝒏 (≥ 𝟐) from a uniform distribution on [−𝜽, 𝜽], where 𝜽 ∈ (𝟎, ∞). Let 𝑿(𝟏) = 𝐦𝐢𝐧{ 𝑿𝟏, 𝑿𝟐, … , 𝑿𝒏} and 𝑿(𝒏) = 𝐦𝐚𝐱{ 𝑿𝟏, 𝑿𝟐, … , 𝑿𝒏}. Then which of the following statementsis/are true?P : (𝑿(𝟏), 𝑿(𝒏)) is a complete statistic. Q : 𝑿(𝒏) − 𝑿(𝟏)is an ancillary statistic.
(A)
P only
(B)
Q only
(C)
Both P and Q
(D)
Neither P nor Q
Q.41
Let {𝑿𝒏}≥𝟏be a sequence of independent and identically distributed random variables having common distribution function 𝑭(⋅). Let 𝒂 < 𝒃 be two real numbers such that 𝑭(𝒙) = 𝟎 for all 𝒙 ≤ 𝒂, 𝟎 < 𝑭(𝒙) < 𝟏 for all𝒂 < 𝒙 < 𝒃 and (𝒙) = 𝟏 for all 𝒙 ≥ 𝒃. Let 𝑺(𝒙) be the empirical distribution function at 𝒙 based on 𝑿𝟏, 𝑿𝟐, … , 𝑿𝒏, 𝒏 ≥ 𝟏. Then which one of the following statements is NOT true?
(A)
(B)
(C)
(D)
Q.42
(A)
(B)
(C)
(D)
Q.43
Let 𝒀 follow 𝑵(𝟎, 𝑰𝟖) distribution, where 𝑰𝟖is the 𝟖 × 𝟖 identity matrix. Let 𝒀𝑻𝚺𝟏𝒀 and 𝒀𝑻𝚺𝟐𝒀 be independent and follow central chi-square distributions with 𝟑 and 𝟒 degrees of freedom, respectively, where 𝚺𝟏and𝚺𝟐are 𝟖 × 𝟖 matrices and 𝒀𝑻denotes transpose of 𝒀. Then which of the following statements is/are true?P : 𝚺𝟏and 𝚺𝟐are idempotent.Q : 𝚺𝟏𝚺𝟐 = 𝟎, where 𝟎 is the 𝟖 × 𝟖 zero matrix.
(A)
P only
(B)
Q only
(C)
Both P and Q
(D)
Neither P nor Q
Q.44 – Q.55 Numerical Answer Type (NAT), carry TWO mark each (no negative marks).
Q.44
Let (𝑿, 𝒀) have a bivariate normal distribution with the joint probability density functionfX,Y(x,y)=1e (3/2xy – 25/32x2 – 2y2), -<x,y<Then 𝟖 𝑬(𝑿𝒀) equals _________.
Q.45
Let 𝒇: ℝ × ℝ → ℝ be defined by (𝒙, 𝒚) = 𝟖𝒙𝟐 − 𝟐𝒚, where ℝ denotes the set of all real numbers. If 𝑴 and 𝒎 denote the maximum and minimum values of 𝒇, respectively, on the set {(𝒙, 𝒚) ∈ ℝ × ℝ ∶ 𝒙𝟐 + 𝒚𝟐 = 𝟏}, then𝑴 − 𝒎 equals_________(round off to 𝟐 decimal places).
Q.46
Let 𝑨 = [𝒂 𝒖𝟏 𝒖𝟐 𝒖𝟑], 𝑩 = [𝒃 𝒖𝟏 𝒖𝟐 𝒖𝟑] and 𝑪 = [𝒖𝟐 𝒖𝟑 𝒖𝟏 𝒂 + 𝒃] be three 𝟒 × 𝟒 real matrices, where 𝒂, 𝒃, 𝒖𝟏, 𝒖𝟐 𝐚𝐧𝐝 𝒖𝟑are 𝟒 × 𝟏 real column vectors. Let 𝐝𝐞(𝑨), 𝐝𝐞𝐭(𝑩) and 𝐝𝐞𝐭(𝑪) denote the determinants of the matrices 𝑨, 𝑩 and 𝑪, respectively. If 𝐝𝐞𝐭(𝑨) = 𝟔 and 𝐝𝐞𝐭(𝑩) = 𝟐, then𝐝𝐞𝐭(𝑨 + 𝑩) − 𝐝𝐞𝐭(𝑪) equals ________
Q.47
Let 𝑿 be a random variable having the moment generating function
Then (𝑿 > 𝟏) equals_____(round off to 𝟐 decimal places).
Q.48
Let {𝑿𝒏}≥𝟏be a sequence of independent and identically distributed random variables each having uniform distribution on [𝟎, 𝟑]. Let 𝒀 be a random variable, independent of {𝑿𝒏}𝒏≥𝟏, having probability mass function
Then 𝑷(𝐦𝐚𝐱{𝑿𝟏, 𝑿𝟐, … , 𝑿𝒀} ≤ 𝟏) equals(round off to decimal places).
Q.49
Let {𝑿𝒏}𝒏≥𝟏be a sequence of independent and identically distributed random variables each having probability density functionLet (𝒏) = 𝐦𝐚𝐱{𝑿𝟏, 𝑿𝟐, … , 𝑿𝒏} for 𝒏 ≥ 𝟏. If 𝒁 is the random variable to which {𝑿(𝒏) − 𝐥𝐨𝐠𝒆 𝒏}𝒏≥𝟏 converges in distribution, as 𝒏 → ∞, then the median of 𝒁 equals ______________(round off to 𝟐 decimal places).
Q.50
Consider an amusement park where visitors are arriving according to a Poisson process with rate 𝟏. Upon arrival, a visitor spends a random amount of time in the park and then departs. The time spent by the visitors are independent of one another, as well as of the arrival process, and have commonprobabilitydensityfunctionIf at a given time point, there are 𝟏𝟎 visitors in the park and 𝒑 is the probability that there will be exactly two more arrivals before the next departure,then 1/p equals ________.
Q.51
Let {𝟎. 𝟗𝟎, 𝟎. 𝟓𝟎, 𝟎. 𝟎𝟏, 𝟎. 𝟗𝟓} be a realization of a random sample of size 4 from the probability density function3
where 𝟎. 𝟓 ≤ 𝜽 < 𝟏. Then the maximum likelihood estimate of 𝜽 based on the observed sample equals__________(round off to 𝟐 decimal places).
Q.52
Let a random sample of size 𝟏𝟎𝟎 from a normal population with unknown mean 𝝁 and variance 𝟗 give the sample mean 𝟓. 𝟔𝟎𝟖. Let (⋅) denote the distribution function of the standard normal random variable. If (𝟏. 𝟗𝟔) = 𝟎. 𝟗𝟕𝟓, 𝚽(𝟏. 𝟔𝟒) = 𝟎. 𝟗𝟓 and the uniformly most powerful unbiased test based on sample mean is used to test 𝑯𝟎: 𝝁 = 𝟓. 𝟎𝟐 against 𝑯𝟏: 𝝁 ≠ 𝟓. 𝟎𝟐, then the 𝒑-value equals____(round off to 𝟑 decimal places).
Q.52
Let a random sample of size 𝟏𝟎𝟎 from a normal population with unknown mean 𝝁 and variance 𝟗 give the sample mean 𝟓. 𝟔𝟎𝟖. Let (⋅) denote the distribution function of the standard normal random variable. If (𝟏. 𝟗𝟔) = 𝟎. 𝟗𝟕𝟓, 𝚽(𝟏. 𝟔𝟒) = 𝟎. 𝟗𝟓 and the uniformly most powerful unbiased test based on sample mean is used to test 𝑯𝟎: 𝝁 = 𝟓. 𝟎𝟐 against 𝑯𝟏: 𝝁 ≠ 𝟓. 𝟎𝟐, then the 𝒑-value equals____(round off to 𝟑 decimal places).
Q.54
Let 𝑿𝟏, 𝑿𝟐, … , 𝑿𝟏𝟎be a random sample from a probability density function𝒇𝜽(𝒙) = 𝒇(𝒙 − 𝜽), −∞ < 𝒙 < ∞,where −∞ < 𝜽 < ∞ and 𝒇(−𝒙) = 𝒇(𝒙) for −∞ < 𝒙 < ∞. For testing 𝑯𝟎: 𝜽 = 𝟏. 𝟐 against 𝑯𝟏: 𝜽 ≠ 𝟏. , let 𝑻+denote the Wilcoxson Signed- rank test statistic. If 𝜼 denotes the probability of the event {𝑻+ < 𝟓𝟎} under 𝑯𝟎, then 𝟑𝟐 𝜼 equals____(round off to 2 decimal places).
Q.55
Consider the multiple linear regression model𝒀𝒊 = 𝖰𝟎 + 𝖰𝟏𝒙𝟏, + 𝖰𝟐𝒙𝟐,𝒊 + ⋯ + 𝖰𝟐𝟐𝒙𝟐𝟐,𝒊 + 𝝐𝒊, 𝒊 = 𝟏, 𝟐, … , 𝟏𝟐𝟑, where, for 𝒋 = 𝟎, 𝟏, 𝟐, … , 𝟐𝟐, 𝖰𝒋’s are unknown parameters and 𝝐𝒊’s are independent and identically distributed 𝑵(𝟎, 𝝈𝟐), 𝝈 > 𝟎,random variables.If the sum of squares due to regression is 𝟑𝟑𝟖. 𝟗𝟐, the total sum of squares is 𝟓𝟐𝟐. 𝟑𝟎 and 𝑹𝟐denotes the value of adjusted 𝑹𝟐, then 𝟏𝟎𝟎 𝑹𝟐equal _____(round off to 𝟐 decimal places).
Q.1
Let (𝒏) = 𝐦𝐚𝐱{𝑿𝟏, 𝑿𝟐, … , 𝑿𝒏} for 𝒏 ≥ 𝟏. If 𝒁 is the random variable to which {𝑿(𝒏) − 𝐥𝐨𝐠𝒆 𝒏}𝒏≥𝟏 converges in distribution, as 𝒏 → ∞, then the median of 𝒁 equals ______________(round off to 𝟐 decimal places).
If at a given time point, there are 𝟏𝟎 visitors in the park and 𝒑 is the probability that there will be exactly two more arrivals before the next departure, then 1/p equals ________.
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