hw-11

Homework 11 in STAT400 @ CSU

Assignment

Be sure to set.seed(400) at the beginning of your homework. Please use \(2000\) bootstrap samples in the problems below.

1. Nike has hired you to help analyze their data on their customers who run. They want to make sure that you understand how their running gear fits their customers. A sample of 25 randomly selected customers was selected, and the customers were asked to submit their weights. The data:

     wt <- c(149, 136, 139, 117, 137, 132, 122, 130, 134, 153, 140, 151, 203, 143, 145, 123, 127, 146, 139, 118, 143, 123, 133, 153, 136)

   1. Calculate the sample standard deviation \(s\) for these weights.
   2. To do the following, use the boot and simpleboot packages as shown in the class handouts.
      1. Compute the bootstrap bias and standard error for \(s\).
      2. Plot a histogram and qq-plot of the bootstrap distribution.
      3. Based on these results: (1) Is there evidence of bias and skewness of the bootstrap distribution for \(s\)? (2) Is it appropriate to assume that the distribution of \(s\) is normally distributed?
      4. Construct 4 types of intervals that we discussed in class by using type=c("norm","basic", "perc", "bca") in the boot.ci command
      5. Plot the four intervals onto a histogram of the sampling distribution using the command geom_segment.
   3. Construct a “studentized” bootstrap \(t\) CI and also plot it onto your histogram of the sampling distribution.
   4. What final result would you report to Nike? Explain your reasoning.

2. For the Verizon dataset from the class handouts construct a 95% CI for the difference of two medians. Construct 4 types of intervals that we discussed in class by using type=c("norm","basic", "perc", "bca") in the boot.ci command.

   1. Are the intervals similar for all the methods? Why or why not?
   2. Let \(\tilde{\mu}_1\) = the population median repair time for ILEC customers and \(\tilde{\mu}_2\) = the population median of repair time for CLEC customers. Based on the results of the BCa interval, would you reject this hypothesis? Explain your answer. \[ H_0: \tilde{\mu}_1 = \tilde{\mu}_2 \\ H_a: \tilde{\mu}_1 \not= \tilde{\mu}_2 \\ \]

3. This data set is the cars data in R. The goal is to create a regression model about the relationship between stopping distance (dist) and speed (speed) in cars.

   1. Create a scatter plot of speed vs. dist in mpg and describe the relationship.
   2. Fit a simple linear model of dist on speed. Describe the result, including diagnostic plots, and create 95% CIs for the coefficients.
   3. Perform the paired bootstrap and compute the bootstrap bias and standard error for the coefficients. Construct 4 types of intervals that we discussed in class by using type=c("norm","basic", "perc", "bca") in the boot.ci command.
   4. Perform the bootstrap using the residuals and compute the bootstrap bias and standard error for the coefficients. Construct 4 types of intervals that we discussed in class by using type=c("norm","basic", "perc", "bca") in the boot.ci command.
   5. Which method (simple linear regression, paired bootstrap, or bootstrap using the residuals) would you use for this dataset and why?

Turn in in a pdf of your homework to canvas. Your .Rmd file on rstudio.cloud will also be used in grading, so be sure they are identical and reproducible.