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.

  1. Write the equation of the regression line. From the table, we know intercept is 120.07 and parity is -1.93 \[ y=120.07-1.93x \]

  2. Interpret the slope in this context, and calculate the predicted birth weight of babies born to smoker and non-smoker mothers. For first born

x<-1
y<-120.07-1.93*x
y
## [1] 118.14

For second born

x<-0
y<-120.07-1.93*x
y
## [1] 120.07
  1. Is there a statistically significant relationship between the average birth weight and smoking? Ho: B1=0 Ha: B2 not 0 Choose alpha at 0.05 We can gather from the table, the p value is 0.1052. The p value is greater than the alpha, hence there is no significant relationship.

8.4 Absenteeism. 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 SouthWales, Australia, in a particular school year. Below are three observations from this data set.

  1. Write the equation of the regression line. \[ y=18.93-9.11{ x }_{ 1 }+3.10{ x }_{ 2 }+2.15{ x }_{ 3 }y \]

y=days x1=eth x2=sex x3=lm

  1. Interpret each one of the slopes in this context. B1 is the estimated number of days on males is higher than the estimated number of days on their female counter parts B2 estimates the number of days for males is higher than the number of days for females B3 says that the estimated number of days on someone with learner status is lower than the number of days for an average learner

  2. 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.

#Lets change the variables to make it easier 
x<-0
y<-1
z<-1
y<-18.93-(9.11*x)+(3.10*y)+(2.15*z)
y
## [1] 24.18
residual<-2-y
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.
R2 <- 1 - (240.57)/(264.17)
R2.Adj <- 1 - ( (240.57/264.17)*((146-1)/(146-3-1)) )
R2;R2.Adj
## [1] 0.08933641
## [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). 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? Based on the adjusted R square values, the no learner status predictor should be removed 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 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.

  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 -The columns represent a shuttle mission. There were 23 shuttle missions total (23 columns) . As for the relationship between temperatures and damaged O-Rings, the temperatures marked as coldest had damanged O rings more than higher temperatures. We can infer that the colder the temperature, the more likely the O ring would be damaged.

  2. 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 table provides the results of a logistic regression. The ouput is almost similar to that of a regular regression, with the exception that the response variable is a binary operator (1 or 0). The coefficients can be exponentiated to produce an odds ratio.

  3. Write out the logistic model using the point estimates of the model parameters. $$ log( )=11.6630-0.2162(temp)

$$

  1. Based on the model, do you think concerns regarding O-rings are justified? Explain. -The low p value seems to suggest that the O ring readings are justified. The low p value is evidence of statistical significance.

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.

  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.
temp1 <- 51
logit1 <- 11.6630 - 0.2162 * temp1
o.ring1 <- exp(logit1)/(1 + exp(logit1))
o.ring1
## [1] 0.6540297
temp2 <- 53
logit2 <- 11.6630 - 0.2162 * temp2
o.ring2 <- exp(logit2)/(1 + exp(logit2))
o.ring2
## [1] 0.5509228
temp3 <- 55
logit3 <- 11.6630 - 0.2162 * temp3
o.ring3 <- exp(logit3)/(1 + exp(logit3))
o.ring3
## [1] 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.
temp.x <- seq(from = 51, to = 71, by = 2)
y <- c(round(o.ring1, 4), round(o.ring2, 4), round(o.ring3, 4), 0.341, 0.251, 0.179, 0.124, 0.084, 0.056, 0.037, 0.024)
plot(temp.x, y, type = "o", col = "blue")

  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. Based on the number of observations, it might be difficult to check if x has a linear relationship to the logit when predictors held constant. We might also has difficulty checking independence of outcomes.