Correlation Coefficients and Linear Association Quiz

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Questions and Answers

Which variable is specified to be the dependent/response variable in regression analysis?

  • Dependent variable (correct)
  • Predictor variable
  • Independent variable
  • Explanatory variable

What is the main purpose of regression analysis?

  • To predict the value of the dependent variable
  • To draw inferences about the relationship between variables
  • To determine the strength of the relationship between variables
  • To examine the functional relationships between variables (correct)

In which situation are both the independent variable(s) and the dependent variable random variables?

  • When examining the functional relationships between variables
  • When using survey data (correct)
  • When selecting cases randomly from the population
  • When predicting the value of the dependent variable

What can be used to predict the value of the dependent variable in regression analysis?

<p>Prediction intervals (C)</p> Signup and view all the answers

Which of the following is a property of the fitted regression line?

<p>All of the above (D)</p> Signup and view all the answers

What is the formula for the estimated regression line?

<p>Y = b0 + b1X (A)</p> Signup and view all the answers

What is the assumption made about the error terms in regression analysis?

<p>The error terms are normally distributed (A)</p> Signup and view all the answers

What is the formula for the confidence interval of the regression coefficient β1?

<p>b1 - tα/2s(b1) &lt; β1 &lt; b1 + tα/2s(b1) (C)</p> Signup and view all the answers

Which of the following statements is true about a negative correlation?

<p>As one variable increases, the other variable decreases (B)</p> Signup and view all the answers

What does a correlation coefficient of 0 indicate?

<p>There is no linear relationship between the variables (D)</p> Signup and view all the answers

What does the sample correlation coefficient (r) estimate?

<p>The population correlation coefficient (ρ) (A)</p> Signup and view all the answers

What is the purpose of conducting a t-test for correlation?

<p>To test if there is a linear relationship between two variables (C)</p> Signup and view all the answers

Which of the following is the correct equation for the simple linear regression model?

<p>$Y = \beta_0 + \beta_1X + \epsilon$ (B)</p> Signup and view all the answers

What is the role of the random error term in the regression model?

<p>To represent the sum of two components (B)</p> Signup and view all the answers

What is the least squares method used for in regression analysis?

<p>To estimate the sample parameters (D)</p> Signup and view all the answers

What is the Gauss-Markov theorem?

<p>The least squares estimators are unbiased and have minimum variance (D)</p> Signup and view all the answers

Which of the following formulas can be used to calculate the coefficient of determination, R2?

<p>R2 = SSR / SS(Total) (A)</p> Signup and view all the answers

What is the purpose of constructing a regression equation?

<p>To obtain a tool for predicting the value of the dependent variable (D)</p> Signup and view all the answers

Which of the following is a measure of how well the regression line fits the sample?

<p>Coefficient of determination (B)</p> Signup and view all the answers

What is the range of values that the correlation coefficient, ρ, can take?

<p>-1 to 1 (C)</p> Signup and view all the answers

Which distribution can be used to establish a critical region for testing the null hypothesis H0: β0 = 0 against a suitable alternative?

<p>t distribution (C)</p> Signup and view all the answers

Which statement is true about the sampling distributions of b0 and b1 in the presence of departures from normality?

<p>They will be approximately normal (D)</p> Signup and view all the answers

What is the difference between a confidence interval and a prediction interval?

<p>A confidence interval is an estimate of the mean response, while a prediction interval is a statement about the value of a new observation (B)</p> Signup and view all the answers

Which assumption of the simple linear regression model is violated if the spread of the residuals increases or decreases with the values of the independent variable or with the predicted values?

<p>Assumption of homogeneity of variance (D)</p> Signup and view all the answers

In time series data or repeated measures data, what problem often occurs in relation to the dependence between successive error terms?

<p>Violation of the assumption of independence (B)</p> Signup and view all the answers

What is the purpose of plotting the observed distribution of residuals against the expected distribution under the assumption of normality?

<p>To compare the two distributions (B)</p> Signup and view all the answers

What should be observed in the scatter plot of a simple linear regression if the assumptions of linearity and homogeneity of variance are met?

<p>A random distribution of residuals around the horizontal line through zero (C)</p> Signup and view all the answers

Which assumption of the simple linear regression model assumes that the error terms associated with each xi have a normal distribution?

<p>Assumption of normality (B)</p> Signup and view all the answers

What assumption of the simple linear regression model is violated if there is a clear pattern observed in the scatter plot of the residuals against the predicted values?

<p>Assumption of linearity (A)</p> Signup and view all the answers

What is the main purpose of regression analysis?

<p>To examine the functional relationships between two or more variables (D)</p> Signup and view all the answers

Which assumption of the simple linear regression model assumes that the error terms associated with each Xi have a normal distribution?

<p>Normality assumption (D)</p> Signup and view all the answers

What is the formula for the estimated regression line?

<p>$Y = \beta_0 + \beta_1X$ (D)</p> Signup and view all the answers

What can be used to predict the value of the dependent variable in regression analysis?

<p>The estimated regression line (C)</p> Signup and view all the answers

Which of the following statements is true about the correlation coefficient (ρ)?

<p>Variables may be perfectly correlated in a curvilinear relationship, even if ρ = 0. (A)</p> Signup and view all the answers

What is the purpose of conducting a t-test for correlation?

<p>To test whether there is a significant linear relationship between two variables. (B)</p> Signup and view all the answers

What assumption is violated if there is a clear pattern observed in the scatter plot of the residuals against the predicted values?

<p>Homoscedasticity assumption (B)</p> Signup and view all the answers

What is the formula for the sample correlation coefficient (r)?

<p>r = SSxy/ [(SSxx)(SSyy)]1/2 (D)</p> Signup and view all the answers

Which of the following is the correct equation for the simple linear regression model?

<p>Y = β0 + β1X + ε (B)</p> Signup and view all the answers

What is the purpose of fitting a line to a sample of points in regression analysis?

<p>To estimate the parameters β0 and β1 (B)</p> Signup and view all the answers

What is the formula for the least squares estimators b0 and b1 in the simple linear regression model?

<p>$b_1 = SS_{xy} / SS_{xx}, b_0 = \bar{Y} -b_1 \bar{X}$ (B)</p> Signup and view all the answers

Which of the following is a property of the least squares estimators in the regression model?

<p>They are unbiased estimators (A)</p> Signup and view all the answers

Which of the following is the correct formula for the coefficient of determination, R2?

<p>$R^2 = 1 - SSE / SST$ (A)</p> Signup and view all the answers

What is the purpose of the ANOVA Table in regression analysis?

<p>To test the significance of the whole set of independent variables (A)</p> Signup and view all the answers

What is the formula for the F-statistic in the ANOVA Table for simple linear regression?

<p>$F = (n-2)R^2 / (1-R^2)$ (A)</p> Signup and view all the answers

What is the purpose of constructing a scatter plot in regression analysis?

<p>To test the assumptions of linearity and homogeneity of variance (D)</p> Signup and view all the answers

Which assumption of the simple linear regression model is violated if there is a clear pattern observed in the scatter plot of the residuals against the predicted values?

<p>Assumption of linearity (C)</p> Signup and view all the answers

What is the purpose of plotting the observed distribution of residuals against the expected distribution under the assumption of normality?

<p>To determine if the residuals are normally distributed (D)</p> Signup and view all the answers

What is the role of the random error term in the regression model?

<p>To account for sampling variation (C)</p> Signup and view all the answers

What is the formula for the estimated regression line?

<p>$y = \beta0 + \beta1x$ (C)</p> Signup and view all the answers

What is the purpose of conducting a t-test for correlation?

<p>To determine if there is a significant linear relationship between two variables (A)</p> Signup and view all the answers

What can be used to predict the value of the dependent variable in regression analysis?

<p>The estimated regression line (B)</p> Signup and view all the answers

Which of the following is a property of the least squares method (LSM) in regression analysis?

<p>The LSM provides unbiased point estimators of the population parameters. (C)</p> Signup and view all the answers

What is the formula for the standard error of the regression coefficient β1?

<p>$s(b1) = \sqrt{\frac{s^2},{SS_{xx}}}$ (C)</p> Signup and view all the answers

What is the purpose of conducting a t-test for the regression coefficient β1?

<p>To test the null hypothesis that β1 = 0 against a suitable alternative. (B)</p> Signup and view all the answers

What is the formula for the confidence interval of the regression coefficient β0?

<p>$b0 - t_{\alpha/2}s(b0) &lt; β0 &lt; b0 + t_{\alpha/2}s(b0)$ (C)</p> Signup and view all the answers

Which distribution is used to establish a critical region for testing the null hypothesis $H_0: \beta_0 = 0$ against a suitable alternative?

<p>t distribution (A)</p> Signup and view all the answers

What is the (1 - α) 100% confidence interval for the mean response $E(Y_i)$?

<p>$i - t_{\alpha/2, (n-2)} s(i) &lt; E(Y_i) &lt; i + t_{\alpha/2, (n-2)} s(i)$ (C)</p> Signup and view all the answers

What is the formula for the (1 - α) 100% prediction interval for a new observation $Y_i(new)$?

<p>$i - t_{\alpha/2, (n-2)} s(pred_i) &lt; Y_i(new) &lt; i + t_{\alpha/2, (n-2)} s(pred_i)$ (C)</p> Signup and view all the answers

What is the formula for the unbiased estimator of the variance of $i$, the error term in the regression model?

<p>$var(pred_i) = s^2 + s^2(i) = s^2[1+ (1/n) + (X_i - \overline{X})^2/SS_{xx}]$ (B)</p> Signup and view all the answers

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