Model Question Paper and Answers KTU GNEST305 AI and Data Science

 

Q1. List the types of machine learning systems with one example each. (CO1)

1. Supervised Learning – The model is trained on labeled data (input–output pairs) to predict outcomes.
   Example: Spam email detection (spam or not spam).

2. Unsupervised Learning – The model finds hidden patterns or groupings in unlabeled data.
   Example: Customer segmentation in marketing.

3. Reinforcement Learning – The model learns by interacting with an environment and receiving rewards or penalties.
   Example: Self-driving cars improving navigation by trial and error.

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Q2. What is the difference between classification and regression? (CO1)

• Classification: Predicts categorical or discrete outcomes. The output is a label or class.
  Example: Predicting whether a patient has a disease (Yes/No).

• Regression: Predicts continuous numerical outcomes. The output is a real value.
  Example: Predicting house prices or stock values.

Key point: Classification → categories; Regression → numbers.

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Q3. What is the importance of linear algebra in data representation? (CO2)

Linear algebra provides the foundation for representing and processing data in machine learning:

• Data is stored in vectors and matrices for efficient manipulation.

• Algorithms like PCA, SVD, and neural networks rely on matrix operations.

• It enables compact representation of large datasets and fast computations.

Thus, linear algebra is essential for data transformations, feature extraction, and optimization in ML.

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Q4. Define Singular Value Decomposition (SVD) with its component matrix and its relevance with reference to the rows and columns of the data matrix. (CO2)

SVD decomposes a matrix A into three matrices:

$$A = U \, \Sigma \, V^T$$

• U → left singular vectors (represent row features).

• Σ → diagonal matrix of singular values (importance of components).

• Vᵀ → right singular vectors (represent column features).

Relevance:

• Captures relationships among rows and columns.

• Reduces dimensionality by keeping only top singular values.

• Widely used in data compression, noise reduction, and recommendation systems.

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Q5. Define a random variable with an example. (CO3)

A random variable is a function that assigns numerical values to outcomes of a random process or experiment.

• Discrete random variable: Takes countable values.
  Example: Rolling a die → values {1, 2, 3, 4, 5, 6}.

• Continuous random variable: Takes infinite possible values within a range.
  Example: Measuring height of students.

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Q6. Differentiate between correlation and regression in terms of their applicability in statistical modelling. (CO3)

• Correlation:

  • Measures the strength and direction of association between two variables.

  • Does not imply causation.

  • Example: Height and weight correlation.

• Regression:

  • Establishes a functional or predictive relationship between variables.

  • One variable is dependent, the other(s) independent.

  • Example: Predicting sales based on advertising expenditure.

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Q7. What are the main benefits of data science in modern industries? (CO4)

• Improved decision-making: Data-driven insights help managers choose better strategies.

• Efficiency and automation: AI/ML models automate repetitive tasks.

• Personalization: Enhances customer experience (e.g., product recommendations).

• Innovation: Helps in discovering new business opportunities and products.

Thus, data science boosts competitiveness and growth in industries.

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Q8. What is meant by Big Data, and how is it related to data science? (CO4)

• Big Data: Refers to massive datasets characterized by the 3Vs:

  • Volume (large size),

  • Velocity (speed of generation),

  • Variety (different formats like text, images, videos).

• Relation to Data Science:
  Data science provides methods and algorithms to process, analyze, and extract insights from Big Data, which cannot be handled by traditional tools.

            PART B

Q9 (a). In binary classification, a model outputs a probability of p = 0.8. If the decision boundary is 0.5, state the predicted class. Justify your reasoning. (CO1)

• Since p = 0.8 > 0.5, the model predicts Class 1.

• Reason: If probability ≥ 0.5, the instance is classified as positive (Class 1); otherwise, negative (Class 0).

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Q9 (b). Apply one step of K-means clustering to the dataset {2, 4, 8, 10} with k = 2 and initial centroids 2 and 8. Show new clusters. (CO1)

• Initial centroids: C1 = 2, C2 = 8

• Distance from data points:

  • 2 → C1(0), C2(6) → Cluster 1

  • 4 → C1(2), C2(4) → Cluster 1

  • 8 → C1(6), C2(0) → Cluster 2

  • 10 → C1(8), C2(2) → Cluster 2

• New clusters:

  • Cluster 1 = {2, 4}

  • Cluster 2 = {8, 10}

• New centroids:

  • C1 = (2+4)/2 = 3

  • C2 = (8+10)/2 = 9

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Q9 (c). A dataset has 100 samples. A classification model misclassifies 12 samples. Compute its accuracy. (CO1)

• Total samples = 100

• Correct predictions = 100 – 12 = 88

• Accuracy = Correct / Total = 88 / 100 = 0.88 = 88%

Q10 (a). For input x = [1, 1], weights w = [0.5, 0.5], and bias b = 0, compute the perceptron output before applying activation. (CO1)

• Formula: $z = w \cdot x + b$

• Calculation: $z = (0.5 \times 1) + (0.5 \times 1) + 0 = 0.5 + 0.5 = 1.0$

• Output before activation = 1.0

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Q10 (b). Explain with an example how a Multi-Layer Perceptron (MLP) can solve a problem that a single perceptron cannot. (CO1)

• A single perceptron can only solve linearly separable problems.

• XOR problem is not linearly separable → cannot be solved by one perceptron.

• An MLP with hidden layers applies nonlinear transformations, making it possible to classify XOR correctly.

• Example: MLP with two hidden neurons can separate XOR points.

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Q10 (c). Logistic regression is used to predict whether a patient has a disease (Yes/No). Explain how the probability output can be used for decision-making. (CO1)

• Logistic regression outputs a probability between 0 and 1.

• A decision threshold (e.g., 0.5) is set:

  • If probability ≥ 0.5 → predict “Disease: Yes”.

  • If probability < 0.5 → predict “Disease: No”.

• This allows doctors to make binary decisions based on risk level.

Q11 (a). Find the eigenvalues of $A=\left(\begin{array}{cc}2& 0\\ 0& 3\end{array}\right)\mathrm{.<span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mord">.</span></span></span></span>\; (CO2)}$

Matrix is diagonal, so its eigenvalues are the diagonal entries: ${\lambda }_1=2,{\lambda }_2=\mathrm{3<strong\; data-end="292"\; data-start="253">.</strong>}$

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Q11 (b). Apply spectral decomposition to the symmetric matrix $A=\begin{pmatrix}1 & 2\\[4pt]2 & 4\end{pmatrix}.$ (CO2)

1. Compute eigenvalues: characteristic equation gives $\lambda (5-\lambda )=\mathrm{0\; \Rightarrow }$$\lambda_1=0,\; \lambda_2=5$
   

2. Corresponding eigenvectors (unnormalized): for $\lambda=0$: $v_1=\left(\begin{array}{c}-2\\ 1\end{array}\right)$ for $\lambda=5$: $v_2 = \begin{pmatrix}1\\[4pt]2\end{pmatrix}$
   

3. Normalize (both have norm $\sqrt{5}$
   $u_1=\frac{1}{\sqrt{5}}\left(\begin{array}{c}-2\\ 1\end{array}\right),u_2=\frac{1}{\sqrt{5}}\left(\begin{array}{c}1\\ 2\end{array}\right)\mathrm{.}$

4. Spectral decomposition: $A = Q \Lambda Q^{T}$ where

$$Q=\begin{pmatrix}u_1 & u_2\end{pmatrix} =\frac{1}{\sqrt{5}}\begin{pmatrix}-2 & 1\\[4pt]1 & 2\end{pmatrix},\quad \Lambda=\begin{pmatrix}0 & 0\\[4pt]0 & 5\end{pmatrix}.$$

Thus $A = Q\Lambda Q^{T}$. (This expresses $A$ as sum of rank-1 components: $A=0\cdot u_1u_1^{T}+5\cdot u_2u_2^{T}$

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Q11 (c). For the covariance matrix $A=\begin{pmatrix}4 & 2\\[4pt]2 & 3\end{pmatrix}$ will the eigenvectors be orthogonal? (CO2)

Yes. $A$ is symmetric, and symmetric (real) matrices always have eigenvectors that can be chosen orthogonal. (So the covariance matrix’s eigenvectors are orthogonal; this is why PCA uses them.)

Q12 (a). A dataset has 100 features. After applying PCA, only 10 features are retained. What does this imply about dimensionality reduction and why might it be useful in Machine Learning? (CO2)

Implication: The top 10 principal components capture the majority of the data variance (most information) while the remaining 90 components contribute little.
Why useful: reduces feature dimensionality → less storage and faster training, lowers risk of overfitting, removes noise/redundant features, and simplifies visualization and interpretation.

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Q12 (b). A noisy image matrix is approximated using only the first two largest singular values in its SVD. Explain computationally why this reduces storage and how it helps in image compression. (CO2)

• SVD form: for an $m\times n$ image $A$, full SVD stores $U_{m\times m},\ \Sigma_{m\times n},\ V^T_{n\times n}$.

• k-rank approximation (k = 2): store only $U_{m\times 2},\ \Sigma_{2\times 2}$(or its 2 singular values), and $V^T_{2\times n}$.

• Storage cost: roughly $m\cdot2 + 2 + 2\cdot n = 2(m+n)+1$ numbers instead of $m\cdot n$. For large images this is a big reduction when $2\ll \min (m,n)$

• Compression & denoising: keeping only largest singular values preserves the main image structure (high energy) and discards small singular values that often correspond to noise—so the image is compressed and less noisy.

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Q12 (c). For the matrix $A=\begin{pmatrix}0 & 1\\[4pt]1 & 0\end{pmatrix}$, find $A^T A$ and compute singular values of $A$. (CO2)

• $A^T = A$ (matrix is symmetric).

• $A^T A = A A = \begin{pmatrix}0 & 1\\[4pt]1 & 0\end{pmatrix}\begin{pmatrix}0 & 1\\[4pt]1 & 0\end{pmatrix} = \begin{pmatrix}1 & 0\\[4pt]0 & 1\end{pmatrix} = I.$
  

• Eigenvalues of $A^T A$ are $\{1,1\}$.

• Singular values = square roots of these eigenvalues → ${\sigma }_1=1,{\sigma }_2=\mathrm{1<strong\; data-end="2022"\; data-start="1985"><br></strong>}$

Q13 (a). A coin is tossed 3 times. List outcomes and probability distribution for at least two heads. (CO3)

• Sample space (8 outcomes): {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

• Event $E$: at least 2 heads = {HHH, HHT, HTH, THH}

  • P(E) = 4/8 = 0.5

• Probability distribution (X = no. of heads):

• $P(X=2) = 3/8$, $P(X=3) = 1/8$
  

• So total $P(X\ge 2)=3\mathrm{/}8+1\mathrm{/}8=0.5$

Q13 (b).A bag contains 3 red and 2 blue balls. One ball is drawn at random. Find the probability that it is
(i) red
(ii) blue.

• Total balls = 3 red + 2 blue = 5

(i) Probability of red = $\dfrac{3}{5} = 0.6$
(ii) Probability of blue = $\dfrac{2}{5} = 0.4$

✅ Final Answer: P(Red) = 0.6, P(Blue) = 0.4

Q13(c).A factory has 3 machines A, B, C producing 30%, 50%, and 20% of items,respectively. Their defect rates are 2%, 3%, and 4%. If a randomly chosen

item is defective, find the probability it was produced by machine B.

Factory machines problem (Bayes’ theorem). (CO3)

• Machine contributions: A = 30%, B = 50%, C = 20%

• Defect rates: A = 2%, B = 3%, C = 4%

Step 1. Total defective probability:

$$P(D) = (0.3)(0.02) + (0.5)(0.03) + (0.2)(0.04) = 0.006 + 0.015 + 0.008 = 0.029$$

Step 2. Probability defective from B:

$$P(B|D) = \frac{P(B)P(D|B)}{P(D)} = \frac{0.5 \times 0.03}{0.029} = \frac{0.015}{0.029} \approx 0.517$$

Answer: Probability ≈ 51.7%.

Q14(a)The marks of 5 students are: 50, 60, 70, 80, 90. Compute the mean and variance. Use these parameters to interpret the data distribution. (CO3)

Answer (a)

• Mean: $\bar{x}=\dfrac{50+60+70+80+90}{5}=\dfrac{350}{5}=70.$
  

• Deviations from mean: $-20,-10,0,10,20.$ Squared deviations sum = $400+100+0+100+400=1000.$

  • Population variance: ${\sigma }^2={\displaystyle \frac{1000}{5}}=200.$

  • Sample variance (unbiased): $s^{2}=\dfrac{1000}{4}=250.$
    

• Interpretation: Mean 70 is the center. Variance 200 (std ≈ 14.14) shows moderate spread around the mean. Data are symmetric and evenly spaced about 70 (no skew), so distribution is centered and fairly dispersed.

Q14(b).The data for $X=\{1, 2, 3, 4, 5\},\ Y=\{2, 4, 6, 8, 10\}$. Compute the correlation coefficient. What does the result indicate?

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Step 1: Recall formula

The Pearson correlation coefficient $r$ is:

$$r = \frac{\text{Cov}(X,Y)}{\sigma_X \cdot \sigma_Y}$$

where

• $\text{Cov}(X,Y) = \frac{1}{n}\sum_{i=1}^n (x_i-\bar X)(y_i-\bar Y)$
  

• ${\sigma }_X=\sqrt{\frac{1}{n}\sum (x_i-\bar{X}{)}^{\mathrm{2}}}$

• $\sigma_Y = \sqrt{\frac{1}{n}\sum (y_i-\bar Y)^2}$

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Step 2: Compute means

$$\bar X = \frac{1+2+3+4+5}{5} = \frac{15}{5} = 3$$
$$\bar Y = \frac{2+4+6+8+10}{5} = \frac{30}{5} = 6$$

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Step 3: Compute covariance

$$\text{Cov}(X,Y) = \frac{1}{5}\sum (x_i-\bar X)(y_i-\bar Y)$$

Make a table:

$x_i$	$y_i$	$x_i-\bar X$	$y_i-\bar Y$	($$xi−Xˉ)(yi−Yˉ)(x_i-\bar X)(y_i-\bar Y)
1	2	-2	-4	8
2	4	-1	-2	2
3	6	0	0	0
4	8	+1	+2	2
5	10	+2	+4	8

Sum = 20

$$\text{Cov}(X,Y) = \frac{20}{5} = 4$$

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Step 4: Compute standard deviations

For $X$:

$$\sigma_X^2 = \frac{1}{5}\sum (x_i-\bar X)^2 = \frac{(-2)^2+(-1)^2+0^2+1^2+2^2}{5} = \frac{10}{5} = 2$$
$$\sigma_X = \sqrt{2}$$

For $Y$:

$$\sigma_Y^2 = \frac{1}{5}\sum (y_i-\bar Y)^2 = \frac{(-4)^2+(-2)^2+0^2+2^2+4^2}{5} = \frac{40}{5} = 8$$
$$\sigma_Y = \sqrt{8} = 2\sqrt{2}$$

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Step 5: Correlation coefficient

$$r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} = \frac{4}{\sqrt{2}\cdot 2\sqrt{2}} = \frac{4}{4} = 1$$

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Interpretation

• $r=1$ means a perfect positive linear correlation.

• As $X$ increases, $Y$ increases in exact proportion ($Y = 2X$).

• Graphically, the points lie exactly on a straight line with slope 2

Q14(c)A sample of size $n$ is drawn from a normal distribution $N(\mu,\sigma^2)$ with known variance. Show that the sample mean is the MLE of $\mu$. (CO3)

Answer (c)

• Likelihood: $L(\mu)=\prod_{i=1}^n \frac{1}{\sqrt{2\pi\sigma^2}}\exp\!\big(-\frac{(x_i-\mu)^2}{2\sigma^2}\big).$

• Log-likelihood: $\ell(\mu) = -\frac{n}{2}\ln(2\pi\sigma^2) - \frac{1}{2\sigma^2}\sum_{i=1}^n (x_i-\mu)^2.$
  

• Derivative w.r.t. $\mu$: $\dfrac{d\ell}{d\mu} = -\frac{1}{2\sigma^2}\cdot 2\sum ( \mu - x_i) = -\frac{1}{\sigma^2}\big(n\mu - \sum x_i\big).$
  

• Set to zero → $n\mu - \sum x_i =0 \Rightarrow \hat\mu = \dfrac{1}{n}\sum_{i=1}^n x_i$
  

• So the MLE of $\mu$ is the sample mean $\bar{x}$
  

Q15 (a)A bank wants to predict whether a loan applicant will default (Yes/No). The dataset contains: Age, Income, Loan Amount, Previous Default (Yes/No).

(i) Write the steps to build a classification model using machine learning.

Steps in building the model:

1. Data Collection – Gather applicant data (Age, Income, Loan Amount, Previous Default, and Default label).

2. Data Preprocessing – Handle missing values, encode categorical variables (e.g., Yes/No → 0/1), normalize features if needed.

3. Feature Selection/Engineering – Use relevant predictors like income-to-loan ratio, past default history.

4. Train-Test Split – Divide dataset into training set (to build model) and test set (to evaluate performance).

5. Model Training – Train a classification algorithm on the training data.

6. Prediction & Evaluation – Use the model on test data and evaluate with metrics like accuracy, precision, recall, F1-score.

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(ii) Suggest which algorithm (Logistic Regression / Decision Tree) could be used and why.

• Logistic Regression is simple, interpretable, and works well if the relationship between features and default is mostly linear.

• Decision Tree is better when the dataset has non-linear patterns or mixed categorical + numerical variables, and it provides clear decision rules.

In practice, Decision Tree may be more useful here since financial default often depends on non-linear conditions (e.g., low income + high loan amount + previous default).

15 (b) Discuss any two applications of machine learning in data science

1. Healthcare

   Machine learning is widely used to improve patient care and medical research.

• By analyzing patient records (age, blood pressure, glucose level, medical history, etc.), classification models can predict whether a person is at risk of diseases such as diabetes, heart disease, or cancer.

• This allows early diagnosis and preventive care before the disease becomes severe.

• ML also assists in medical imaging (e.g., detecting tumors in X-rays/MRI scans) and in personalized treatment plans by learning from large datasets of patient outcomes.

• Overall, ML in healthcare leads to better decision-making, reduced costs, and improved patient survival rates.

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2. E-commerce

   Machine learning powers most modern e-commerce platforms.

• Recommender systems suggest products to users by studying purchase history, browsing behavior, and ratings (e.g., “Customers who bought this also bought…”).

• ML helps in dynamic pricing, where prices are adjusted in real time based on demand, user location, and buying patterns.

• It is also used in fraud detection by identifying unusual transactions, and in customer segmentation, allowing targeted marketing campaigns.

• These applications increase customer satisfaction, sales, and overall business efficiency.

Q16 (a) A university wants to analyze students’ performance data – attendance, assignment scores, exam scores. Describe how the data science process would be applied here.

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Step 1: Data Collection

• Gather data from attendance records, assignment submissions, and exam scores for all students.

• Include relevant attributes such as student ID, course, and semester.

Step 2: Data Preprocessing

• Handle missing values (e.g., missing attendance or exam scores).

• Normalize/standardize scores if needed to bring them to a common scale.

• Encode categorical data (e.g., grade levels or pass/fail) if required.

Step 3: Data Exploration & Visualization

• Use summary statistics (mean, variance) to understand trends.

• Plot histograms, boxplots, or scatterplots to detect patterns or outliers.

Step 4: Feature Engineering

• Derive new features like average score, assignment-to-exam ratio, or attendance percentage.

Step 5: Modeling & Analysis

• Apply predictive models (e.g., regression to predict exam scores) or classification models (e.g., pass/fail prediction).

• Use clustering to group students by performance levels.

Step 6: Evaluation

• Evaluate models using metrics like accuracy, RMSE, or F1-score.

• Validate findings with cross-validation or on a test dataset.

Step 7: Interpretation & Decision Making

• Identify factors affecting performance (e.g., low attendance → lower scores).

• Provide insights to improve teaching strategies, student engagement, and interventions for at-risk students.

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Summary: The data science process turns raw student data into actionable insights for academic improvement through systematic collection, cleaning, modeling, and interpretation.

Q16(b).A bank receives the following types of data daily: 500,000 transaction records (numerical), 50,000 customer feedback forms (text), ATM surveillance videos (image/video).

(a) Identify the Big Data characteristics present.
(b) Briefly explain why traditional methods fail and how data science techniques overcome this.

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(a) Big Data Characteristics (3Vs)

Characteristic	Explanation in this case
Volume	Huge amounts of data: 500,000 transactions, 50,000 feedback forms, video streams.
Velocity	Data arrives continuously, especially transactions and videos, requiring fast processing.
Variety	Data is multi-format: numerical (transactions), text (feedback), images/videos (ATM cameras).
(Optional) Veracity	Data may contain errors/noise, e.g., incomplete forms or unclear video frames.
(Optional) Value	Extracting insights can improve banking services and fraud detection.

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(b) Why traditional methods fail & how data science helps

• Traditional methods fail:

  • Relational databases and spreadsheets cannot handle massive volume efficiently.

  • Structured-only methods cannot process unstructured data (text, images, video).

  • Real-time analysis of high velocity data is not feasible with traditional tools.

• Data science techniques overcome this:

  • Big Data frameworks (Hadoop, Spark) handle large-scale storage and processing.

  • Machine learning and NLP analyze text feedback automatically.

  • Computer vision algorithms process surveillance videos for security/fraud detection.

  • Predictive analytics enables real-time decision-making and insights from diverse data types.

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Summary: The bank’s daily data exemplifies Big Data (Volume, Velocity, Variety), and data science provides scalable, automated methods to extract actionable insights where traditional methods fail.