Question 2 A random sample of size 8 from an unknown distribution with a population mean ,u has yielded the following observations: 0.35 1.15 0.40 0.08 0.03 0.20 0.10 0.15 The sample mean and standard deviation were calculated as 0.3075 and 0.3641 respectively. The sample averages for each of 1000 bootstrap samples were calculated, and then ordered from smallest to largest. The following are some of the ordered results obtained: [a] Use the bootstrap results to nd an exact pvalue for a test of the null hypothesis Ho: ,u = 0.50 against H1: ,u. 5:5 0.50. 1What would you conclude? Sample No. 1 10 25 50 100 Sample Ave. 0.076 0.106 0.125 0.139 0.163 Sample No. 900 950 975 990 1000 Sample Ave. 0.470 0.541 0.579 0.651 0.813 (b) Construct a 95% condence interval for the population mean based on the bootstrap data provided. [(3) Compare the interval obtained in the previous question with a 95% condence interval ob tained using standard normal theory. (d) R was used to generate 20 bootstrap samples of size 8 and their means were calculated as shown below: 0.10 0.35 0.16 0.21 0.16 0.45 0.07 0.22 0.15 0.18 0.17 0.30 0.13 0.17 0.17 0.14 0.17 0.10 0.47 0.41 i) Calculate the bootstrap estimate of the bias of the sample mean. ii) Hence calculate the bootstrap bias corrected mean estimate