Figure from the original question paper Figure from the original question paper Figure from the original question paper Figure from the original question paper Figure from the original question paper

(A) Useful Data has been mentioned above.

(B) This data attachment is just for a reference & not for an evaluation.

Stats 2 End Term PYQ May 2025: 14 Sept 2025 (Set 2) · IITM BS

Question 1

+1 markOne correct option

A
Useful Data has been mentioned above.
B
This data attachment is just for a reference & not for an evaluation.

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Correct answer

A
Useful Data has been mentioned above.

Question 1 on its own page

Question 2

+3 marksWritten answer

Let be i.i.d. Exp . Find the value of . Enter the answer correct to two decimal places.

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Question 3

+3 marksOne correct option

Let be i.i.d. samples from with unknown parameters and . Consider an estimator of defined as

If is an unbiased estimator of , then find the value of and Risk( ) .

A
B
C
D

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Correct answer

A

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Question 4

+0.5 marksWritten answer

Setup : A machine is set to fill packets with an average of 500g of flour with a known standard deviation of 20g. Let the null and alternative hypotheses be and . To verify this, a quality team selects a random sample of 64 packets and defined the critical region as . Find the power of the test against the alternative . (See the "Useful data" given in question number 2 for values. ) Solution: Since the population standard deviation is known, the sampling distribution of under the null hypothesis will be distribution with mean and variance . To find the power of the test, we will solve: Consider the following options for the blanks.

Answer the given sub-questions. Note: Enter only the correct option number (e.g. 8) without any brackets or punctuation.

Enter the correct option number for .

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Question 4 on its own page

Question 5

+0.5 marksWritten answer

Setup : A machine is set to fill packets with an average of 500g of flour with a known standard deviation of 20g. Let the null and alternative hypotheses be and . To verify this, a quality team selects a random sample of 64 packets and defined the critical region as . Find the power of the test against the alternative . (See the "Useful data" given in question number 2 for values. ) Solution: Since the population standard deviation is known, the sampling distribution of under the null hypothesis will be distribution with mean and variance . To find the power of the test, we will solve: Consider the following options for the blanks.

Answer the given sub-questions. Note: Enter only the correct option number (e.g. 8) without any brackets or punctuation.

Enter the correct option number for .

Show answer

A written answer, not marked automatically.

Question 5 on its own page

Question 6

+0.5 marksWritten answer

Setup : A machine is set to fill packets with an average of 500g of flour with a known standard deviation of 20g. Let the null and alternative hypotheses be and . To verify this, a quality team selects a random sample of 64 packets and defined the critical region as . Find the power of the test against the alternative . (See the "Useful data" given in question number 2 for values. ) Solution: Since the population standard deviation is known, the sampling distribution of under the null hypothesis will be distribution with mean and variance . To find the power of the test, we will solve: Consider the following options for the blanks.

Answer the given sub-questions. Note: Enter only the correct option number (e.g. 8) without any brackets or punctuation.

Enter the correct option number for .

Show answer

A written answer, not marked automatically.

Question 6 on its own page

Question 7

+0.5 marksWritten answer

Setup : A machine is set to fill packets with an average of 500g of flour with a known standard deviation of 20g. Let the null and alternative hypotheses be and . To verify this, a quality team selects a random sample of 64 packets and defined the critical region as . Find the power of the test against the alternative . (See the "Useful data" given in question number 2 for values. ) Solution: Since the population standard deviation is known, the sampling distribution of under the null hypothesis will be distribution with mean and variance . To find the power of the test, we will solve: Consider the following options for the blanks.

Answer the given sub-questions. Note: Enter only the correct option number (e.g. 8) without any brackets or punctuation.

Enter the correct option number for .

Show answer

A written answer, not marked automatically.

Question 7 on its own page

Question 8

+0.5 marksWritten answer

Setup : A machine is set to fill packets with an average of 500g of flour with a known standard deviation of 20g. Let the null and alternative hypotheses be and . To verify this, a quality team selects a random sample of 64 packets and defined the critical region as . Find the power of the test against the alternative . (See the "Useful data" given in question number 2 for values. ) Solution: Since the population standard deviation is known, the sampling distribution of under the null hypothesis will be distribution with mean and variance . To find the power of the test, we will solve: Consider the following options for the blanks.

Answer the given sub-questions. Note: Enter only the correct option number (e.g. 8) without any brackets or punctuation.

Enter the correct option number for .

Show answer

A written answer, not marked automatically.

Question 8 on its own page

Question 9

+0.5 marksWritten answer

Setup : A machine is set to fill packets with an average of 500g of flour with a known standard deviation of 20g. Let the null and alternative hypotheses be and . To verify this, a quality team selects a random sample of 64 packets and defined the critical region as . Find the power of the test against the alternative . (See the "Useful data" given in question number 2 for values. ) Solution: Since the population standard deviation is known, the sampling distribution of under the null hypothesis will be distribution with mean and variance . To find the power of the test, we will solve: Consider the following options for the blanks.

Answer the given sub-questions. Note: Enter only the correct option number (e.g. 8) without any brackets or punctuation.

Enter the correct option number for .

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A written answer, not marked automatically.

Question 9 on its own page

Question 10

+3 marksOne correct option

The number of complaints per hour at a call center is modeled by a Poisson distribution. The observed number of complaints over different periods of five hour shifts are 3, 2, 4, 1 and 2. Assume the prior distribution of to be Gamma( ). Find the posterior distribution of .

A
Gamma(16,5)
B
Beta(16, 6)
C
Gamma(6,16)
D
Gamma(16,6)

Show answer

Correct answer

D
Gamma(16,6)

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Question 11

+3 marksWritten answer

Suppose Normal( with an unknown variance . For = 25 i.i.d. samples of , the observed sample mean is 52.4, and the sample standard deviation is 4. Let the null and alternative hypotheses be and . Consider a test that rejects if for some constant at 5% level of significance. Find the value of . Enter the answer correct to two decimal places.

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Question 12

+2 marksWritten answer

Let be the number of questions asked by a randomly selected student during a live session. The probability mass function of is given by

Based on the above data, answer the given subquestions.

Given that a student had asked at least one question, what is the probability that the student asked exactly 2 questions? Enter the answer correct to two decimal places.

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Question 13

+3 marksWritten answer

Let be the number of questions asked by a randomly selected student during a live session. The probability mass function of is given by

Based on the above data, answer the given subquestions.

Define a new random variable . Find the value of .

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Question 14

+3 marksWritten answer

Let Bernoulli(0.6) and Bernoulli(0.3) be independent random variables. Define a new random variable . Based on the above data, answer the given subquestions.

Find the value of . Enter the answer correct to two decimal places.

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Question 15

+2 marksOne correct option

Let Bernoulli(0.6) and Bernoulli(0.3) be independent random variables. Define a new random variable . Based on the above data, answer the given subquestions.

Find the moment generating function (MGF) of .

A
—
B
—
C
—
D
—

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Correct answer

D
—

Question 15 on its own page

Question 16

+2 marksOne correct option

Let be a discrete random variable such that {0, 1} with where is an unknown parameter. Consider a random sample (0, 1, 1, 0, 1) from . Based on the above data, answer the given subquestions.

Find the value of method of moment estimate of .

A
B
C
D

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Correct answer

B

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Question 17

+3 marksWritten answer

Let be a discrete random variable such that {0, 1} with where is an unknown parameter. Consider a random sample (0, 1, 1, 0, 1) from . Based on the above data, answer the given subquestions.

Find the difference between the value of the method of moment estimate and the maximum likelihood estimate of .

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Question 18

+3 marksOne correct option

A student is attempting to solve logic puzzles on an online platform. The number of attempts required to solve a puzzle for the first time is modeled as a Geometric distribution. The student's attempts observed for solving 4 different puzzles are 2, 3, 1, and 2. Assume prior distribution of to be Beta . Based on the above data, answer the given subquestions.

Find the posterior distribution of .

A
Gamma
B
Beta
C
Beta
D
Beta

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Correct answer

B
Beta

Question 18 on its own page

Question 19

+2 marksWritten answer

A student is attempting to solve logic puzzles on an online platform. The number of attempts required to solve a puzzle for the first time is modeled as a Geometric distribution. The student's attempts observed for solving 4 different puzzles are 2, 3, 1, and 2. Assume prior distribution of to be Beta . Based on the above data, answer the given subquestions.

Find the posterior mean. Enter the answer correct to two decimal places.

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Question 20

+2 marksOne correct option

A pharmaceutical company claims that its new drug reduces systolic blood pressure by an average of 10 mmHg, with a known standard deviation of 2 mmHg. A medical researcher believes that the drug may be less effective than claimed. To test this claim, she conducts a study on a random sample of 64 patients and measures the reduction in systolic blood pressure after using the drug for 2 weeks. The researcher decides to reject the company's claim if sample mean and accept it otherwise. Based on the above data, answer the given subquestions.

Define null and alternative hypotheses.

A
—
B
—
C
—
D
—

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Correct answer

B
—

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Question 21

+3 marksWritten answer

A pharmaceutical company claims that its new drug reduces systolic blood pressure by an average of 10 mmHg, with a known standard deviation of 2 mmHg. A medical researcher believes that the drug may be less effective than claimed. To test this claim, she conducts a study on a random sample of 64 patients and measures the reduction in systolic blood pressure after using the drug for 2 weeks. The researcher decides to reject the company's claim if sample mean and accept it otherwise. Based on the above data, answer the given subquestions.

Find the power of the test for . Enter the answer correct to two decimal places.

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Question 22

+1 markWritten answer

Setup : A machine is set to fill packets with an average of 500g of flour with a known standard deviation of 20g. Let the null and alternative hypotheses be and . To verify this, a quality team selects a random sample of 64 packets and defined the critical region as . Find the power of the test against the alternative . (See the "Useful data" given in question number 2 for values. ) Solution: Since the population standard deviation is known, the sampling distribution of under the null hypothesis will be distribution with mean and variance . To find the power of the test, we will solve: Consider the following options for the blanks.

Answer the given sub-questions. Note: Enter only the correct option number (e.g. 8) without any brackets or punctuation.

Show answer

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Question 22 on its own page

Question 23

+1 markWritten answer

Let be the number of questions asked by a randomly selected student during a live session. The probability mass function of is given by

Based on the above data, answer the given subquestions.

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Question 24

+1 markWritten answer

Let Bernoulli(0.6) and Bernoulli(0.3) be independent random variables. Define a new random variable . Based on the above data, answer the given subquestions.

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Question 25

+1 markWritten answer

Let be a discrete random variable such that {0, 1} with where is an unknown parameter. Consider a random sample (0, 1, 1, 0, 1) from . Based on the above data, answer the given subquestions.

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Question 26

+1 markWritten answer

A student is attempting to solve logic puzzles on an online platform. The number of attempts required to solve a puzzle for the first time is modeled as a Geometric distribution. The student's attempts observed for solving 4 different puzzles are 2, 3, 1, and 2. Assume prior distribution of to be Beta . Based on the above data, answer the given subquestions.

Show answer

A written answer, not marked automatically.

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Question 27

+1 markWritten answer

A pharmaceutical company claims that its new drug reduces systolic blood pressure by an average of 10 mmHg, with a known standard deviation of 2 mmHg. A medical researcher believes that the drug may be less effective than claimed. To test this claim, she conducts a study on a random sample of 64 patients and measures the reduction in systolic blood pressure after using the drug for 2 weeks. The researcher decides to reject the company's claim if sample mean and accept it otherwise. Based on the above data, answer the given subquestions.

Show answer

A written answer, not marked automatically.

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