question exercise

In Chapter 4, we used logistic regression to predict the probability of
default using income and balance on the Default data set. We will
now estimate the test error of this logistic regression model using the
validation set approach. Do not forget to set a random seed before
beginning your analysis.

(a) Fit a logistic regression model that uses income and balance to
predict default.

(b) Using the validation set approach, estimate the test error of this
model. In order to do this, you must perform the following steps:

i. Split the sample set into a training set and a validation set.

ii. Fit a multiple logistic regression model using only the training observations.

iii. Obtain a prediction of default status for each individual in
the validation set by computing the posterior probability of
default for that individual, and classifying the individual to
the default category if the posterior probability is greater
than 0.5.
iv. Compute the validation set error, which is the fraction of
the observations in the validation set that are misclassified.

(c) Repeat the process in (b) three times, using three different splits
of the observations into a training set and a validation set. Comment on the results obtained.

(d) Now consider a logistic regression model that predicts the probability of default using income, balance, and a dummy variable
for student. Estimate the test error for this model using the validation set approach. Comment on whether or not including a
dummy variable for student leads to a reduction in the test error
rate.

2) We will now consider the Boston housing data set, from the MASS
library.

(a) Based on this data set, provide an estimate for the population
mean of medv. Call this estimate ˆ?.

(b) Provide an estimate of the standard error of ˆ?. Interpret this
result.
Hint: We can compute the standard error of the sample mean by
dividing the sample standard deviation by the square root of the
number of observations.

(c) Now estimate the standard error of ˆ? using the bootstrap. How
does this compare to your answer from (b)?

(d) Based on your bootstrap estimate from (c), provide a 95 % confidence interval for the mean of medv. Compare it to the results
obtained using t.test(Boston$medv).
Hint: You can approximate a 95 % confidence interval using the
formula [ˆ? ? 2SE(ˆ?), ?ˆ + 2SE(ˆ?)].

be aware to use R or R studio