data/polynomial_regression.json

{
  "title": "Polynomial Regression Mastery: 100 MCQs",
  "description": "A comprehensive set of 100 multiple-choice questions designed to teach and test your understanding of Polynomial Regression, starting from basic Linear Regression concepts to advanced ideas like model evaluation, bias-variance tradeoff, and overfitting.",
  "questions": [
    {
      "id": 1,
      "questionText": "What is the main goal of Linear Regression?",
      "options": [
        "To find clusters in data",
        "To compress the data",
        "To find a straight-line relationship between variables",
        "To predict categories"
      ],
      "correctAnswerIndex": 2,
      "explanation": "Linear Regression tries to find the best straight line that shows the relationship between input and output variables."
    },
    {
      "id": 2,
      "questionText": "In Linear Regression, what kind of relationship is modeled between X and Y?",
      "options": [
        "Polynomial",
        "Linear",
        "Circular",
        "Exponential"
      ],
      "correctAnswerIndex": 1,
      "explanation": "Linear Regression assumes a straight-line (linear) relationship between the independent and dependent variables."
    },
    {
      "id": 3,
      "questionText": "What does the slope in Linear Regression represent?",
      "options": [
        "The change in Y for a one-unit change in X",
        "The error of the model",
        "The average of all Y values",
        "The value of Y when X is 0"
      ],
      "correctAnswerIndex": 0,
      "explanation": "The slope tells us how much Y changes when X increases by 1 unit."
    },
    {
      "id": 4,
      "questionText": "What is the intercept in a Linear Regression equation?",
      "options": [
        "The number of data points",
        "The steepness of the line",
        "The point where the line crosses the Y-axis",
        "The residual value"
      ],
      "correctAnswerIndex": 2,
      "explanation": "The intercept is the Y value when X equals 0. It’s where the line meets the Y-axis."
    },
    {
      "id": 5,
      "questionText": "What does a residual represent in regression?",
      "options": [
        "The slope of the line",
        "The average of predictions",
        "Difference between actual and predicted values",
        "The standard deviation"
      ],
      "correctAnswerIndex": 2,
      "explanation": "A residual is the difference between the actual value and the predicted value. It shows how far the model’s prediction is from reality."
    },
    {
      "id": 6,
      "questionText": "What method is commonly used to fit a Linear Regression line?",
      "options": [
        "Gradient Ascent",
        "Residual Addition",
        "Ordinary Least Squares",
        "Mean Minimization"
      ],
      "correctAnswerIndex": 2,
      "explanation": "Ordinary Least Squares (OLS) minimizes the sum of squared residuals to find the best-fitting line."
    },
    {
      "id": 7,
      "questionText": "What happens if residuals are not randomly distributed?",
      "options": [
        "There may be a pattern not captured by the model",
        "It increases accuracy",
        "The slope becomes 0",
        "The model is perfect"
      ],
      "correctAnswerIndex": 0,
      "explanation": "If residuals show a pattern, it means the model missed some relationship in the data."
    },
    {
      "id": 8,
      "questionText": "What type of variable does Linear Regression predict?",
      "options": [
        "Continuous",
        "Integer only",
        "Categorical",
        "Binary"
      ],
      "correctAnswerIndex": 0,
      "explanation": "Linear Regression is used for predicting continuous numerical values like height, weight, or prices."
    },
    {
      "id": 9,
      "questionText": "Which assumption is true for Linear Regression?",
      "options": [
        "All features are independent",
        "Residuals are normally distributed",
        "Data must be categorical",
        "Output is binary"
      ],
      "correctAnswerIndex": 1,
      "explanation": "One assumption of Linear Regression is that residuals should follow a normal distribution."
    },
    {
      "id": 10,
      "questionText": "What problem occurs when data is not linear?",
      "options": [
        "Lower variance",
        "Perfect prediction",
        "Poor model fit",
        "Balanced output"
      ],
      "correctAnswerIndex": 2,
      "explanation": "Linear Regression works best for linear data. If data is curved, it won’t fit well, leading to high error."
    },
    {
      "id": 11,
      "questionText": "What is Polynomial Regression used for?",
      "options": [
        "Modeling curved relationships",
        "Modeling straight-line relationships",
        "Finding clusters",
        "Reducing dimensionality"
      ],
      "correctAnswerIndex": 0,
      "explanation": "Polynomial Regression models non-linear or curved relationships between input and output variables."
    },
    {
      "id": 12,
      "questionText": "Polynomial Regression is an extension of which model?",
      "options": [
        "Decision Tree",
        "Linear Regression",
        "Logistic Regression",
        "Naive Bayes"
      ],
      "correctAnswerIndex": 1,
      "explanation": "Polynomial Regression is an extension of Linear Regression where input features are raised to powers."
    },
    {
      "id": 13,
      "questionText": "In Polynomial Regression, we add what kind of terms to the model?",
      "options": [
        "Cubic roots only",
        "Squared and higher power terms of input",
        "Logarithmic terms",
        "Exponential terms"
      ],
      "correctAnswerIndex": 1,
      "explanation": "Polynomial Regression includes higher power terms like x², x³, etc., to capture curves in the data."
    },
    {
      "id": 14,
      "questionText": "What shape can a second-degree Polynomial Regression model represent?",
      "options": [
        "Circle",
        "Parabola",
        "Zigzag",
        "Straight line"
      ],
      "correctAnswerIndex": 1,
      "explanation": "A second-degree polynomial creates a parabola-shaped curve, allowing the model to fit U-shaped data."
    },
    {
      "id": 15,
      "questionText": "What is the general form of a Polynomial Regression equation with one variable?",
      "options": [
        "y = b0 + b1x + b2x² + ... + bkx^k",
        "y = mx + b",
        "y = b0 + b1x",
        "y = bx + c"
      ],
      "correctAnswerIndex": 0,
      "explanation": "Polynomial Regression includes terms of increasing power: x, x², x³, etc., up to the desired degree k."
    },
    {
      "id": 16,
      "questionText": "What happens when you increase the degree of a polynomial too much?",
      "options": [
        "The model becomes linear",
        "The model may overfit the data",
        "The model becomes simpler",
        "The error increases on training data"
      ],
      "correctAnswerIndex": 1,
      "explanation": "A high-degree polynomial can overfit by fitting noise in the training data rather than the true pattern."
    },
    {
      "id": 17,
      "questionText": "Overfitting in Polynomial Regression leads to what?",
      "options": [
        "Lower variance",
        "Simpler equations",
        "Better generalization",
        "Poor performance on new data"
      ],
      "correctAnswerIndex": 3,
      "explanation": "Overfitting means the model performs well on training data but fails to generalize to unseen data."
    },
    {
      "id": 18,
      "questionText": "What is underfitting?",
      "options": [
        "When the model is too simple to capture patterns",
        "When training accuracy is 100%",
        "When residuals are 0",
        "When the model is too complex"
      ],
      "correctAnswerIndex": 0,
      "explanation": "Underfitting happens when the model is too simple and cannot capture the underlying structure of the data."
    },
    {
      "id": 19,
      "questionText": "Which term describes the trade-off between bias and variance in a polynomial model?",
      "options": [
        "Regularization",
        "Feature Scaling",
        "Gradient Descent",
        "Bias-Variance Tradeoff"
      ],
      "correctAnswerIndex": 3,
      "explanation": "Bias-Variance Tradeoff explains how increasing model complexity reduces bias but increases variance."
    },
    {
      "id": 20,
      "questionText": "What is the degree of a polynomial?",
      "options": [
        "Number of variables",
        "Highest power of the input variable",
        "Sum of all coefficients",
        "Number of residuals"
      ],
      "correctAnswerIndex": 1,
      "explanation": "The degree of a polynomial is the highest exponent of the input variable in the equation."
    },
    {
      "id": 21,
      "questionText": "Which type of relationship can Polynomial Regression handle that Linear Regression cannot?",
      "options": [
        "Categorical",
        "Binary",
        "Constant",
        "Non-linear"
      ],
      "correctAnswerIndex": 3,
      "explanation": "Polynomial Regression can model curved, non-linear relationships, unlike simple linear regression."
    },
    {
      "id": 22,
      "questionText": "What does increasing the polynomial degree do?",
      "options": [
        "Simplifies computation",
        "Decreases coefficients",
        "Removes noise",
        "Adds more curve flexibility"
      ],
      "correctAnswerIndex": 3,
      "explanation": "A higher degree polynomial gives the model more flexibility to follow the data's shape."
    },
    {
      "id": 23,
      "questionText": "What kind of curve does a third-degree polynomial create?",
      "options": [
        "Straight line",
        "S-shape",
        "U-shape",
        "Flat line"
      ],
      "correctAnswerIndex": 1,
      "explanation": "A cubic polynomial (degree 3) can create an S-shaped curve that changes direction once."
    },
    {
      "id": 24,
      "questionText": "Which library in Python is commonly used to create polynomial features?",
      "options": [
        "NumPy",
        "scikit-learn",
        "Pandas",
        "Matplotlib"
      ],
      "correctAnswerIndex": 1,
      "explanation": "The PolynomialFeatures class from scikit-learn is used to generate higher-degree input features."
    },
    {
      "id": 25,
      "questionText": "What function in scikit-learn is used to transform data into polynomial features?",
      "options": [
        "create_poly_data()",
        "PolynomialFeatures()",
        "poly_transform()",
        "make_polynomial()"
      ],
      "correctAnswerIndex": 1,
      "explanation": "The PolynomialFeatures() function expands input features into polynomial combinations."
    },
    {
      "id": 26,
      "questionText": "Which of the following problems is Polynomial Regression best suited for?",
      "options": [
        "Linear relationships only",
        "Categorical output prediction",
        "Curved relationships between variables",
        "Time series forecasting only"
      ],
      "correctAnswerIndex": 2,
      "explanation": "Polynomial Regression is best used when data shows a curved or non-linear pattern between input and output."
    },
    {
      "id": 27,
      "questionText": "If the degree of the polynomial is 1, what does Polynomial Regression become?",
      "options": [
        "Logistic Regression",
        "Linear Regression",
        "Decision Tree",
        "Ridge Regression"
      ],
      "correctAnswerIndex": 1,
      "explanation": "When the degree is 1, Polynomial Regression is the same as simple Linear Regression."
    },
    {
      "id": 28,
      "questionText": "What happens when you use a degree that is too low for Polynomial Regression?",
      "options": [
        "No bias",
        "Underfitting",
        "Perfect fit",
        "Overfitting"
      ],
      "correctAnswerIndex": 1,
      "explanation": "Using a degree that is too low may cause the model to miss patterns, leading to underfitting."
    },
    {
      "id": 29,
      "questionText": "What kind of error increases with a high-degree polynomial?",
      "options": [
        "Noise",
        "Correlation",
        "Bias",
        "Variance"
      ],
      "correctAnswerIndex": 3,
      "explanation": "High-degree polynomials often increase variance, meaning the model becomes sensitive to small data changes."
    },
    {
      "id": 30,
      "questionText": "What is the main goal when choosing the degree of a polynomial?",
      "options": [
        "To balance bias and variance",
        "To reduce coefficients",
        "To fit as many points as possible",
        "To maximize error"
      ],
      "correctAnswerIndex": 0,
      "explanation": "The degree should be chosen to balance bias (simplicity) and variance (complexity) for good generalization."
    },
    {
      "id": 31,
      "questionText": "What technique can help prevent overfitting in Polynomial Regression?",
      "options": [
        "Adding more features",
        "Increasing polynomial degree",
        "Removing training data",
        "Regularization"
      ],
      "correctAnswerIndex": 3,
      "explanation": "Regularization methods like Ridge or Lasso Regression can reduce overfitting by penalizing large coefficients."
    },
    {
      "id": 32,
      "questionText": "What is Ridge Regression also known as?",
      "options": [
        "Variance Reduction",
        "L2 Regularization",
        "Elastic Net",
        "L1 Regularization"
      ],
      "correctAnswerIndex": 1,
      "explanation": "Ridge Regression uses L2 Regularization, which penalizes the sum of squared coefficients."
    },
    {
      "id": 33,
      "questionText": "What is Lasso Regression also known as?",
      "options": [
        "L1 Regularization",
        "Bias Correction",
        "L2 Regularization",
        "Polynomial Fitting"
      ],
      "correctAnswerIndex": 0,
      "explanation": "Lasso Regression uses L1 Regularization, which penalizes the absolute values of coefficients."
    },
    {
      "id": 34,
      "questionText": "What is the main difference between Ridge and Lasso?",
      "options": [
        "Ridge can remove features, Lasso cannot",
        "Ridge uses L1, Lasso uses L2",
        "Both remove coefficients equally",
        "Lasso can make some coefficients zero, Ridge cannot"
      ],
      "correctAnswerIndex": 3,
      "explanation": "Lasso can shrink some coefficients to exactly zero, performing feature selection, while Ridge cannot."
    },
    {
      "id": 35,
      "questionText": "What evaluation metric measures how well the model explains the variance of the data?",
      "options": [
        "Mean Absolute Error",
        "Mean Squared Error",
        "R-squared",
        "Root Mean Square Deviation"
      ],
      "correctAnswerIndex": 2,
      "explanation": "R-squared measures the proportion of variance in the target variable explained by the model."
    },
    {
      "id": 36,
      "questionText": "What is the range of R-squared values?",
      "options": [
        "0 to 1",
        "1 to infinity",
        "0 to 100",
        "-1 to 1"
      ],
      "correctAnswerIndex": 0,
      "explanation": "R-squared ranges from 0 to 1, where 1 means perfect prediction and 0 means no predictive power."
    },
    {
      "id": 37,
      "questionText": "Which error metric squares the difference between actual and predicted values?",
      "options": [
        "Correlation Coefficient",
        "R-squared",
        "Mean Absolute Error",
        "Mean Squared Error"
      ],
      "correctAnswerIndex": 3,
      "explanation": "Mean Squared Error (MSE) calculates the average of squared prediction errors."
    },
    {
      "id": 38,
      "questionText": "Why is Root Mean Squared Error (RMSE) preferred over MSE?",
      "options": [
        "It gives larger values",
        "It reduces overfitting",
        "It is in the same units as the target variable",
        "It lowers variance"
      ],
      "correctAnswerIndex": 2,
      "explanation": "RMSE is the square root of MSE, giving error values in the same unit as the dependent variable."
    },
    {
      "id": 39,
      "questionText": "What can be a sign of overfitting when comparing training and test errors?",
      "options": [
        "Training error is low but test error is high",
        "Both errors are low",
        "Both errors are high",
        "Test error is lower than training error"
      ],
      "correctAnswerIndex": 0,
      "explanation": "If the training error is much lower than test error, it indicates the model has memorized the training data."
    },
    {
      "id": 40,
      "questionText": "Which plot is useful to visualize Polynomial Regression fit?",
      "options": [
        "Scatter plot with curve",
        "Line plot",
        "Bar plot",
        "Pie chart"
      ],
      "correctAnswerIndex": 0,
      "explanation": "Scatter plots with a fitted curve help visualize how well the polynomial model fits the data."
    },
    {
      "id": 41,
      "questionText": "How can you check if adding polynomial terms improves your model?",
      "options": [
        "By visualizing the curve",
        "By comparing R-squared values",
        "By adding random features",
        "By increasing degree blindly"
      ],
      "correctAnswerIndex": 1,
      "explanation": "Comparing R-squared and validation errors helps decide if extra polynomial terms improve model accuracy."
    },
    {
      "id": 42,
      "questionText": "What is multicollinearity in Polynomial Regression?",
      "options": [
        "When residuals are independent",
        "When regularization is applied",
        "When output is non-linear",
        "When input features are highly correlated"
      ],
      "correctAnswerIndex": 3,
      "explanation": "Polynomial features (x, x², x³, etc.) are often correlated, causing multicollinearity, which affects coefficient stability."
    },
    {
      "id": 43,
      "questionText": "Which method can help reduce multicollinearity in polynomial models?",
      "options": [
        "Adding noise",
        "Increasing degree",
        "Regularization",
        "Ignoring correlations"
      ],
      "correctAnswerIndex": 2,
      "explanation": "Regularization (Ridge or Lasso) reduces coefficient sensitivity caused by multicollinearity."
    },
    {
      "id": 44,
      "questionText": "What is the purpose of feature scaling in Polynomial Regression?",
      "options": [
        "To make data categorical",
        "To prevent large coefficient values",
        "To remove outliers",
        "To increase variance"
      ],
      "correctAnswerIndex": 1,
      "explanation": "Feature scaling ensures that polynomial features with large values do not dominate during training."
    },
    {
      "id": 45,
      "questionText": "Which scaling method is commonly used before Polynomial Regression?",
      "options": [
        "Min-Max Scaling",
        "Text Vectorization",
        "One-Hot Encoding",
        "Label Encoding"
      ],
      "correctAnswerIndex": 0,
      "explanation": "Min-Max Scaling is often used to bring features within a small range, improving numerical stability."
    },
    {
      "id": 46,
      "questionText": "What is the main advantage of Polynomial Regression over Linear Regression?",
      "options": [
        "Faster computation",
        "Easier interpretation",
        "Ability to fit curved patterns",
        "Less data needed"
      ],
      "correctAnswerIndex": 2,
      "explanation": "Polynomial Regression can model curved, non-linear data patterns that linear models cannot handle."
    },
    {
      "id": 47,
      "questionText": "Which curve fitting problem can Polynomial Regression solve?",
      "options": [
        "Fitting U-shaped and S-shaped data",
        "Fitting straight lines",
        "Finding text patterns",
        "Classifying images"
      ],
      "correctAnswerIndex": 0,
      "explanation": "Polynomial Regression is effective for U-shaped or S-shaped curves that need flexibility in fitting."
    },
    {
      "id": 48,
      "questionText": "Which of these statements about high-degree polynomials is true?",
      "options": [
        "They are simple to interpret",
        "They generalize well",
        "They may oscillate wildly between points",
        "They reduce variance"
      ],
      "correctAnswerIndex": 2,
      "explanation": "High-degree polynomials may fluctuate too much between data points, reducing stability."
    },
    {
      "id": 49,
      "questionText": "What type of regularization combines L1 and L2?",
      "options": [
        "Ridge",
        "Dropout",
        "Elastic Net",
        "Lasso"
      ],
      "correctAnswerIndex": 2,
      "explanation": "Elastic Net combines both L1 (Lasso) and L2 (Ridge) regularization techniques."
    },
    {
      "id": 50,
      "questionText": "What does the alpha parameter control in Ridge and Lasso Regression?",
      "options": [
        "The learning rate",
        "The model degree",
        "The regularization strength",
        "The intercept"
      ],
      "correctAnswerIndex": 2,
      "explanation": "Alpha controls how strongly the model penalizes large coefficient values. Higher alpha means stronger regularization."
    },
    {
      "id": 51,
      "questionText": "What happens if the polynomial degree is set too high on a small dataset?",
      "options": [
        "Perfect fitting always",
        "Underfitting",
        "Overfitting",
        "No change in accuracy"
      ],
      "correctAnswerIndex": 2,
      "explanation": "A high-degree polynomial can memorize the training data, leading to overfitting and poor generalization."
    },
    {
      "id": 52,
      "questionText": "Which of the following helps reduce overfitting in Polynomial Regression?",
      "options": [
        "Using regularization",
        "Using fewer data points",
        "Adding noise to labels",
        "Increasing polynomial degree"
      ],
      "correctAnswerIndex": 0,
      "explanation": "Regularization penalizes large coefficients, which helps reduce overfitting."
    },
    {
      "id": 53,
      "questionText": "What does feature scaling do before applying polynomial features?",
      "options": [
        "Ensures all features contribute equally",
        "Removes outliers",
        "Increases model degree",
        "Makes coefficients smaller"
      ],
      "correctAnswerIndex": 0,
      "explanation": "Feature scaling ensures all input features have similar ranges, preventing domination by one feature."
    },
    {
      "id": 54,
      "questionText": "Why is Polynomial Regression still considered a linear model?",
      "options": [
        "Because it ignores nonlinear patterns",
        "Because coefficients are linear in parameters",
        "Because data must be linear",
        "Because it uses straight lines"
      ],
      "correctAnswerIndex": 1,
      "explanation": "Despite nonlinear features, the model remains linear in terms of its coefficients."
    },
    {
      "id": 55,
      "questionText": "Which sklearn class is used to generate polynomial features?",
      "options": [
        "PolynomialFeatures",
        "PolyScaler",
        "FeatureGenerator",
        "PolynomialModel"
      ],
      "correctAnswerIndex": 0,
      "explanation": "PolynomialFeatures from sklearn.preprocessing expands input data to include polynomial terms."
    },
    {
      "id": 56,
      "questionText": "What is the main disadvantage of using very high-degree polynomials?",
      "options": [
        "Simpler model",
        "Overfitting and numerical instability",
        "Lower computation time",
        "Underfitting"
      ],
      "correctAnswerIndex": 1,
      "explanation": "High-degree polynomials can overfit and suffer from large coefficient swings causing instability."
    },
    {
      "id": 57,
      "questionText": "In Polynomial Regression, which term represents the intercept?",
      "options": [
        "x^n term",
        "x^1 term",
        "x^0 term",
        "x^2 term"
      ],
      "correctAnswerIndex": 2,
      "explanation": "The x^0 term represents the constant (intercept) of the polynomial equation."
    },
    {
      "id": 58,
      "questionText": "What will happen if we skip PolynomialFeatures but use degree > 1 in LinearRegression?",
      "options": [
        "It will use polynomial terms automatically",
        "The model will fail",
        "It will regularize coefficients",
        "It will behave like linear regression"
      ],
      "correctAnswerIndex": 3,
      "explanation": "LinearRegression does not create polynomial terms automatically. Without PolynomialFeatures, it stays linear."
    },
    {
      "id": 59,
      "questionText": "Which cross-validation technique is useful to choose polynomial degree?",
      "options": [
        "Train-Test Split only",
        "Random Sampling",
        "Leave-One-Out CV",
        "No validation needed"
      ],
      "correctAnswerIndex": 2,
      "explanation": "Leave-One-Out Cross Validation works well to find the optimal polynomial degree for small datasets."
    },
    {
      "id": 60,
      "questionText": "How does increasing polynomial degree affect bias and variance?",
      "options": [
        "Increases bias and decreases variance",
        "Decreases bias and increases variance",
        "Increases both",
        "Decreases both"
      ],
      "correctAnswerIndex": 1,
      "explanation": "Higher degrees reduce bias (fit training data better) but increase variance (sensitive to noise)."
    },
    {
      "id": 61,
      "questionText": "What does the term 'interaction features' mean in Polynomial Regression?",
      "options": [
        "Features multiplied together",
        "Random noise features",
        "Unrelated features",
        "Features added together"
      ],
      "correctAnswerIndex": 0,
      "explanation": "Interaction features are created by multiplying original features, capturing combined effects."
    },
    {
      "id": 62,
      "questionText": "What happens to training error as we increase polynomial degree?",
      "options": [
        "Always increases",
        "Usually decreases",
        "Becomes random",
        "Stays constant"
      ],
      "correctAnswerIndex": 1,
      "explanation": "A higher-degree polynomial fits the training data better, reducing training error."
    },
    {
      "id": 63,
      "questionText": "Which step comes immediately after generating polynomial features?",
      "options": [
        "Scaling",
        "Model fitting",
        "Data shuffling",
        "Feature selection"
      ],
      "correctAnswerIndex": 1,
      "explanation": "After generating polynomial features, the next step is fitting the regression model."
    },
    {
      "id": 64,
      "questionText": "What is a typical symptom of overfitting in Polynomial Regression?",
      "options": [
        "Identical train and test results",
        "Low training accuracy",
        "High training accuracy but low test accuracy",
        "High test accuracy"
      ],
      "correctAnswerIndex": 2,
      "explanation": "Overfitting happens when a model performs very well on training data but poorly on unseen data."
    },
    {
      "id": 65,
      "questionText": "How can we make polynomial regression less sensitive to outliers?",
      "options": [
        "Use regularization",
        "Add more noise",
        "Ignore scaling",
        "Increase degree"
      ],
      "correctAnswerIndex": 0,
      "explanation": "Regularization like Ridge or Lasso limits large coefficient values, making the model less sensitive to outliers."
    },
    {
      "id": 66,
      "questionText": "Which metric is least suitable for measuring polynomial regression performance?",
      "options": [
        "R-squared",
        "Confusion Matrix",
        "Mean Absolute Error",
        "Mean Squared Error"
      ],
      "correctAnswerIndex": 1,
      "explanation": "Confusion Matrix is used for classification problems, not regression."
    },
    {
      "id": 67,
      "questionText": "What is the shape of the curve in quadratic regression?",
      "options": [
        "Circle",
        "Parabola",
        "Hyperbola",
        "Line"
      ],
      "correctAnswerIndex": 1,
      "explanation": "A second-degree polynomial forms a parabola."
    },
    {
      "id": 68,
      "questionText": "What does PolynomialFeatures(degree=3) generate for input x?",
      "options": [
        "x^2 only",
        "x^3 only",
        "x, x^2, x^3",
        "x"
      ],
      "correctAnswerIndex": 2,
      "explanation": "It expands the feature set to include x, x^2, and x^3 terms."
    },
    {
      "id": 69,
      "questionText": "When should we use Polynomial Regression over Linear Regression?",
      "options": [
        "When relationship is clearly nonlinear",
        "When slope is constant",
        "When data has many missing values",
        "When data is categorical"
      ],
      "correctAnswerIndex": 0,
      "explanation": "Polynomial Regression captures nonlinear relationships between input and output."
    },
    {
      "id": 70,
      "questionText": "Why does feature scaling matter more for higher-degree polynomials?",
      "options": [
        "Because it reduces intercept",
        "Because it helps visualization",
        "Because polynomial terms grow rapidly",
        "Because it ignores bias"
      ],
      "correctAnswerIndex": 2,
      "explanation": "High-degree terms like x^5 or x^6 can produce large numeric values; scaling keeps them manageable."
    },
    {
      "id": 71,
      "questionText": "What is the main effect of high-degree polynomials on model complexity?",
      "options": [
        "Increases complexity",
        "Keeps complexity same",
        "Removes features",
        "Reduces complexity"
      ],
      "correctAnswerIndex": 0,
      "explanation": "High-degree polynomials add more terms, increasing model complexity and flexibility."
    },
    {
      "id": 72,
      "questionText": "Which method helps select the optimal polynomial degree?",
      "options": [
        "Cross-validation",
        "Using only training error",
        "Trial and error",
        "Random selection"
      ],
      "correctAnswerIndex": 0,
      "explanation": "Cross-validation evaluates model performance on unseen data to choose the best polynomial degree."
    },
    {
      "id": 73,
      "questionText": "What is bias in the context of polynomial regression?",
      "options": [
        "Error due to noise",
        "Error due to large coefficients",
        "Error due to model simplicity",
        "Random fluctuation"
      ],
      "correctAnswerIndex": 2,
      "explanation": "Bias measures the error caused by approximating a complex relationship with a simple model."
    },
    {
      "id": 74,
      "questionText": "What is variance in the context of polynomial regression?",
      "options": [
        "Error due to bias",
        "Error due to sensitivity to training data",
        "Error due to model simplicity",
        "Error due to missing features"
      ],
      "correctAnswerIndex": 1,
      "explanation": "Variance is the error caused when the model changes too much with small changes in the training data."
    },
    {
      "id": 75,
      "questionText": "Which combination of bias and variance is ideal?",
      "options": [
        "High bias, low variance",
        "Low bias, high variance",
        "High bias, high variance",
        "Low bias, low variance"
      ],
      "correctAnswerIndex": 3,
      "explanation": "The ideal model has low bias (accurate on training) and low variance (stable on new data)."
    },
    {
      "id": 76,
      "questionText": "How can we detect overfitting visually?",
      "options": [
        "By examining coefficients only",
        "By looking at training vs test error",
        "By plotting residuals",
        "By plotting polynomial degree only"
      ],
      "correctAnswerIndex": 1,
      "explanation": "Overfitting is indicated when training error is very low but test error is high."
    },
    {
      "id": 77,
      "questionText": "Which method reduces model complexity while keeping fit reasonable?",
      "options": [
        "Regularization",
        "Adding more polynomial terms",
        "Ignoring validation data",
        "Increasing dataset noise"
      ],
      "correctAnswerIndex": 0,
      "explanation": "Regularization penalizes large coefficients, simplifying the model and reducing overfitting."
    },
    {
      "id": 78,
      "questionText": "Why is L1 regularization useful in polynomial regression?",
      "options": [
        "Increases variance",
        "Makes polynomial degree higher",
        "Removes features automatically",
        "Decreases bias only"
      ],
      "correctAnswerIndex": 2,
      "explanation": "L1 regularization (Lasso) can shrink some coefficients to zero, effectively selecting important features."
    },
    {
      "id": 79,
      "questionText": "Why is L2 regularization useful in polynomial regression?",
      "options": [
        "Removes features",
        "Reduces large coefficient impact",
        "Increases polynomial degree",
        "Increases training error only"
      ],
      "correctAnswerIndex": 1,
      "explanation": "L2 regularization (Ridge) penalizes large coefficients to make the model more stable."
    },
    {
      "id": 80,
      "questionText": "Which visualization helps check polynomial fit?",
      "options": [
        "Histogram",
        "Box plot",
        "Scatter plot with fitted curve",
        "Bar chart"
      ],
      "correctAnswerIndex": 2,
      "explanation": "Scatter plots with fitted curves show how well the polynomial captures data patterns."
    },
    {
      "id": 81,
      "questionText": "What does R-squared indicate in polynomial regression?",
      "options": [
        "Mean squared error",
        "Training time",
        "Number of features",
        "Proportion of variance explained"
      ],
      "correctAnswerIndex": 3,
      "explanation": "R-squared measures how much of the target variance is captured by the model."
    },
    {
      "id": 82,
      "questionText": "Which error metric gives average magnitude of prediction errors?",
      "options": [
        "Mean Absolute Error",
        "Variance",
        "R-squared",
        "Mean Squared Error"
      ],
      "correctAnswerIndex": 0,
      "explanation": "Mean Absolute Error calculates the average absolute difference between predicted and actual values."
    },
    {
      "id": 83,
      "questionText": "Which metric penalizes large errors more heavily?",
      "options": [
        "MSE",
        "MAE",
        "R-squared",
        "Correlation coefficient"
      ],
      "correctAnswerIndex": 0,
      "explanation": "MSE squares the errors, giving higher weight to large deviations."
    },
    {
      "id": 84,
      "questionText": "Why is cross-validation important in polynomial regression?",
      "options": [
        "To ignore overfitting",
        "To fit data perfectly",
        "To evaluate model on unseen data",
        "To increase polynomial degree"
      ],
      "correctAnswerIndex": 2,
      "explanation": "Cross-validation tests model performance on unseen data, helping select optimal degree and reduce overfitting."
    },
    {
      "id": 85,
      "questionText": "Which technique can combine multiple polynomial models for better prediction?",
      "options": [
        "Single model fitting",
        "L1 regularization only",
        "Feature scaling",
        "Bagging"
      ],
      "correctAnswerIndex": 3,
      "explanation": "Bagging combines predictions from multiple models to reduce variance and improve accuracy."
    },
    {
      "id": 86,
      "questionText": "Which problem arises if polynomial degree is too low?",
      "options": [
        "Feature scaling",
        "Underfitting",
        "Regularization",
        "Overfitting"
      ],
      "correctAnswerIndex": 1,
      "explanation": "A low-degree polynomial may fail to capture data patterns, causing underfitting."
    },
    {
      "id": 87,
      "questionText": "Which method automatically selects important polynomial terms?",
      "options": [
        "Lasso Regression",
        "Ridge Regression",
        "Cross-validation only",
        "Standard Linear Regression"
      ],
      "correctAnswerIndex": 0,
      "explanation": "Lasso regression can shrink some coefficients to zero, selecting the most important features."
    },
    {
      "id": 88,
      "questionText": "Which is a symptom of multicollinearity in polynomial regression?",
      "options": [
        "Low variance",
        "High R-squared always",
        "Unstable coefficients",
        "Zero training error"
      ],
      "correctAnswerIndex": 2,
      "explanation": "Polynomial terms are often correlated, making coefficients unstable and sensitive to small data changes."
    },
    {
      "id": 89,
      "questionText": "Which of these is an advantage of polynomial regression?",
      "options": [
        "Fits linear data only",
        "Can fit nonlinear patterns",
        "Removes outliers automatically",
        "Reduces training data needed"
      ],
      "correctAnswerIndex": 1,
      "explanation": "Polynomial regression captures nonlinear relationships between variables."
    },
    {
      "id": 90,
      "questionText": "Which is a common step before polynomial regression on real data?",
      "options": [
        "Removing target variable",
        "Feature scaling",
        "Increasing polynomial degree blindly",
        "Random noise addition"
      ],
      "correctAnswerIndex": 1,
      "explanation": "Feature scaling ensures all polynomial terms are on a similar scale for stable model training."
    },
    {
      "id": 91,
      "questionText": "Which model would you choose for a U-shaped data trend?",
      "options": [
        "Linear Regression",
        "Logistic Regression",
        "Quadratic Polynomial Regression",
        "Cubic Regression"
      ],
      "correctAnswerIndex": 2,
      "explanation": "Quadratic (degree 2) polynomial regression is ideal for U-shaped patterns."
    },
    {
      "id": 92,
      "questionText": "Which model would you choose for an S-shaped trend?",
      "options": [
        "Quadratic Regression",
        "Cubic Regression",
        "Linear Regression",
        "Logistic Regression"
      ],
      "correctAnswerIndex": 1,
      "explanation": "Cubic (degree 3) polynomial regression can fit S-shaped trends with one inflection point."
    },
    {
      "id": 93,
      "questionText": "Which is an indicator of underfitting in polynomial regression?",
      "options": [
        "Low bias",
        "High variance",
        "Low training error and high test error",
        "High training error and high test error"
      ],
      "correctAnswerIndex": 3,
      "explanation": "Underfitting shows both training and test errors are high due to a too-simple model."
    },
    {
      "id": 94,
      "questionText": "What is the effect of regularization on polynomial coefficients?",
      "options": [
        "Increases bias only",
        "Reduces magnitude of coefficients",
        "Increases all coefficients",
        "Removes training data"
      ],
      "correctAnswerIndex": 1,
      "explanation": "Regularization penalizes large coefficients to reduce overfitting."
    },
    {
      "id": 95,
      "questionText": "Which method can evaluate polynomial regression stability across datasets?",
      "options": [
        "Only visualization",
        "Train-test split",
        "Cross-validation",
        "Random coefficient assignment"
      ],
      "correctAnswerIndex": 2,
      "explanation": "Cross-validation tests the model on multiple data splits to check stability and generalization."
    },
    {
      "id": 96,
      "questionText": "Why should we avoid excessively high-degree polynomials?",
      "options": [
        "They increase overfitting",
        "They always improve R-squared",
        "They reduce bias",
        "They remove noise automatically"
      ],
      "correctAnswerIndex": 0,
      "explanation": "Excessively high-degree polynomials may fit noise rather than the actual pattern, causing overfitting."
    },
    {
      "id": 97,
      "questionText": "Which method can simplify a polynomial regression model?",
      "options": [
        "Ignoring validation",
        "Increasing degree",
        "Adding noise",
        "Regularization"
      ],
      "correctAnswerIndex": 3,
      "explanation": "Regularization reduces large coefficients and can simplify the model."
    },
    {
      "id": 98,
      "questionText": "Which of the following is true about polynomial regression predictions?",
      "options": [
        "Always linear",
        "Independent of input",
        "Always quadratic",
        "Can be nonlinear even with linear coefficients"
      ],
      "correctAnswerIndex": 3,
      "explanation": "Predictions can be nonlinear because the input features are polynomial terms, even if the model is linear in coefficients."
    },
    {
      "id": 99,
      "questionText": "Which is a good strategy for selecting polynomial degree?",
      "options": [
        "Ignoring training error",
        "Always using degree 5",
        "Using cross-validation",
        "Trial and error without validation"
      ],
      "correctAnswerIndex": 2,
      "explanation": "Cross-validation helps find a degree that balances underfitting and overfitting."
    },
    {
      "id": 100,
      "questionText": "What is the final goal of polynomial regression?",
      "options": [
        "To remove features",
        "To increase variance",
        "To predict continuous values with nonlinear patterns",
        "To classify data"
      ],
      "correctAnswerIndex": 2,
      "explanation": "Polynomial regression aims to predict continuous outcomes while capturing nonlinear relationships."
    }
  ]
}