We Helped With This Statistics with R Programming Homework: Have A Similar One?

1. This problem is a follow up on problem 1 of assignment 5 about the regression through the origin model of the form

Y =β₁x+ε, where ε is the standard normal distribution.

.      (a)  First, generate a predictor x as a random vector of length n = 100 from the standard normal and a response y as follows. 
y = 2x + ε


where ε is also a random vector of length n = 100 from the standard normal 


that takes exactly the same values as x. 


.            (b)  Perform a simple linear regression of y onto x, without an intercept.

Report the coefficient estimate βˆ₁, the standard error of this coefficient estimate, and the t-statistic and p-value associated with the null hypothesis H₀ : β₁ = 0. Comment on these results. (You can perform regression without an intercept using the command lm(y∼ x + 0).) 


.      (c)  Now perform a simple linear regression of x onto y without an intercept. Report the coefficient estimate βˆ₁, the standard error of this coefficient estimate, and the t-statistic and p-value associated with the null hypothesis H₀ : β₁ = 0. Comment on these results. 


.      (d)  What are the relationships between (a), the results obtained in (b) and the ones in (c)? 


.           (e)  For the regression of Y onto X without an intercept, the t-statistic for

H₀ : 


β₁ = 0 takes the form

T= B₁ − β₁ / S/ √ S_xx,

where


S²= S S E / n-1    and

Note that these formulas are slightly different from those given in (slide 29, LinearRegression-1), since here we are performing regression without an in- tercept.

(f) Show algebraically (to receive bonus points), and confirm numerically in R, that the t-statistic above can be written as (values)

.         (g)  Using the results from (d), argue that the t-statistic for the regression of y onto 


x is the same as the t-statistic for the regression of x onto y. 


.      (h)  In R, show that when regression is performed with an intercept, the tstatistic forH₀ :β₁ =0 is the same for the regression of y onto x as it is for the regression of x onto y. 


4. The data set low bwt.txt contains information for a sample of 100 low birth weight infants. The variables are

sbp : maternal systolic blood pressure
sex : gender of the baby


toxemia : toxemia during pregnancy (yes or no) germ.hem : germinal matrix hemorrhage (yes or no) gest.age : gestational age in weeks
apgar5 : five-minute APGAR score

.      (a)  Using germinal matrix hemorrhage as the response, fit a logistic regression model where the predictor variable x₁ is the 5-minute APGAR score. Write the equation and interpret β₁, the estimated coefficient of

Apgar score. 


.         (b)  What is the estimate and 95% confidence interval for the slope

(coefficient for apgar5) in the odds ratio scale? Interpret the estimate

(what does the odds ratio mean?). 


.      (c)  At the 0.05 level of significance, test the null hypothesis: H₀ : β₁ = 0 where β₁ is the coefficient for apgar5. 


.      (d)  If a new infant from the population has an APGAR score of 3 what is the predicted probability that this child will experience a brain hemorrhage? What is the probability if the child’s score is 7?