## Question

1. A researcher is interested in better understanding variability in birthweights of children whose mothers suffer from a particular disease. She takes a SRS of size 12 and records the resulting birthweight in pounds (given below).

3.5 5.4 4.2 4.0 4.8 4.8 3.4 3.9 4.7 4.9 4.5 4.5

a) She decides to use interquartile range (IQR = Q3 – Q1) to measure variability. Calculate the IQR for these data. Note that IQR can be calculated in R via:

quantile(data,.75) - quantile(data,.25).

b) She also wishes to construct a 95% confidence interval for IQR. Use R to create the interval using the percentile method with B = 2,000. You may use the R boot package.

c) Now, use R to create the same interval using the BCA method with B=2,000. You will want to use the R boot package.

2. A researcher conducts a study to decide whether support groups improve academic performance for at-risk high school students. Ten such students are randomly selected to take part in the support group for a semester, while the other 10 at-risk students serve as a control group. At the end of the semester, the improvement in GPA versus the previous semester is recorded for each student.

Support Group: 0.5 0.8 0.7 0.7 -0.1 0.2 0.4 0.4 0.5 0.4

Control Group: -0.3 0.0 -0.1 0.2 -0.1 -0.2 -0.2 0.0 -0.1 0.1

a) At the 5% level, test whether there is evidence of a difference in variance between the two groups. Assume normality for this test. You may use R for to conduct the test, but you need to write your hypotheses, the test statistic, the p-value, and the decision/conclusion in the context of the problem.

b) At the 10% level, test whether there is evidence that the true mean GPA increase is higher for the students who were in the support group. Use the results of part a) to inform your decision of whether to pool (or not). You may use R for to conduct the test, but you need to write your hypotheses, the test statistic, the p-value, and the decision/conclusion in the context of the problem.

c) At the 10% level, use the Wilcoxon-Mann-Whitney test to compare the two groups. You may use R for to conduct the test, but you need to write your hypotheses, the test statistic, the p- value, and the decision/conclusion in the context of the problem. Comment on how this test differs from the test in part b).

d) At the 10% level, use R to compare the two groups using a permutation test (with 100,000 randomly generated permutations). You need to write your hypotheses, the test statistic, the p- value, and the decision/conclusion in the context of the problem. Comment on how this test differs from the test in part b).

## Solution Preview

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# Question 1data <- c(3.5, 5.4, 4.2, 4.0, 4.8, 4.8, 3.4, 3.9, 4.7, 4.9, 4.5, 4.5)

## 1 (a)

IQR_range <- quantile(data,.75) - quantile(data,.25)

## 1 (b)

library(boot)

boot.out <- boot(data=data, statistic = function(dat,inds){median(dat[inds])},

R=2000)

boot.ci(boot.out,type=c("perc"))

## 1 (c)

boot.out...

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