Lugiai_HW3

.docx

School

George Washington University *

*We aren’t endorsed by this school

Course

1011

Subject

Statistics

Date

Jan 9, 2024

Type

docx

Pages

Uploaded by PrivateScience8970 on coursehero.com

1. In each of the following settings, give a 95% confidence interval for the coefficient of x1. (SE stands for Standard Error) Answer on attached file 2. For each of the settings in the previous exercise, test the null hypothesis that the coefficient of x1 is zero versus the two-sided alternative. Answers on attached file 3. In each of the following situations, explain what is wrong and why. (a) One of the assumptions for multiple regression is that the distribution of each explanatory variable is Normal. This statement is incorrect. Multiple regression does not assume that the distribution of each explanatory variable is Normal. Instead, it assumes that the errors or residuals (the differences between the observed and predicted values) are normally distributed, not the individual explanatory variables. This is known as the Normality of residuals assumption, not the Normality of the explanatory variables. (b) The smaller the P-value for the ANOVA F test, the greater the explanatory power of the model. This statement is incorrect. A smaller P-value for the ANOVA F test indicates that there is evidence that at least one of the explanatory variables in the model has a significant effect on the response variable. However, it does not quantify the strength of the relationship or the explanatory power of the model. The coefficient of determination (R-squared) is a measure of the proportion of variation in the response variable explained by the model, and it provides a better indication of the model's explanatory power. (c)All explanatory variables that are significantly correlated with the response variable will have a statistically significant regression coefficient in the multiple regression model. This statement is incorrect. While it is true that statistically significant correlations between explanatory variables and the response variable are a good indicator of their potential importance in the model, it does not guarantee that the regression coefficients for those variables will be statistically significant. The significance of individual coefficients depends on other factors, such as multicollinearity and the sample size.

(d) The multiple correlation gives the proportion of the variation in the response variable that is explained by the explanatory variables. This statement is generally correct. The multiple correlation coefficient (also known as the multiple R) quantifies the strength and direction of the linear relationship between the response variable and the set of explanatory variables collectively. However, it does not directly give the proportion of variation explained. To get that, you would square the multiple correlation coefficient to obtain the coefficient of determination (R-squared). (e) In a multiple regression with a sample size of 35 and 4 explanatory variables, the test statistic for the null hypothesis H0: b2 = 0 is a t statistic that follows the t (30) distribution when the null hypothesis is true. The statement is partially incorrect as it states that that the test statistic follows a t(30) distribution ONLY when the null hypothesis is true. The test statistic follows a t-distribution with degrees of freedom equal to a specific value, regardless of whether the null hypothesis is true or not. (f) A small P-value for the ANOVA F test implies that all explanatory variables are statistically different from zero. This statement is incorrect. A small P-value for the ANOVA F test indicates that at least one of the explanatory variables in the model has a significant effect on the response variable, but it does not necessarily mean that all of them are statistically different from zero. Some explanatory variables may still have coefficients that are not significantly different from zero, especially if there are multicollinearity issues or if the sample size is small. 4. The corresponding plot if residuals against predicted values y^ is shown. Interpret the plot (pick the right option(s) from below). A) It appears that the data contain an outlier. B) It appears that the variance of ε is not constant. C) It appears that a quadratic model would be a better fit. D) The residuals appear to be randomly scattered so that no model modifications are necessary. This residual plot shows a slight tree pattern, which indicates that the variance of the error is slowly increasing.

4. A regression model was fit and the following residual plot was observed. Which of the following assumptions appears violated based on this plot? A) The variance of the errors is constant B) The errors are independent C) The mean of the errors is zero D) The errors are normally distributed

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Recommended textbooks for you

Calculus For The Life Sciences

Calculus

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:Pearson Addison Wesley,

College Algebra

Algebra

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:Cengage Learning

Algebra and Trigonometry (MindTap Course List)

Algebra

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:Cengage Learning

College Algebra

Algebra

ISBN:9781938168383

Author:Jay Abramson

Publisher:OpenStax

Glencoe Algebra 1, Student Edition, 9780079039897...

Algebra

ISBN:9780079039897

Author:Carter

Publisher:McGraw Hill

Functions and Change: A Modeling Approach to Coll...

Algebra

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Cengage Learning

SEE MORE TEXTBOOKS

Recommended textbooks for you

Calculus For The Life Sciences
Calculus
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:Pearson Addison Wesley,
College Algebra
Algebra
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:Cengage Learning
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:Cengage Learning
College Algebra
Algebra
ISBN:9781938168383
Author:Jay Abramson
Publisher:OpenStax
Glencoe Algebra 1, Student Edition, 9780079039897...
Algebra
ISBN:9780079039897
Author:Carter
Publisher:McGraw Hill
Functions and Change: A Modeling Approach to Coll...
Algebra
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Cengage Learning