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We Helped With This Statistics with R Programming Homework: Have A Similar One?
|Subject||R | R Studio|
|More Info||Assignment Statistics|
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
β1 = 0 takes the form
T= B1 − β1 / S/ √ Sxx,
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
. (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?