Question

2) Airline accidents: According to the U.S. National Transportation Safety Board, the number of airline accidents by year from 1983 to 2006 were 23, 16, 21, 24, 34, 30, 28, 24, 26, 18, 23, 23, 36, 37, 49, 50, 51, 56, 46, 41, 54, 30, 40, and 31.

a. For the sample data, compute the mean and its standard error (from the standard deviation), and the median.

b. Using R, compute bootstrap estimates of the mean, median and 25% trimmed mean with estimates of their standard errors, using B = 1000 resamples. (Use set.seed(1) to allow your results to be reproduced.) c.

Compare parts a and b. How do the estimates compare?

3) Consider a population that has a normal distribution with mean µ = 36, standard deviation σ = 8. a. The sampling distribution of ?̅ for samples of size 500 will have what distribution, mean and standard error? b. Use R to draw a random sample of size 500 from thus population. Conduct exploratory data analysis on your sample. c. Compute the bootstrap distribution for your sample and note the bootstrap mean and standard error. d. Compare the bootstrap distribution to the theoretical sampling distribution and create a table as given below. Mean Standard Deviation Population Sample distribution of ?̅ Sample Bootstrap distribution e. Repeat for sample of sizes n = 50 and n = 10. Carefully describe your observations about the effects of sample size on the bootstrap distribution.

4) Suppose we have a random sample from the Binomial distribution. Here X represents the number of successes in n trials with probability of success given by θ. Data: 6 5 5 5 7 4 ∼ binomial(8,θ) a. Estimate θ. Think what will be a parametric estimate for θ based on the given sample. b. Write out the R code to generate data of 100 parametric bootstrap samples and compute an 80% percentile confidence interval for θ. (Hint: You will generate random samples of size equal to the given sample size from a Binomial with n = 8 and using the estimate of θ that you get in

part a.)a. Estimate θ. Think what will be a parametric estimate for θ based on the given sample. b. Write out the R code to generate data of 100 parametric bootstrap samples and compute an 80% percentile confidence interval for θ. (Hint: You will generate random samples of size equal to the given sample size from a Binomial with n = 8 and using the estimate of θ that you get in part a.)

5) Will students wait times for the arrival of an instructor who is a full professor differ from the wait times for an instructor who is a graduate student? This question was investigated by counting how many minutes undergraduate students waiting in two small seminar classes, one taught by a full professor and one taught by a graduate student. Suppose we want to examine if the time waited is different for the full professor than for the graduate student. a. Propose a parametric approach to answer this question. Mention clearly all assumptions for such an approach b. Compare your approach in part a. to a bootstrap approach for a two-sample problem. Clearly state your conlusions in relation to this problem. Graduate Student instructor 9 11 14 14 16 19 37 Full Professor 28 23 18 16 15 15 13 31 31

Answer 1

the number of airline accidents by year from 1983 to 2006 were

23, 16, 21, 24, 34, 30, 28, 24, 26, 18, 23, 23, 36, 37, 49, 50, 51, 56, 46, 41, 54, 30, 40, and 31.

a. mean of sample data is

x̄ = ( sum of observations ) / ( number of observations)

= ( 23+16+21+24+34+30+28+24+26+18+23+23+36+37+49+50+51+56+46+41+54+30+40+31)/24

= 811/24

= 33.7917

the average number of airline accidents by year 1983 to 2006 is 33.7917

standard error = σ/√n

  σ = √(3347.86/24)

= √(139.4941)

= 11.8107

standard error = (11.8107)/(√24)

= 2.4108

standard error of geven sample data is 2.4108

median : arrange the data in ascending order

16, 18, 21, 23, 23, 23, 24, 24, 26, 28, 30,30, 31, 34, 36, 37, 40, 41, 46, 49, 50, 51, 54, 56,

median = ( n+1)/2 th value

= (30+31)/2

= 30.5

median of given sample data is 30.5