library(DATA606)

8.2 Baby weights, Part II.

Exercise 8.1 introduces a data set on birth weight of babies. Another variable we consider is parity, which is 0 if the child is the first born, and 1 otherwise.

  1. Write the equation of the regression line.

bweight = 120.07 - 1.93 * parity

  1. Interpret the slope in this context, and calculate the predicted birth weight of first borns and others.

First born babies weighs 1.93 ounces heavier than non-first borns

_First born weight = 120.07_

_Not first born weight = 120.07 - 19.3 = 118.14_
  1. Is there a statistically significant relationship between the average birth weight and parity?

Since the p-value of 0.1052 is greater than 0.05, we cannot reject the null hypothesis and conclude that there is no significant relationship between weight and parity

8.4 Absenteeism, Part I.

Researchers interested in the relationship between absenteeism from school and certain demographic characteristics of children collected data from 146 randomly sampled students in rural New South Wales, Australia, in a particular school year.

The summary table below shows the results of a linear regression model for predicting the average number of days absent based on ethnic background (eth: 0 - aboriginal, 1 - not aboriginal), sex (sex: 0 - female, 1 - male), and learner status (lrn: 0 - average learner, 1 - slow learner).

  1. Write the equation of the regression line.

days_absent = 18.93 - 9.11 * eth + 3.10 * sex + 2.15 * lrn

  1. Interpret each one of the slopes in this context.

The ethnic background slope predict that a non-aboriginal student would have a higher average number of days absent

The sex slope predict that a male student would have a higher average number of days absent

The learner status slope predict that a slow learner student would have a higher average number of days absent

  1. Calculate the residual for the first observation in the data set: a student who is aboriginal,male, a slow learner, and missed 2 days of school.
days_absent <- 18.93 - (9.11 * 0) + (3.10 * 1) + (2.15 * 1); days_absent
## [1] 24.18
residual <- 2 - days_absent; residual
## [1] -22.18
  1. The variance of the residuals is 240.57, and the variance of the number of absent days for all students in the data set is 264.17. Calculate the R2 and the adjusted R2. Note that there are 146 observations in the data set.
varE <- 240.57
varY <- 264.17
n <- 146
k <- 3

R2 <- 1 - (varE / varY); R2
## [1] 0.08933641
R2adj <- 1 - ((varE / varY) * (n-1) / (n-k-1)); R2adj
## [1] 0.07009704

8.8 Absenteeism, Part II.

Exercise 8.4 considers a model that predicts the number of days absent using three predictors: ethnic background (eth), gender (sex), and learner status (lrn). Which, if any, variable should be removed from the model first?

Having the highest adjusted R2 of 0.0723, learner statu (lrn) should be removed from the model first

8.16 Challenger disaster, Part I.

On January 28, 1986, a routine launch was anticipated for the Challenger space shuttle. Seventy-three seconds into the flight, disaster happened: the shuttle broke apart, killing all seven crew members on board. An investigation into the cause of the disaster focused on a critical seal called an O-ring, and it is believed that damage to these O-rings during a shuttle launch may be related to the ambient temperature during the launch. The table summarizes observational data on O-rings for 23 shuttle missions, where the mission order is based on the temperature at the time of the launch. Temp gives the temperature in Fahrenheit,Damaged represents the number of damaged O-rings, and Undamaged represents the number of O-rings.

  1. Each column of the table above represents a different shuttle mission. Examine these data and describe what you observe with respect to the relationship between temperatures and damaged O-rings.

It appears that of the 23 shuttle missions, the one with lowest temperature had the most number of damaged O-rings

  1. Failures have been coded as 1 for a damaged O-ring and 0 for an undamaged O-ring, and a logistic regression model was fit to these data. A summary of this model is given below. Describe the key components of this summary table in words.

An intercept of 11.6630 suggests the probability of damaged O-rings at 0 temperature. The negative temperature coefficient suggests that an increase of 1 degree in temperature reduces the number of damaged O-rings by 0.2162

  1. Write out the logistic model using the point estimates of the model parameters.

log(p/(1-p)) = 116630 - 0.2162 * temperature

  1. Based on the model, do you think concerns regarding O-rings are justified? Explain.

Yes, ambient temperature contributes to the number of damaged O-rings and critical to the success/failure of shutlle mission launch

8.18 Challenger disaster, Part II.

Exercise 8.16 introduced us to O-rings that were identified as a plausible explanation for the breakup of the Challenger space shuttle 73 seconds into takeo off in 1986. The investigation found that the ambient temperature at the time of the shuttle launch was closely related to the damage of O-rings, which are a critical component of the shuttle.

  1. Use the model to calculate the probability that an O-ring will become damaged at each of the following ambient temperatures: 51, 53, and 55 degrees Fahrenheit.
c_temp <- c(51,53,55)
damaged_prob <- exp(11.6630-(0.2162*c_temp))/(1+exp(11.6630-(0.2162*c_temp)))
damaged_prob
## [1] 0.6540297 0.5509228 0.4432456
  1. Add the model-estimated probabilities from part (a) on the plot, then connect these dots using a smooth curve to represent the model-estimated probabilities.
temps <- seq(from = 51, to = 81)
predicted_prob <- exp(11.6630-(0.2162*temps))/(1+exp(11.6630-(0.2162*temps)))
dtemp <- as.data.frame(cbind(temps, predicted_prob))
plot(dtemp$temps, dtemp$predicted_prob)

  1. Describe any concerns you may have regarding applying logistic regression in this application, and note any assumptions that are required to accept the model’s validity.