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. The summary table below shows the results of a linear regression model for predicting the average birth weight of babies, measured in ounces, from parity.
- Write the equation of the regression line.
\[\widehat{weight} = 120.07 - 1.93 * {parity}\]
- Interpret the slope in this context, and calculate the predicted birth weight of first borns and others.
The estimated body weight of non-first-born is 1.62 oz lower than first born
- Is there a statistically significant relationship between the average birth weight and parity?
since the p-value is high (0.1052), we fail to reject the null hypothesis. The data do not provide strong evidence that there is a difference in weight between first-born and non-first-born babies.
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 sam-pled students in rural New South Wales, Australia, in a particular school year. Below are three observations from this data set.
- Write the equation of the regression line.
\[\widehat{days_{absent}} = 18.93 - 9.11eth + 3.10sex + 2.15lrn\]
- Interpret each one of the slopes in this context.
- The model predicts that non-aboriginal students will have 9.11 less absences.
- The model predicts that males will be absent 3.1 days more than females
- The model predicts that slow learners will be absent 2.15 times more than average learners.
- 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.
eth <- 0
sex <- 1
lrn <- 1
yhat <- 18.93 - 9.11 * eth + 3.1 * sex + 2.15 * lrn
2 - yhat## [1] -22.18- 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 observations in the data set.
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). The table below shows the adjusted R-squared for the model as well as adjusted R-squared values for all models we evaluate in the first step of the backwards elimination process.
Which, if any, variable should be removed from the model first?
The learner status.
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 below 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 that were not damaged.
- Each column of the table above represents a di???erent shuttle mission. Examine these data and describe what you observe with respect to the relationship between temperatures and damaged O-rings.
There appears to show a negative realtionship.
temperature <- c(53,57,58,63,66,67,67,67,68,69,70,70,70,70,72,73,75,75,76,76,78,79,81)
damaged <- c(5,1,1,1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,0,0,0,0)
undamaged <- c(1,5,5,5,6,6,6,6,6,6,5,6,5,6,6,6,6,5,6,6,6,6,6)
ShuttleMission <- data.frame(temperature, damaged, undamaged)
plot(ShuttleMission)
- 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.
the y-intercept is 11.7 and the point estimate is 0.22
- Write out the logistic model using the point estimates of the model parameters.
log(p-hat/ (1 - p-hat)) = 11.6630 - 0.2162 * Temperature
- Based on the model, do you think concerns regarding O-rings are justified? Explain.
Yes. The chance of a dmged o-ring is >40%
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??? 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. See this earlier exercise if you would like to browse the original data.
- 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.
library(ggplot2)
ggplot(ShuttleMission,aes(x=temperature,y=damaged)) + geom_point() +
stat_smooth(method = 'lm', family = 'binomial')## Warning: Ignoring unknown parameters: family
- 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.
Need to check for constant variability of residuals, and residuals are independent of the order they were collected.