The data set lowbwt contains information for the sample of 100 low birth weight infants born in Boston, Massachusetts [81]. The variable grmhem is a dichotomous random variable indicating whether an infant experienced a germinal matrix hemorrhage. The value 1 designates that a hemorrhage occurred and 0 that it did not. The infants' five-minute apgar scores are saved under the name apgars, and indicators of preeclampsia - where 1 represents a diagnosis of preeclampsia during pregnancy for the child's mother and 0 no such diagnosis - under the variable name preeclampsia.

(a) Using germinal matrix hemorrhage as the response, fit a logistic regression model of the form

where $x_1$ is five-minute apgar score. Interpret $\beta_1$, the estimated coefficient of apgar score.

(b) If a particular child has a five-minute 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?

(c) At the 0.

05 level of significance, test the null hypothesis that the population parameter $\beta_1$ is equal to 0.

What do you conclude?

(d) Now fit the regression model

where $x_2$ represents preeclampsia status. Interpret $\beta_2$, the estimated coefficient of preeclampsia.

(e) For a child whose mother was diagnosed with preeclampsia during pregnancy, what is the predicted probability of experiencing a germinal matrix hemorrhage? What is the probability for a child whose mother was not diagnosed?

(f) What are the estimated odds of suffering a germinal matrix hemorrhage for children whose mothers were diagnosed with preeclampsia relative to children whose mothers were not?

Construct a 95% confidence interval for the population odds ratio associated with preeclampsia status. Does this interval contain the value 1? What does this tell you?

In 1-p. = [] Bo + B1x1