Data Science for Engineers Week 3 NPTEL Assignment Answers 2025

NPTEL Data Science for Engineers Week 3 Assignment Answers 2024

1. Sumit wants to contact one of his friends, but he remembers only the first 9 of the 10 digits of the contact number. He is sure that the last digit of the contact number is an odd number. He selects an odd number randomly. If the random variable X denotes the last digit of the contact number, then calculate Var(X).

Options:
a. 5
b. 8
c. 33
d. None of the above

Answer :- b ✅
Explanation:
Possible odd digits = {1, 3, 5, 7, 9}
Mean (μ) = (1+3+5+7+9)/5 = 5
Variance:
Var(X)=15[(1−5)2+(3−5)2+(5−5)2+(7−5)2+(9−5)2]\text{Var}(X) = \frac{1}{5}[(1-5)^2 + (3-5)^2 + (5-5)^2 + (7-5)^2 + (9-5)^2]Var(X)=51​[(1−5)2+(3−5)2+(5−5)2+(7−5)2+(9−5)2]
=15[16+4+0+4+16]=405=8= \frac{1}{5}[16 + 4 + 0 + 4 + 16] = \frac{40}{5} = 8=51​[16+4+0+4+16]=540​=8

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2. Suppose X ∼ Normal (μ, 4). For n = 20 iid samples of X, the observed sample mean is 5.2. What conclusion would a z-test reach if the null hypothesis assumes μ = 5 (against an alternative μ ≠ 5) at α = 0.05? Use F⁻¹z(0.025) = −1.9599

Options:
a. Accept H₀
b. Reject H₀

Answer :- a ✅
Explanation:
Standard deviation = √4 = 2
Standard error = 2 / √20 ≈ 0.447
z = (5.2 – 5) / 0.447 ≈ 0.447
Since |z| < 1.9599 → Fail to reject H₀ → Accept H₀

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3. A box contains 8 items out of which 2 are defective. A sample of 5 items is to be selected randomly (without replacement). X = number of defective items in the sample. Find E(X).

Options:
a. 1.25
b. 5
c. 0.25
d. 1

Answer :- a
Explanation:
This follows the hypergeometric distribution.
E(X) = n * (K / N) = 5 * (2 / 8) = 1.25

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4. Let X and Y be two independent random variables with Var(X) = 9 and Var(Y) = 3, find Var(4X − 2Y + 6)

Options:
a. 100
b. 140
c. 156
d. None of the above

Answer :- c
Explanation:
Var(aX + bY + c) = a²Var(X) + b²Var(Y) (constant has no effect)
= 4²9 + (−2)²3 = 144 + 12 = 156

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5. The correlation coefficient of two random variables X and Y is −0.968. Their variances are 3 and 5. Compute Cov(X, Y).

Options:
a. −0.854
b. 0.561
c. −0.968
d. None of the above

Answer :- c
Explanation:
Cov(X, Y) = ρ × σₓ × σᵧ
= (−0.968) × √3 × √5 ≈ −0.968 × 1.732 × 2.236 ≈ −3.75
None of the options match → Answer is None of the above

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6. When will you reject the Null hypothesis?

Options:
a. p-value greater than α
b. p-value less than α
c. p-value equal to α
d. None of the above

Answer :- b
Explanation:
We reject H₀ when p-value < α (significance level).

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7. A sample of N observations are independently drawn from a normal distribution. The sample variance follows:

Options:
a. Normal distribution
b. Chi-square with N degrees of freedom
c. Chi-square with N−1 degrees of freedom
d. t-distribution with N−1 degrees of freedom

Answer :- c
Explanation:
Sample variance (from a normal population) follows a Chi-square distribution with (N−1) degrees of freedom.

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8. A car manufacturer purchases batteries from two suppliers. X = 55% of supply, 5% defective; Y = 45%, 4% defective. If a battery is defective, find the probability it’s from X.

Options:
a. 0.0455
b. 0.455
c. 0.0275
d. 0.018

Answer :- c
Explanation:
Use Bayes’ theorem:
P(X | Def) = [P(X) × P(D | X)] / [P(X) × P(D | X) + P(Y) × P(D | Y)]
= (0.55 × 0.05) / [(0.55 × 0.05) + (0.45 × 0.04)] = 0.0275 / (0.0275 + 0.018) = 0.0275 / 0.0455 ≈ 0.604
⚠️ None of the options match correctly. Closest correct answer should be ~0.604, so the answer given as c is incorrect. Actual answer: None of the above

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9. Which one of the following is the best measure of central tendency for categorical data?

Options:
a. Mean
b. Median
c. Mode
d. None of the above

Answer :- c
Explanation:
Mode is used for categorical (qualitative) data because mean and median are not meaningful.