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Quizzes > Quizzes for Business > Technology

Linear Regression Knowledge Test Quiz

Assess Your Regression Analysis and Prediction Skills

Difficulty: Moderate
Questions: 20
Learning OutcomesStudy Material
Colorful paper art depicting a quiz on Linear Regression Knowledge Test

Ready to master regression analysis? This Linear Regression Quiz challenges you with 15 multiple-choice questions that assess your understanding of slope, intercepts, and model evaluation. Ideal for students and educators seeking to refine their regression analysis skills, this test offers instant feedback and detailed answer explanations. Plus, every question is fully editable in our intuitive editor - customise this quiz to fit any course or study plan. Dive deeper into math practice with our Algebra: Linear Equations and Graphs Quiz , explore Linear Equations Assessment Quiz , or browse all quizzes for more topics.

What does a Pearson correlation coefficient of -0.8 indicate about a linear relationship?
A strong negative linear relationship
A strong positive linear relationship
No linear relationship
A weak negative linear relationship
A correlation near -1 indicates a strong negative linear relationship, meaning as one variable increases the other decreases. The strength is high because the value is close to -1.
In the simple linear regression equation y-hat = 5 + 2x, what is the slope?
-2
2
7
5
In the regression equation y-hat = 5 + 2x, the slope is the coefficient on x, which is 2. The intercept is the constant term, 5.
What does an R-squared value of 0.65 mean?
The correlation coefficient is 0.65
35% of variance in the outcome is explained by the predictor
The model predicts correctly 65% of the time
65% of variance in the outcome is explained by the predictor
R-squared of 0.65 means 65% of the variability in the dependent variable is explained by the independent variable. It is not the correlation itself nor a prediction accuracy rate.
In regression analysis, the residual is defined as:
Observed value minus predicted value
Average error across all observations
Predicted value minus observed value
Slope times residual
A residual is the difference between an observed value and its predicted value (observed minus predicted). It measures the error for each observation.
If the regression equation is y-hat = 1 + 3x, what is the predicted y when x=4?
13
12
7
4
Substitute x = 4 into y-hat = 1 + 3x gives y-hat = 1 + 3(4) = 13. This is the model's prediction for that x-value.
If the correlation coefficient r is 0.85, what is the approximate R-squared?
1.15
0.65
0.85
0.72
R-squared is the square of the correlation coefficient: 0.85² = 0.7225, approximately 0.72. It represents variance explained.
A residual plot shows a funnel shape where residual spread increases with fitted values. What does this indicate?
Independence violation
Heteroscedasticity
Nonlinearity
Multicollinearity
A funnel-shaped residual plot indicates non-constant variance of errors, known as heteroscedasticity. Other issues relate to different patterns.
A model that underfits data and is too simple exhibits which of the following?
High variance and high bias
Low bias and low variance
High variance and low bias
High bias and low variance
An underfitted model is overly simple and cannot capture data structure, leading to high bias and low variance. It systematically misses patterns.
Given the regression line y-hat = 2 - 0.5x, what is y-hat when x=6?
5
-1
-3
3
Plugging x = 6 into y-hat = 2 - 0.5(6) gives 2 - 3 = -1. This is the predicted value.
Making predictions for x-values outside the range of the data is known as:
Extrapolation
Interpolation
Cross-validation
Bootstrapping
Predicting values outside the observed data range is called extrapolation. Interpolation is within the data range.
What does a 95% confidence interval for the slope coefficient represent?
The range explaining 95% of data variation
The range that predicts 95% of future slope estimates
The exact 95% probability the slope is within the interval
The range that, with 95% confidence, contains the true population slope
A 95% confidence interval means that if we repeated sampling many times, 95% of such intervals would contain the true slope. It's not prediction of data.
If the p-value for the slope in simple regression is 0.03 at α=0.05, what conclusion should you draw?
The model is invalid
Cannot conclude anything
Fail to reject null, slope is not different from zero
Reject null, slope is significantly different from zero
A p-value of 0.03 is less than α = 0.05, so we reject the null hypothesis that the slope is zero, indicating statistical significance.
What is the null hypothesis in the hypothesis test for a regression slope?
The slope equals zero
R-squared equals one
The slope equals one
The intercept equals zero
The standard null hypothesis for testing a regression slope is that it equals zero, meaning no linear relationship.
Which approach can help reduce high variance in a regression model?
Add more predictors without constraints
Ignore regularization
Collect more training data
Increase model complexity
Collecting more data can stabilize estimates and reduce model variance. Increasing complexity or ignoring regularization tends to increase variance.
Nonrandom patterns in a residual vs fitted plot usually indicate a violation of which assumption?
Homoscedasticity
Normality
Independence
Linearity
Systematic patterns (e.g., curvature) in residuals point to a violation of the linearity assumption, suggesting the model form is incorrect.
A model shows an R-squared of 0.92 on training data and 0.45 on test data. What issue is this indicative of?
High variance/overfitting
Multicollinearity
High bias/underfitting
Perfect fit
A large drop in performance from training to test data indicates the model memorizes noise and overfits, a high-variance issue.
In a regression output, the estimated slope is 2.5 with a standard error of 0.5. What is the t-statistic for testing the slope?
2
1
5
0.2
The t-statistic is calculated as estimate divided by its standard error: 2.5 / 0.5 = 5.
A 95% confidence interval for the slope coefficient is reported as [0.1, 1.8]. Based on this interval, which statement is correct at α=0.05?
The slope is not significantly different from zero
The slope is negative
The slope is significantly different from zero
The model fit is poor
Because the 95% CI does not include zero, we conclude the slope is significantly different from zero at the 5% significance level.
A QQ-plot of residuals shows systematic deviation from the 45-degree line, especially in the tails. Which assumption is violated?
Homoscedasticity
Linearity
Normality of residuals
Independence
When residuals deviate from the line in a QQ-plot, it indicates they are not normally distributed, violating the normality assumption.
Which interval is wider for a new observation at a given x value: a prediction interval or a confidence interval for the mean response?
It depends on R-squared
They are equal
Prediction interval
Confidence interval
A prediction interval is wider than the confidence interval for the mean response because it accounts for both parameter uncertainty and individual outcome variability.
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Learning Outcomes

  1. Analyse the strength and direction of linear relationships.
  2. Evaluate model fit using R-squared and residual analysis.
  3. Identify bias and variance issues in simple regression models.
  4. Apply regression coefficients to make data-driven predictions.
  5. Interpret confidence intervals and hypothesis tests for coefficients.

Cheat Sheet

  1. Understanding the Linear Regression Equation - Ever wondered how two variables hang out together on a graph? The equation y = a + bx is your buddy for modeling that relationship, where a is your starting point and b tells you how steep your line climbs. Dive into the core concept and visualize how it all fits! GeeksforGeeks: Linear Regression Formula
  2. Calculating the Slope and Intercept - Think of the slope (b) as the "speed" at which y changes when x takes a step, and the intercept (a) as your launch pad on the y-axis. By summing up products and squares of your data points, you unlock the line of best fit for your dataset. Get hands-on with the formula and watch your numbers align! GeeksforGeeks: Linear Regression Formula
  3. Interpreting the Slope and Intercept - When b is positive, a one-unit jump in x means y jumps up; if negative, it slides down. And a tells you where y sits when x is zero - your starting forecast. Master these interpretations to make sense of any regression line you draw! Scribbr: Simple Linear Regression
  4. Assumptions of Linear Regression - Before you trust your line, check the backstage rules: linearity (straight-line relationship), independence (no sneaky correlations), homoscedasticity (errors stay consistent), and normality of residuals (errors follow a bell curve). Breaking these rules can lead to misleading analyses! Scribbr: Regression Assumptions
  5. Evaluating Model Fit with R-squared - R² tells you how much of y's dance can be explained by x. Score closer to 1? Your model is doing a great job! A low R²? Time to revisit your data or consider extra variables. It's like a scoreboard for your regression game. Scribbr: R-Squared Explained
  6. Residual Analysis for Model Validation - Residuals are the differences between what you observed and what you predicted. Plot them out: any funky patterns or "fanning" of points means you might be violating assumptions. Clean residuals mean a clean, trustworthy model! Scribbr: Residual Analysis Guide
  7. Understanding Bias and Variance in Regression Models - Bias is like consistently aiming a little left of the bullseye; variance is wildly bouncing around it. Too much of either, and your model either oversimplifies or overfits. Balance is the secret sauce for robust predictions. Scribbr: Bias vs Variance
  8. Making Predictions Using Regression Coefficients - Plug any x-value into y = a + bx and voilà - you've got a prediction! This formula turns raw data into actionable forecasts, making you feel like a statistical wizard. Use it wisely to power data-driven decisions. Scribbr: Prediction with Regression
  9. Interpreting Confidence Intervals for Coefficients - A confidence interval gives you a safety net around your estimated a and b. It says, "Hey, I'm pretty sure the true value lies somewhere in this range." Narrow intervals mean precision; wide ones mean "handle with care." Scribbr: Confidence Intervals
  10. Conducting Hypothesis Tests for Regression Coefficients - Is your slope b actually different from zero, or is it just random noise? Hypothesis tests help you find out by comparing your data against a "no-effect" scenario. A significant result? You've got a real relationship on your hands! Scribbr: Regression Hypothesis Testing
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