Master Simple Linear Regression: A Comprehensive Guide for PhD Students

Question 1

Which formula calculates the least squares estimator of the slope parameter in simple linear regression?

Accepted Answer

b = Σ((x - x̄)(y - ȳ)) / Σ((x - x̄)²))

Answer

b = Σ(xy) / Σ(x²)

Answer

b = (ȳ - x̄) / (Σ(xy) / Σ(x²))

Answer

b = (x̄ - ȳ) / (Σ(xy) / Σ(x²))

Question 2

In simple linear regression, which measure indicates the proportion of variation in the dependent variable explained by the independent variable?

Accepted Answer

R-squared

Answer

Adjusted R-squared

Answer

F-statistic

Answer

Durbin-Watson statistic

Question 3

What is the null hypothesis for a t-test of the slope parameter in simple linear regression?

Accepted Answer

H₀: β = 0

Answer

H₀: β ≠ 0

Answer

H₀: β > 0

Answer

H₀: β < 0

Question 4

The Durbin-Watson statistic is used to detect:

Accepted Answer

Autocorrelation

Answer

Heteroskedasticity

Answer

Multicollinearity

Answer

Normality

Question 5

In simple linear regression, the model is Y = β0 + β1X + ε. How do the sign and magnitude of the slope parameter (β1) relate to the direction and strength of the relationship between X and Y?

Accepted Answer

The sign of β1 indicates the direction, and the magnitude indicates the strength.

Answer

The magnitude of β1 indicates the direction, and the sign indicates the strength.

Answer

Both the sign and magnitude of β1 indicate the direction.

Question 6

Which of the following is NOT a fundamental concept in simple linear regression?

Accepted Answer

Hypothesis testing

Answer

Model specification

Answer

Parameter estimation

Answer

Goodness-of-fit measures

Question 7

Identify the correct equation for a simple linear regression model:

Accepted Answer

y = b0 + b1x

Answer

y = b1x + b0

Answer

y = b0 + b1x^2

Answer

y = b1x^2 + b0

Question 8

Which method is employed to estimate parameters in simple linear regression?

Accepted Answer

Method of least squares

Answer

Maximum likelihood estimation

Answer

Bayesian inference

Answer

Monte Carlo simulation

Question 9

Define the coefficient of determination (R-squared) in simple linear regression:

Accepted Answer

The proportion of variance in the dependent variable explained by the independent variable

Answer

The standard deviation of the residuals

Answer

The slope of the regression line

Answer

The intercept of the regression line

Question 10

What is the primary purpose of conducting hypothesis testing in simple linear regression?

Accepted Answer

To determine whether the slope of the regression line is significantly different from zero

Answer

To estimate the standard deviation of the residuals

Answer

To calculate the coefficient of determination

Answer

To make predictions about the dependent variable

Question 11

What is the purpose of employing regularization techniques in simple linear regression?

Accepted Answer

To mitigate overfitting and improve predictive performance

Answer

To increase the complexity of the model

Answer

To enhance the interpretability of the model

Answer

To reduce the number of features in the model

Question 12

What does the y-intercept of the regression line in simple linear regression represent?

Accepted Answer

The predicted value of the response variable when the predictor variable is equal to zero

Answer

The average value of the response variable

Answer

The difference between the observed and predicted values of the response variable

Answer

The slope of the regression line

Question 13

What is the purpose of conducting a residual analysis in simple linear regression?

Accepted Answer

To identify potential violations of model assumptions

Answer

To improve the accuracy of parameter estimates

Answer

To test the significance of the regression coefficients

Answer

To make predictions for new data points

Question 14

What is the critical value in hypothesis testing?

Accepted Answer

The value of the test statistic that determines the rejection region for a given significance level

Answer

The value of the test statistic calculated from the sample data

Answer

The probability of rejecting the null hypothesis when it is true

Answer

The power of the test

Question 15

What is the purpose of the least squares method in simple linear regression?

Accepted Answer

To find the line of best fit that minimizes the sum of squared residuals

Answer

To find the mean of the response variable

Answer

To find the standard deviation of the response variable

Answer

To find the correlation coefficient

Question 16

What is the interpretation of the slope coefficient in a simple linear regression model?

Accepted Answer

The change in the response variable for a one-unit increase in the predictor variable, holding other variables constant

Answer

The mean of the predictor variable

Answer

The standard deviation of the predictor variable

Answer

The correlation coefficient between the predictor and response variables

Question 17

When is it appropriate to use a simple linear regression model?

Accepted Answer

When the relationship between the predictor and response variables is approximately linear

Answer

When the predictor variable is categorical

Answer

When the response variable is not normally distributed

Answer

When there is a non-linear relationship between the predictor and response variables

Question 18

What is the difference between a simple linear regression model and a multiple linear regression model?

Accepted Answer

Simple linear regression has one predictor variable, while multiple linear regression has two or more predictor variables

Answer

Simple linear regression assumes a linear relationship, while multiple linear regression does not

Answer

Simple linear regression is used for prediction, while multiple linear regression is used for explanation

Answer

There is no difference between a simple linear regression model and a multiple linear regression model

Question 19

The equation of a simple linear regression model is:

Accepted Answer

y = mx + b

Answer

y = mx - b

Answer

y = bx + m

Answer

y = b - mx

Question 20

The assumption of linearity in simple linear regression means that:

Accepted Answer

The relationship between x and y is linear

Answer

The residuals are normally distributed

Answer

The variance of the residuals is constant

Answer

The data points are independent

Question 21

A potential limitation of simple linear regression is that it:

Accepted Answer

Can only model linear relationships

Answer

Assumes a normal distribution of residuals

Answer

Requires a large sample size

Answer

Is not sensitive to outliers

Question 22

The difference between a population regression line and a sample regression line is that:

Accepted Answer

The population regression line represents the true relationship, while the sample regression line is an estimate

Answer

The sample regression line represents the true relationship, while the population regression line is an estimate

Answer

There is no difference between them

Answer

The population regression line is linear, while the sample regression line may not be

Question 23

To test the significance of the slope coefficient in a simple linear regression model, you can use:

Accepted Answer

t-test

Answer

F-test

Answer

chi-square test

Answer

regression analysis

Question 24

The coefficient of determination in a simple linear regression model measures:

Accepted Answer

The proportion of variance in y explained by x

Answer

The strength of the linear relationship between x and y

Answer

The predicted value of y for a given value of x

Answer

The significance of the slope coefficient

Question 25

What is the slope of the least squares regression line?

Accepted Answer

The average change in the dependent variable for a one-unit increase in the independent variable

Answer

The correlation coefficient between the independent and dependent variables

Answer

The y-intercept of the regression line

Answer

The standard deviation of the residuals