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January 2026 term · Statistics for Data Science II · BSMA1004

Statistics for Data Science II End Term: 10 May 2026 (January 2026 term)

The IIT Madras BS Statistics for Data Science II (Stats 2) End Term paper sat on 10 May 2026, in the January 2026 term: 51 questions for 93 marks in 180 minutes. Every question is below with its answer. Take it as a timed mock test to be marked, or read it through first.

Questions
51
Marks
93
Duration
180 min
MCQ
11
Written
37
MSQ
3
Take as mock testPractise with answers

Updated

Official paper: Statistics2 06 May 26 · No negative marking.

Question 1

+1 markOne correct option
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
Figure from the original question paper
  1. A

    Useful Data has been mentioned above.

  2. B

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

Show answer

Correct answer

  • A

    Useful Data has been mentioned above.

Question 1 on its own page

Question 2

+1 markOne correct option
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
Figure from the original question paper
  1. A

    Useful Data has been mentioned above.

  2. B

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

Show answer

Correct answer

  • A

    Useful Data has been mentioned above.

Question 2 on its own page

Question 3

+3 marksOne correct option

Suppose are i.i.d random variables with E[ ]= and Var( = for . Let

be an estimator of . Find the mean squared error (MSE) of .

Suppose are i.i.d random variables with E[ ]= and Var( = for . Let
  1. A
    Figure from the original question paper
  2. B
    Figure from the original question paper
  3. C
    Figure from the original question paper
  4. D

    —

Show answer

Correct answer

  • A
    Figure from the original question paper
Question 3 on its own page

Question 4

+3 marksWritten answer

Let . The prior distribution of has the following PMF:

Consider the samples S: { }. Find the posterior probability using Bayes' theorem. Enter the answer correct to two decimal places.

Let . The prior distribution of has the following PMF:
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Question 4 on its own page

Question 5

+3 marksOne correct option

The probability density function of a random variable is given as

Find the value of .

The probability density function of a random variable is given as
  1. A
    Figure from the original question paper
  2. B
    Figure from the original question paper
  3. C
    Figure from the original question paper
  4. D
    Figure from the original question paper
Show answer

Correct answer

  • D
    Figure from the original question paper
Question 5 on its own page

Question 6

+3 marksWritten answer

Let be a sample from a normal distribution having a variance of 25. We wish to test the hypothesis against . Consider a test that rejects for , where is the sample mean. Find the value of at a significance level . Enter the answer correct to two decimal places.

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Question 6 on its own page

Question 7

+0.5 marksWritten answer

A pharmaceutical company produces vaccine vials, and historically, the probability that a vial is defective is . A new sterilization method is introduced, and the company wants to test whether this method has changed the defect rate and, specifically, whether it has increased the probability of defective vials. To test this, 250 vials are produced using the new process and inspected. The company will reject the new sterilization method if the total number of defective vials is greater than 7 out of 250. Using the normal approximation to the binomial distribution, find the significance level of the test.

A pharmaceutical company produces vaccine vials, and historically, the probability that a vial is defective is . A new

Enter the correct option number for .

Show answer

A written answer, not marked automatically.

Question 7 on its own page

Question 8

+3 marksWritten answer

The time (in minutes) passengers spend in the airport security line during a quiet period follows a continuous uniform distribution . The waiting times (in minutes) from a random sample of 5 passengers are as follows: Find the method of moments estimate of .

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Question 8 on its own page

Question 9

+0.5 marksWritten answer

A pharmaceutical company produces vaccine vials, and historically, the probability that a vial is defective is . A new sterilization method is introduced, and the company wants to test whether this method has changed the defect rate and, specifically, whether it has increased the probability of defective vials. To test this, 250 vials are produced using the new process and inspected. The company will reject the new sterilization method if the total number of defective vials is greater than 7 out of 250. Using the normal approximation to the binomial distribution, find the significance level of the test.

A pharmaceutical company produces vaccine vials, and historically, the probability that a vial is defective is . A new

Enter the correct option number for .

Show answer

A written answer, not marked automatically.

Question 9 on its own page

Question 10

+3 marksOne or more correct options

Let be a discrete random variable with PMF as follows:

Suppose i.i.d. . Define a new random variable . Which of the following option(s) is (are) true about the moment generating function of the random variable ?

Let be a discrete random variable with PMF as follows:

Select all that apply.

  1. A
    Figure from the original question paper
  2. B
    Figure from the original question paper
  3. C

    —

  4. D

    —

  5. E
    Figure from the original question paper
Show answer

Correct answers

  • B
    Figure from the original question paper
  • C

    —

Question 10 on its own page

Question 11

+0.5 marksWritten answer

A pharmaceutical company produces vaccine vials, and historically, the probability that a vial is defective is . A new sterilization method is introduced, and the company wants to test whether this method has changed the defect rate and, specifically, whether it has increased the probability of defective vials. To test this, 250 vials are produced using the new process and inspected. The company will reject the new sterilization method if the total number of defective vials is greater than 7 out of 250. Using the normal approximation to the binomial distribution, find the significance level of the test.

A pharmaceutical company produces vaccine vials, and historically, the probability that a vial is defective is . A new

Enter the correct option number for .

Show answer

A written answer, not marked automatically.

Question 11 on its own page

Question 12

+3 marksOne or more correct options

Let be i.i.d. samples from and be i.i.d. samples from . The observed sample means are and . We wish to test the hypotheses: against . Consider a test that rejects if Let denote the significance level of the test and denote the cumulative distribution function (CDF) of the standard normal distribution. Express the critical value in terms of and . Select all the option(s) that apply.

Select all that apply.

  1. A

    —

  2. B
    Figure from the original question paper
  3. C

    —

  4. D
    Figure from the original question paper
Show answer

Correct answers

  • B
    Figure from the original question paper
  • D
    Figure from the original question paper
Question 12 on its own page

Question 13

+0.5 marksWritten answer

A pharmaceutical company produces vaccine vials, and historically, the probability that a vial is defective is . A new sterilization method is introduced, and the company wants to test whether this method has changed the defect rate and, specifically, whether it has increased the probability of defective vials. To test this, 250 vials are produced using the new process and inspected. The company will reject the new sterilization method if the total number of defective vials is greater than 7 out of 250. Using the normal approximation to the binomial distribution, find the significance level of the test.

A pharmaceutical company produces vaccine vials, and historically, the probability that a vial is defective is . A new

Enter the correct option number for .

Show answer

A written answer, not marked automatically.

Question 13 on its own page

Question 14

+2 marksOne correct option

Let be a discrete random variable with probability mass function:

Consider the random sample {1, 2, 1, 2}. Assume that the prior distribution of is uniform on Based on the above data, answer the given subquestions.

Let be a discrete random variable with probability mass function:

Which of the following is the correct likelihood function based on the given sample?

  1. A

    —

  2. B

    —

  3. C

    —

  4. D

    —

Show answer

Correct answer

  • D

    —

Question 14 on its own page

Question 15

+1 markWritten answer

A pharmaceutical company produces vaccine vials, and historically, the probability that a vial is defective is . A new sterilization method is introduced, and the company wants to test whether this method has changed the defect rate and, specifically, whether it has increased the probability of defective vials. To test this, 250 vials are produced using the new process and inspected. The company will reject the new sterilization method if the total number of defective vials is greater than 7 out of 250. Using the normal approximation to the binomial distribution, find the significance level of the test.

A pharmaceutical company produces vaccine vials, and historically, the probability that a vial is defective is . A new

Calculate the value for . Enter the answer correct to three decimal place.

Show answer

A written answer, not marked automatically.

Question 15 on its own page

Question 16

+3 marksWritten answer

Let be a discrete random variable with probability mass function:

Consider the random sample {1, 2, 1, 2}. Assume that the prior distribution of is uniform on Based on the above data, answer the given subquestions.

Let be a discrete random variable with probability mass function:

Find the posterior mean of using the given prior distribution. Enter the answer correct to one decimal place.

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Question 16 on its own page

Question 17

+3 marksOne or more correct options

Let be i.i.d. samples from and be i.i.d. samples from . The observed sample means are and . We wish to test the hypotheses: against Consider a test that rejects if Let denote the significance level of the test and denote the cumulative distribution function (CDF) of the standard normal distribution. Express the critical value in terms of and . Select all the option(s) that apply.

Select all that apply.

  1. A

    —

  2. B
    Figure from the original question paper
  3. C

    —

  4. D
    Figure from the original question paper
Show answer

Correct answers

  • B
    Figure from the original question paper
  • D
    Figure from the original question paper
Question 17 on its own page

Question 18

+3 marksWritten answer

Let be a discrete random variable with PMF as follows:

Define another random variable . Based on the above data, answer the given subquestions.

Let be a discrete random variable with PMF as follows:

What is the standard deviation of ? Enter the answer correct to two decimal places.

Show answer

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Question 19

+3 marksWritten answer

Let be a discrete random variable taking values {1, 2, 3} with respective probabilities

, where is a parameter. Consider the samples {1, 2, 2, 1, 3, 2, 1, 3, 2, 3} from . Using a Uniform[0, 1] prior, find the posterior mean of . Enter the answer correct to two decimal places.

Let be a discrete random variable taking values {1, 2, 3} with respective probabilities
Show answer

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Question 19 on its own page

Question 20

+2 marksWritten answer

Let be a discrete random variable with PMF as follows:

Define another random variable . Based on the above data, answer the given subquestions.

Let be a discrete random variable with PMF as follows:

Find Cov .

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Question 21

+2 marksOne correct option

Let be a discrete random variable with probability mass function:

Consider the random sample {2,3,1}. Assume that the prior distribution of is Gamma . Based on the above data, answer the given subquestions.

Let be a discrete random variable with probability mass function:

Which of the following is the correct likelihood function based on the given sample?

  1. A

    —

  2. B

    —

  3. C

    —

  4. D

    —

Show answer

Correct answer

  • B

    —

Question 21 on its own page

Question 22

+2 marksOne correct option

The joint PDF of two random variables and is given by

Based on the above data, answer the given subquestions.

The joint PDF of two random variables and is given by

Find the value of .

  1. A

    3

  2. B
    Figure from the original question paper
  3. C

    6

  4. D
    Figure from the original question paper
Show answer

Correct answer

  • A

    3

Question 22 on its own page

Question 23

+3 marksWritten answer

Let be a discrete random variable with probability mass function:

Consider the random sample {2,3,1}. Assume that the prior distribution of is Gamma . Based on the above data, answer the given subquestions.

Let be a discrete random variable with probability mass function:

Find the posterior mean of using the given prior distribution.

Show answer

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Question 24

+3 marksWritten answer

The joint PDF of two random variables and is given by

Based on the above data, answer the given subquestions.

The joint PDF of two random variables and is given by

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

Show answer

A written answer, not marked automatically.

Question 24 on its own page

Question 25

+3 marksWritten answer

A discrete random variable has the following probability mass function :

Based on the above data, answer the given subquestions.

A discrete random variable has the following probability mass function :

If , then find the value of .

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Question 25 on its own page

Question 26

+2 marksOne correct option

A factory produces light bulbs. The manufacturer claims that the probability that a randomly selected bulb is defective is 0.1. Ravi suspects that the defect rate is actually higher than claimed. To test this, he randomly selects 10 bulbs from the production line. If two or more bulbs are found to be defective, he rejects the manufacturer's claim; otherwise, he accepts it. Based on the above data, answer the given subquestions.

Define the null hypothesis and the alternative hypothesis.

  1. A

    —

  2. B

    —

  3. C

    —

  4. D

    —

Show answer

Correct answer

  • B

    —

Question 26 on its own page

Question 27

+2 marksWritten answer

A discrete random variable has the following probability mass function :

Based on the above data, answer the given subquestions.

A discrete random variable has the following probability mass function :

Define a new random variable . Find the value of . Enter the answer correct to one decimal place.

Show answer

A written answer, not marked automatically.

Question 27 on its own page

Question 28

+3 marksWritten answer

A factory produces light bulbs. The manufacturer claims that the probability that a randomly selected bulb is defective is 0.1. Ravi suspects that the defect rate is actually higher than claimed. To test this, he randomly selects 10 bulbs from the production line. If two or more bulbs are found to be defective, he rejects the manufacturer's claim; otherwise, he accepts it. Based on the above data, answer the given subquestions.

Find the significance level of the test. Enter the answer correct to three decimal places.

Show answer

A written answer, not marked automatically.

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Question 29

+3 marksWritten answer

Let the random variable represent the number of trials required to achieve a certain outcome in an experiment, where {1, 2, 3, 4, 5, 6}. The probability that the outcome occurs in 1 to 5 trials is each, while the probability that it takes 6 trials is . A random sample of five independent observations of is recorded as: 4, 2, 5, 3, 1 Based on the above data, answer the given subquestions.

Find the method of moments estimate of for the given sample. Enter the answer correct to one decimal place.

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Question 30

+2 marksOne correct option

Let be a random variable with mean and variance . Let and be the sample means of two independent random samples of sizes and , respectively, drawn from the distribution of . Consider the estimator where and are constants such that . Is an unbiased estimator of ?

  1. A

    Yes

  2. B

    No

Show answer

Correct answer

  • A

    Yes

Question 30 on its own page

Question 31

+2 marksOne correct option

Let the random variable represent the number of trials required to achieve a certain outcome in an experiment, where {1, 2, 3, 4, 5, 6}. The probability that the outcome occurs in 1 to 5 trials is each, while the probability that it takes 6 trials is . A random sample of five independent observations of is recorded as: 4, 2, 5, 3, 1 Based on the above data, answer the given subquestions.

The experiment is considered efficient if the expected number of trials required to achieve the outcome is at most 4. Using the estimated value of , determine whether this requirement is satisfied.

  1. A

    The requirement is satisfied.

  2. B

    The requirement is not satisfied.

Show answer

Correct answer

  • A

    The requirement is satisfied.

Question 31 on its own page

Question 32

+0.5 marksWritten answer

Let be a random variable with mean and variance . Let and be the sample means of two independent random samples of sizes and , respectively, drawn from the distribution of . Consider the estimator where and are constants such that .

Let be a random variable with mean and variance . Let and be the sample means of two independent random samples of

Enter the correct option number for .

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Question 33

+2 marksOne correct option

A box contains some green balls and some yellow balls. Meera claims that if a ball is picked randomly from the box, then the probability that the ball will be green is 0.25. Arjun suspects that this probability is too high. To test the claim, he randomly picks 8 balls from the box. If two or more green balls are obtained among the 8 balls, then he accepts Meera's claim; otherwise, he rejects it. Based on the above data, answer the given subquestions.

Define the null hypothesis and the alternative hypothesis.

  1. A

    —

  2. B

    —

  3. C

    —

  4. D

    —

Show answer

Correct answer

  • C

    —

Question 33 on its own page

Question 34

+0.5 marksWritten answer

Let be a random variable with mean and variance . Let and be the sample means of two independent random samples of sizes and , respectively, drawn from the distribution of . Consider the estimator where and are constants such that .

Let be a random variable with mean and variance . Let and be the sample means of two independent random samples of

Enter the correct option number for .

Show answer

A written answer, not marked automatically.

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Question 35

+3 marksWritten answer

A box contains some green balls and some yellow balls. Meera claims that if a ball is picked randomly from the box, then the probability that the ball will be green is 0.25. Arjun suspects that this probability is too high. To test the claim, he randomly picks 8 balls from the box. If two or more green balls are obtained among the 8 balls, then he accepts Meera's claim; otherwise, he rejects it. Based on the above data, answer the given subquestions.

Find the significance level of the test. Enter the answer correct to three decimal places.

Show answer

A written answer, not marked automatically.

Question 35 on its own page

Question 36

+0.5 marksWritten answer

Let be a random variable with mean and variance . Let and be the sample means of two independent random samples of sizes and , respectively, drawn from the distribution of . Consider the estimator where and are constants such that .

Let be a random variable with mean and variance . Let and be the sample means of two independent random samples of

Enter the correct option number for .

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Question 37

+2 marksWritten answer

Suppose . Based on the above data, answer the given subquestions.

Find the value of for which . Enter the answer correct to one decimal place.

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Question 38

+0.5 marksWritten answer

Let be a random variable with mean and variance . Let and be the sample means of two independent random samples of sizes and , respectively, drawn from the distribution of . Consider the estimator where and are constants such that .

Let be a random variable with mean and variance . Let and be the sample means of two independent random samples of

Enter the correct option number for .

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Question 39

+3 marksWritten answer

Suppose . Based on the above data, answer the given subquestions.

Find . Enter the answer correct to two decimal places.

Show answer

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Question 40

+1 markWritten answer

Let be a random variable with mean and variance . Let and be the sample means of two independent random samples of sizes and , respectively, drawn from the distribution of . Consider the estimator where and are constants such that .

Let be a random variable with mean and variance . Let and be the sample means of two independent random samples of

Calculate the value for . Enter the answer correct to two decimal places.

Show answer

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Question 41

+1 markWritten answer

A pharmaceutical company produces vaccine vials, and historically, the probability that a vial is defective is . A new sterilization method is introduced, and the company wants to test whether this method has changed the defect rate and, specifically, whether it has increased the probability of defective vials. To test this, 250 vials are produced using the new process and inspected. The company will reject the new sterilization method if the total number of defective vials is greater than 7 out of 250. Using the normal approximation to the binomial distribution, find the significance level of the test.

A pharmaceutical company produces vaccine vials, and historically, the probability that a vial is defective is . A new
Show answer

A written answer, not marked automatically.

Question 41 on its own page

Question 42

+1 markWritten answer

Let be a discrete random variable with probability mass function:

Consider the random sample {2,3,1}. Assume that the prior distribution of is Gamma . Based on the above data, answer the given subquestions.

Let be a discrete random variable with probability mass function:
Show answer

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Question 43

+1 markWritten answer

A discrete random variable has the following probability mass function :

Based on the above data, answer the given subquestions.

A discrete random variable has the following probability mass function :
Show answer

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Question 44

+1 markWritten answer

Let the random variable represent the number of trials required to achieve a certain outcome in an experiment, where {1, 2, 3, 4, 5, 6}. The probability that the outcome occurs in 1 to 5 trials is each, while the probability that it takes 6 trials is . A random sample of five independent observations of is recorded as: 4, 2, 5, 3, 1 Based on the above data, answer the given subquestions.

Show answer

A written answer, not marked automatically.

Question 44 on its own page

Question 45

+1 markWritten answer

A box contains some green balls and some yellow balls. Meera claims that if a ball is picked randomly from the box, then the probability that the ball will be green is 0.25. Arjun suspects that this probability is too high. To test the claim, he randomly picks 8 balls from the box. If two or more green balls are obtained among the 8 balls, then he accepts Meera's claim; otherwise, he rejects it. Based on the above data, answer the given subquestions.

Show answer

A written answer, not marked automatically.

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Question 46

+1 markWritten answer

Suppose . Based on the above data, answer the given subquestions.

Show answer

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Question 47

+1 markWritten answer

Let be a discrete random variable with probability mass function:

Consider the random sample {1, 2, 1, 2}. Assume that the prior distribution of is uniform on Based on the above data, answer the given subquestions.

Let be a discrete random variable with probability mass function:
Show answer

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Question 48

+1 markWritten answer

Let be a discrete random variable with PMF as follows:

Define another random variable . Based on the above data, answer the given subquestions.

Let be a discrete random variable with PMF as follows:
Show answer

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Question 49

+1 markWritten answer

The joint PDF of two random variables and is given by

Based on the above data, answer the given subquestions.

The joint PDF of two random variables and is given by
Show answer

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Question 50

+1 markWritten answer

A factory produces light bulbs. The manufacturer claims that the probability that a randomly selected bulb is defective is 0.1. Ravi suspects that the defect rate is actually higher than claimed. To test this, he randomly selects 10 bulbs from the production line. If two or more bulbs are found to be defective, he rejects the manufacturer's claim; otherwise, he accepts it. Based on the above data, answer the given subquestions.

Show answer

A written answer, not marked automatically.

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Question 51

+1 markWritten answer

Let be a random variable with mean and variance . Let and be the sample means of two independent random samples of sizes and , respectively, drawn from the distribution of . Consider the estimator where and are constants such that .

Let be a random variable with mean and variance . Let and be the sample means of two independent random samples of
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