8.2
(a) baby_weight =  120.07 - 1.93 * parity
(b) The estimated body weight of non first born babies is 1.93 ounces
    lower than first born babies.

(first_born <- 120.07 - 1.93 * 0)

## [1] 120.07

(non_first_born <- 120.07 - 1.93 * 1)

## [1] 118.14

(c) H0: b1 = 0 
    HA: b1 not equal to 0
    p value from the table is 0.1052
    p > 0.05, we cannot reject null hypothesis.
    The data provide strong evidence that the true slope parameter is 0 and that
    there is no assosciation between parity and birth weight.

8.4
(a) number_of_days_absent = 18.93 - 9.11 * eth + 3.10 * sex + 2.15 * lrn
(b) b_eth : The model predicts 9.11 days decrease in the predicted absenteeism
    when subject is aboriginal, all else held constant.
    b_sex : The model predicts 3.10 days increase in the predicted absenteeism
    when subject is male, all else held constant.
    b_lrn : The model predicts 2.15 days increase in the predicted absenteeism
    when subject is slow learner, all else held constant.
(c) The model over predicts this subject's absenteeism.   

(number_of_days_absent <- 18.93 - 9.11 * 0 + 3.10 * 1 + 2.15 * 1)

## [1] 24.18

(e <- 2 - number_of_days_absent)

## [1] -22.18

(d)

(r_sq <- 1 - (240.57 / 264.17))

## [1] 0.08933641

(r_sq_adjusted <- 1 - (240.57 / 264.17) * (146 -1) / (146 - 3 -1))

## [1] 0.07009704

8.8 Remove lrn (learner status)

8.16
(a) Damaged O-rings count is inversely proportional to the temperature.
(b) Intercept and coefficient of temperature define the model. z and p value 
    care about understanding of which variables are statistically significant 
    predictors of the response, or if there is interest in producing a simpler 
    model at the potential cost of a little prediction accuracy.
(c) log(e)(p / (1 - 1p)) = 11.6630 - 0.2162 * Temperature
(d) Since the coefficient of tempearature is -ve, the concerns regarding 
    O-rings are justified(inverse relationship).

8.18
(a)

p <- function(temp) {
  exponent <- 11.6630 - 0.2162 * temp
  round(exp(exponent) / (1 + exp(exponent)), 3) 
}
p(51)

## [1] 0.654

p(53)

## [1] 0.551

p(55)

## [1] 0.443

(b)

temp <- seq(from = 51, to = 71, by = 2)
model_predictions <- unlist(lapply(temp, p))
plot(temp, model_predictions, type = "o", col = "blue")

(c) Logistic Regression requires that we fulfill their two key conditions for 
creating this model.
 1. Each predictor x is linearly related to logit p if all other predictors are 
 held constant.
 2. Each outcome Y is independent of the other outcomes.
 Both conditions are difficult to verify. There has only been 23 shuttle 
 missions, which may not be enough of a sample size to adequate see if the first 
 criteria can be satisfied. And for number 2, shuttles are very complicated. It 
 is unclear that the O ring is independent of other outcomes. This needs further 
 investigation. Therefore, we may have difficulty assuming that this logistic 
 regression can be used with the information that we have right now.